Daily Papers

Many machine learning algorithms rely on the Fourier transform, which is also the inner core of many analog computing devices and, specifically, of optical computing. The square of the modulus of the Fourier spectrum naturally maps onto pairwise interactions, as in the renowned Ising model, making it applicable to combinatorial optimization. Here we show how the nonlinear Fourier transform, related to integrable nonlinear partial differential equations, significantly widens the computational capability and also enables a wide variety of computing gates. We study specifically the case of the nonlinear Schrödinger equation; we introduce an input composed by ``tokens'', i.e., batches of information encoded in a multi-step potential, and we show how the nonlinear spectral computing arising in the wave propagation not only generalizes combinatorial optimization to higher order problems, but also realizes boolean gates. In the weakly nonlinear regime, nonlinearity generates higher-order interactions as perturbative corrections to the pairwise terms. In the solitonic regime, the discrete nonlinear spectrum performs digital computation through soliton generation. By using four input tokens, we demonstrate the programmability of all 256 Boolean functions of three variables, including 3-SAT instances. Our results show that integrable nonlinear propagation provides an unprecedented single physical platform for higher-order and logical computation, opening the way to a rich variety of new algorithms for analog computing, machine learning, and classical and quantum information processing.

Logic synthesis, a pivotal stage in chip design, entails optimizing chip specifications encoded in hardware description languages like Verilog into highly efficient implementations using Boolean logic gates. The process involves a sequential application of logic minimization heuristics (``synthesis recipe"), with their arrangement significantly impacting crucial metrics such as area and delay. Addressing the challenge posed by the broad spectrum of design complexities - from variations of past designs (e.g., adders and multipliers) to entirely novel configurations (e.g., innovative processor instructions) - requires a nuanced `synthesis recipe` guided by human expertise and intuition. This study conducts a thorough examination of learning and search techniques for logic synthesis, unearthing a surprising revelation: pre-trained agents, when confronted with entirely novel designs, may veer off course, detrimentally affecting the search trajectory. We present ABC-RL, a meticulously tuned alpha parameter that adeptly adjusts recommendations from pre-trained agents during the search process. Computed based on similarity scores through nearest neighbor retrieval from the training dataset, ABC-RL yields superior synthesis recipes tailored for a wide array of hardware designs. Our findings showcase substantial enhancements in the Quality-of-result (QoR) of synthesized circuits, boasting improvements of up to 24.8% compared to state-of-the-art techniques. Furthermore, ABC-RL achieves an impressive up to 9x reduction in runtime (iso-QoR) when compared to current state-of-the-art methodologies.

The notion of variation is introduced for the Boolean set and based on which Boolean logic backpropagation principle is developed. Using this concept, deep models can be built with weights and activations being Boolean numbers and operated with Boolean logic instead of real arithmetic. In particular, Boolean deep models can be trained directly in the Boolean domain without latent weights. No gradient but logic is synthesized and backpropagated through layers.

We develop a method for training neural networks on Boolean data in which the values at all nodes are strictly pm 1, and the resulting models are typically equivalent to networks whose nonzero weights are also pm 1. The method replaces loss minimization with a nonconvex constraint formulation. Each node implements a Boolean threshold function (BTF), and training is expressed through a divide-and-concur decomposition into two complementary constraints: one enforces local BTF consistency between inputs, weights, and output; the other imposes architectural concurrence, equating neuron outputs with downstream inputs and enforcing weight equality across training-data instantiations of the network. The reflect-reflect-relax (RRR) projection algorithm is used to reconcile these constraints. Each BTF constraint includes a lower bound on the margin. When this bound is sufficiently large, the learned representations are provably sparse and equivalent to networks composed of simple logical gates with pm 1 weights. Across a range of tasks -- including multiplier-circuit discovery, binary autoencoding, logic-network inference, and cellular automata learning -- the method achieves exact solutions or strong generalization in regimes where standard gradient-based methods struggle. These results demonstrate that projection-based constraint satisfaction provides a viable and conceptually distinct foundation for learning in discrete neural systems, with implications for interpretability and efficient inference.

A single two-input gate suffices for all of Boolean logic in digital hardware. No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations. Here I show that a single binary operator, eml(x,y)=exp(x)-ln(y), together with the constant 1, generates the standard repertoire of a scientific calculator. This includes constants such as e, pi, and i; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions. For example, exp(x)=eml(x,1), ln(x)=eml(1,eml(eml(1,x),1)), and likewise for all other operations. That such an operator exists was not anticipated; I found it by systematic exhaustive search and established constructively that it suffices for the concrete scientific-calculator basis. In EML (Exp-Minus-Log) form, every such expression becomes a binary tree of identical nodes, yielding a grammar as simple as S -> 1 | eml(S,S). This uniform structure also enables gradient-based symbolic regression: using EML trees as trainable circuits with standard optimizers (Adam), I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4. The same architecture can fit arbitrary data, but when the generating law is elementary, it may recover the exact formula.

Implementing Boolean functions with circuits consisting of logic gates is fundamental in digital computer design. However, the implemented circuit must be exactly equivalent, which hinders generative neural approaches on this task due to their occasionally wrong predictions. In this study, we introduce a generative neural model, the "Circuit Transformer", which eliminates such wrong predictions and produces logic circuits strictly equivalent to given Boolean functions. The main idea is a carefully designed decoding mechanism that builds a circuit step-by-step by generating tokens, which has beneficial "cutoff properties" that block a candidate token once it invalidate equivalence. In such a way, the proposed model works similar to typical LLMs while logical equivalence is strictly preserved. A Markov decision process formulation is also proposed for optimizing certain objectives of circuits. Experimentally, we trained an 88-million-parameter Circuit Transformer to generate equivalent yet more compact forms of input circuits, outperforming existing neural approaches on both synthetic and real world benchmarks, without any violation of equivalence constraints.

Deep learning is computationally intensive, with significant efforts focused on reducing arithmetic complexity, particularly regarding energy consumption dominated by data movement. While existing literature emphasizes inference, training is considerably more resource-intensive. This paper proposes a novel mathematical principle by introducing the notion of Boolean variation such that neurons made of Boolean weights and inputs can be trained -- for the first time -- efficiently in Boolean domain using Boolean logic instead of gradient descent and real arithmetic. We explore its convergence, conduct extensively experimental benchmarking, and provide consistent complexity evaluation by considering chip architecture, memory hierarchy, dataflow, and arithmetic precision. Our approach achieves baseline full-precision accuracy in ImageNet classification and surpasses state-of-the-art results in semantic segmentation, with notable performance in image super-resolution, and natural language understanding with transformer-based models. Moreover, it significantly reduces energy consumption during both training and inference.

This study leverages the data representation capability of fuzzy based membership-mappings for practical secure distributed deep learning using fully homomorphic encryption. The impracticality issue of secure machine (deep) learning with fully homomorphic encrypted data, arising from large computational overhead, is addressed via applying fuzzy attributes. Fuzzy attributes are induced by globally convergent and robust variational membership-mappings based local deep models. Fuzzy attributes combine the local deep models in a robust and flexible manner such that the global model can be evaluated homomorphically in an efficient manner using a boolean circuit composed of bootstrapped binary gates. The proposed method, while preserving privacy in a distributed learning scenario, remains accurate, practical, and scalable. The method is evaluated through numerous experiments including demonstrations through MNIST dataset and Freiburg Groceries Dataset. Further, a biomedical application related to mental stress detection on individuals is considered.

Dense retrieval, which aims to encode the semantic information of arbitrary text into dense vector representations or embeddings, has emerged as an effective and efficient paradigm for text retrieval, consequently becoming an essential component in various natural language processing systems. These systems typically focus on optimizing the embedding space by attending to the relevance of text pairs, while overlooking the Boolean logic inherent in language, which may not be captured by current training objectives. In this work, we first investigate whether current retrieval systems can comprehend the Boolean logic implied in language. To answer this question, we formulate the task of Boolean Dense Retrieval and collect a benchmark dataset, BoolQuestions, which covers complex queries containing basic Boolean logic and corresponding annotated passages. Through extensive experimental results on the proposed task and benchmark dataset, we draw the conclusion that current dense retrieval systems do not fully understand Boolean logic in language, and there is a long way to go to improve our dense retrieval systems. Furthermore, to promote further research on enhancing the understanding of Boolean logic for language models, we explore Boolean operation on decomposed query and propose a contrastive continual training method that serves as a strong baseline for the research community.

In this work, we introduce Boolformer, the first Transformer architecture trained to perform end-to-end symbolic regression of Boolean functions. First, we show that it can predict compact formulas for complex functions which were not seen during training, when provided a clean truth table. Then, we demonstrate its ability to find approximate expressions when provided incomplete and noisy observations. We evaluate the Boolformer on a broad set of real-world binary classification datasets, demonstrating its potential as an interpretable alternative to classic machine learning methods. Finally, we apply it to the widespread task of modelling the dynamics of gene regulatory networks. Using a recent benchmark, we show that Boolformer is competitive with state-of-the art genetic algorithms with a speedup of several orders of magnitude. Our code and models are available publicly.

Understanding properties of deep neural networks is an important challenge in deep learning. In this paper, we take a step in this direction by proposing a rigorous way of verifying properties of a popular class of neural networks, Binarized Neural Networks, using the well-developed means of Boolean satisfiability. Our main contribution is a construction that creates a representation of a binarized neural network as a Boolean formula. Our encoding is the first exact Boolean representation of a deep neural network. Using this encoding, we leverage the power of modern SAT solvers along with a proposed counterexample-guided search procedure to verify various properties of these networks. A particular focus will be on the critical property of robustness to adversarial perturbations. For this property, our experimental results demonstrate that our approach scales to medium-size deep neural networks used in image classification tasks. To the best of our knowledge, this is the first work on verifying properties of deep neural networks using an exact Boolean encoding of the network.

Partition logics -- non-Boolean event structures obtained by pasting Boolean algebras -- provide a natural language for situations in which a system has a definite latent state but can be accessed and resolved only through mutually incompatible coarse-grained modes of observation. We show that this structure arises in a range of social-science settings by constructing six explicit examples from personnel assessment, survey framing, clinical diagnosis, espionage coordination, legal pluralism, and organizational auditing. For each case we identify the latent state space, the observational contexts as partitions, and the shared atoms that intertwine contexts, yielding instances of the L_{12} bowtie, triangle, pentagon, and automaton partition logics. These examples make precise a notion of social complementarity: different modes of inquiry can be incompatible even though the underlying system remains fully value-definite. Complementarity in this sense does not entail contextuality or ontic indeterminacy. We further compare the classical probabilities generated by convex mixtures of dispersion-free states with the quantum-like Born probabilities available when the same exclusivity graph admits a faithful orthogonal representation. The framework thus separates logical structure from probabilistic realization and suggests empirically testable benchmarks for quantum-cognition models.

This paper presents a quantitative program verification infrastructure for discrete probabilistic programs. Our infrastructure can be viewed as the probabilistic analogue of Boogie: its central components are an intermediate verification language (IVL) together with a real-valued logic. Our IVL provides a programming-language-style for expressing verification conditions whose validity implies the correctness of a program under investigation. As our focus is on verifying quantitative properties such as bounds on expected outcomes, expected run-times, or termination probabilities, off-the-shelf IVLs based on Boolean first-order logic do not suffice. Instead, a paradigm shift from the standard Boolean to a real-valued domain is required. Our IVL features quantitative generalizations of standard verification constructs such as assume- and assert-statements. Verification conditions are generated by a weakest-precondition-style semantics, based on our real-valued logic. We show that our verification infrastructure supports natural encodings of numerous verification techniques from the literature. With our SMT-based implementation, we automatically verify a variety of benchmarks. To the best of our knowledge, this establishes the first deductive verification infrastructure for expectation-based reasoning about probabilistic programs.

Boolean query construction is often critical for medical systematic review literature search. To create an effective Boolean query, systematic review researchers typically spend weeks coming up with effective query terms and combinations. One challenge to creating an effective systematic review Boolean query is the selection of effective MeSH Terms to include in the query. In our previous work, we created neural MeSH term suggestion methods and compared them to state-of-the-art MeSH term suggestion methods. We found neural MeSH term suggestion methods to be highly effective. In this demonstration, we build upon our previous work by creating (1) a Web-based MeSH term suggestion prototype system that allows users to obtain suggestions from a number of underlying methods and (2) a Python library that implements ours and others' MeSH term suggestion methods and that is aimed at researchers who want to further investigate, create or deploy such type of methods. We describe the architecture of the web-based system and how to use it for the MeSH term suggestion task. For the Python library, we describe how the library can be used for advancing further research and experimentation, and we validate the results of the methods contained in the library on standard datasets. Our web-based prototype system is available at http://ielab-mesh-suggest.uqcloud.net, while our Python library is at https://github.com/ielab/meshsuggestlib.

We introduce a quantum neural network, QNN, that can represent labeled data, classical or quantum, and be trained by supervised learning. The quantum circuit consists of a sequence of parameter dependent unitary transformations which acts on an input quantum state. For binary classification a single Pauli operator is measured on a designated readout qubit. The measured output is the quantum neural network's predictor of the binary label of the input state. First we look at classifying classical data sets which consist of n-bit strings with binary labels. The input quantum state is an n-bit computational basis state corresponding to a sample string. We show how to design a circuit made from two qubit unitaries that can correctly represent the label of any Boolean function of n bits. For certain label functions the circuit is exponentially long. We introduce parameter dependent unitaries that can be adapted by supervised learning of labeled data. We study an example of real world data consisting of downsampled images of handwritten digits each of which has been labeled as one of two distinct digits. We show through classical simulation that parameters can be found that allow the QNN to learn to correctly distinguish the two data sets. We then discuss presenting the data as quantum superpositions of computational basis states corresponding to different label values. Here we show through simulation that learning is possible. We consider using our QNN to learn the label of a general quantum state. By example we show that this can be done. Our work is exploratory and relies on the classical simulation of small quantum systems. The QNN proposed here was designed with near-term quantum processors in mind. Therefore it will be possible to run this QNN on a near term gate model quantum computer where its power can be explored beyond what can be explored with simulation.

Real-time EEG classification on edge devices is bottlenecked by the floating-point arithmetic of conventional neural networks. We investigated Differentiable Logic Gate Networks (Diff-Logic) as a hardware-native alternative that compiles models into pure Boolean circuits executable via bitwise CPU operations. Through rigorous iso-parameter experiments across four EEG datasets spanning two classification tasks, binary dementia detection and 3-class emotion recognition, we compared Diff-Logic against matched-capacity Multi-Layer Perceptron (MLP) and Binarized Neural Network (BNN) baselines at four complexity tiers (50k-500k parameters). On dementia screening, Diff-Logic achieved 80.2% Macro F1, outperforming the MLP baseline by 6.8%. On emotion recognition, the MLP retained a moderate performance advantage but incurred a 2.3times higher latency and 14times larger model size when deployed on a power-constrained (7W) Nvidia Jetson Orin Nano CPU (Single-core). Critically, Diff-Logic inference time remained nearly constant across a 10times increase in model scale, achieving a peak speedup of 2.9times over MLPs at the largest complexity tier. Our results establish logic-based neural architectures as a practical paradigm for resource-constrained brain-computer interfaces, achieving competitive or superior performance while natively satisfying the latency and memory constraints of portable edge deployment. Code is available on GitHub: https://github.com/Shyamal-Dharia/eeg-difflogic

We contribute a theoretical and operational framework for neurosymbolic AI called DeepLog. DeepLog introduces building blocks and primitives for neurosymbolic AI that make abstraction of commonly used representations and computational mechanisms used in neurosymbolic AI. DeepLog can represent and emulate a wide range of neurosymbolic systems. It consists of two key components. The first is the DeepLog language for specifying neurosymbolic models and inference tasks. This language consists of an annotated neural extension of grounded first-order logic, and makes abstraction of the type of logic, e.g. boolean, fuzzy or probabilistic, and whether logic is used in the architecture or in the loss function. The second DeepLog component is situated at the computational level and uses extended algebraic circuits as computational graphs. Together these two components are to be considered as a neurosymbolic abstract machine, with the DeepLog language as the intermediate level of abstraction and the circuits level as the computational one. DeepLog is implemented in software, relies on the latest insights in implementing algebraic circuits on GPUs, and is declarative in that it is easy to obtain different neurosymbolic models by making different choices for the underlying algebraic structures and logics. The generality and efficiency of the DeepLog neurosymbolic machine is demonstrated through an experimental comparison between 1) different fuzzy and probabilistic logics, 2) between using logic in the architecture or in the loss function, and 3) between a standalone CPU-based implementation of a neurosymbolic AI system and a DeepLog GPU-based one.

Modern neural networks demonstrate state-of-the-art performance on numerous existing benchmarks; however, their high computational requirements and energy consumption prompt researchers to seek more efficient solutions for real-world deployment. Logic gate networks (LGNs) learns a large network of logic gates for efficient image classification. However, learning a network that can solve a simple problem like CIFAR-10 can take days to weeks to train. Even then, almost half of the network remains unused, causing a discretization gap. This discretization gap hinders real-world deployment of LGNs, as the performance drop between training and inference negatively impacts accuracy. We inject Gumbel noise with a straight-through estimator during training to significantly speed up training, improve neuron utilization, and decrease the discretization gap. We theoretically show that this results from implicit Hessian regularization, which improves the convergence properties of LGNs. We train networks 4.5 times faster in wall-clock time, reduce the discretization gap by 98%, and reduce the number of unused gates by 100%.

Serial verification gates are a core reliability primitive in LLM harnesses: a candidate answer is returned only if k verifier calls all accept it. Under conditionally independent gates, the recent Odds Law (arXiv:2606.15712) shows that posterior log-odds grow linearly in k, so failure decays exponentially, and states that "a tight theory of partially correlated verifier cascades remains open." This note gives a minimal such theory. Modeling the per-instance false-accept rate on the generator's own errors as a latent variable αsim G (de Finetti), the exact cascade posterior is ell_k = ell_0 - ln m_k, with m_k the k-th moment of G. Then: (i) ell_k is concave in k for every non-degenerate G -- the Odds Law is its tangent at the first gate and an upper bound; (ii) for Beta(a,b) latents, failure decays polynomially, 1-r_k asymp k^{-b}, with correlation parameter ρ_v = 1/(a+b+1); (iii) a blind-spot atom of mass 1-π at α=1 caps the evidence extractable from any number of gates at -ln(1-π) nats, so reliability saturates below 1; (iv) letting the true-accept rate also vary (βsim H) yields a trichotomy -- gates eventually always help, plateau, or actively harm -- decided by the upper-tail exponents of G and H, with closed-form crossover k^dagger. The mechanism is survivorship: errors surviving gates are the high-α ones. The theory is measurable: R repeated verdicts per instance identify the first R moments of G, so two verdicts identify ρ_v; beta-binomial likelihood and NPMLE recover the reliability curve and the ill-posed ceiling. In synthetic tests, independence-based extrapolation underestimates failure by 20x at k=5 and ~3000x at k=10; the correlated fit at R=8 tracks held-out depths. The practical lever is decorrelation -- changing model family, modality, or evidence source -- not adding gates.

Tool-using LLM agents can violate the very policies they are deployed to enforce while appearing to complete the task successfully. In policy-permissive environments, a tool may execute any well-formed call even when the corresponding state transition is forbidden by domain policy. The result is a silent wrong state (a booking cancelled, a passenger count changed, a claim acted on without verification) that neither the tool nor the agent's self-report exposes. We study this failure mode in the τ^2-bench airline domain. On a budget agent, 78% of observed failures are silent wrong-state failures with no tool error, and the aggregate failure rate is reproducible across disjoint seeds, not sampling noise. We then evaluate a lightweight intervention: deterministic, read-only pre-execution gates that inspect the proposed call and current state before allowing a write. A four-gate suite raises full-benchmark success from 29.6% to 42.0% on gpt-4o-mini (+12.4pp; paired task-level bootstrap P=0.0012), and the lift reproduces on a disjoint 15-seed set (+12.3pp; P=0.0008). The effect is concentrated where the gates fire: on the 26/50 firing tasks, success rises by +19.2pp, while movement on the 24 non-firing tasks does not exclude zero. Two negative controls (a self-enforcing retail domain and BFCL) bound the mechanism: gates help when tools are policy-permissive and add little where tools already self-enforce. As suggestive evidence, not a central claim, the same failure mode persists at the frontier: gpt-5.2 at default reasoning still attempts policy-violating writes, and the same suite improves success from 61.2% to 71.6% (+10.4pp; P=0.020; n=5, no replication). The contribution is a bounded evaluation and reliability result: deterministic gates do not guarantee task success, but they can deterministically prevent a known class of silent policy-violating writes at the action boundary.

We present AutoBool, a reinforcement learning (RL) framework that trains large language models (LLMs) to generate effective Boolean queries for medical systematic reviews. Boolean queries are the primary mechanism for literature retrieval in this domain and must achieve high recall while maintaining reasonable precision - a challenging balance that existing prompt-based LLM approaches often struggle to achieve. A major limitation in this space is the lack of high-quality ground-truth Boolean queries for each topic, which makes supervised fine-tuning impractical. AutoBool addresses this challenge by using RL to directly optimize query generation with retrieval measures, without requiring target queries. To support this effort, we create and release the largest dataset of its kind: 65588 topics in total for training and evaluating the task of automatic Boolean query formulation. Experiments on our new dataset and two established datasets (CLEF TAR and Seed Collection) show that AutoBool significantly outperforms zero shot/few shot prompting and matches or exceeds the effectiveness of much larger GPT-based models (e.g., GPT-4o, O3) using smaller backbones. It also approaches effectiveness of expert-authored queries while retrieving 10 to 16 times fewer documents. Ablation studies reveal the critical roles of model backbone, size, decoding temperature, and prompt design. Code and data are available at https://github.com/ielab/AutoBool.

Multi-controlled single-target (MC) gates are some of the most crucial building blocks for varied quantum algorithms. How to implement them optimally is thus a pivotal question. To answer this question in an architecture-independent manner, and to get a worst-case estimate, we should look at a linear nearest-neighbor (LNN) architecture, as this can be embedded in almost any qubit connectivity. Motivated by the above, here we describe a method which implements MC gates using no more than sim 4k+8n CNOT gates -- up-to 60% reduction over state-of-the-art -- while allowing for complete flexibility to choose the locations of n controls, the target, and a dirty ancilla out of k qubits. More strikingly, in case k approx n, our upper bound is sim 12n -- the best known for unrestricted connectivity -- and if n = 1, our upper bound is sim 4k -- the best known for a single long-range CNOT gate over k qubits -- therefore, if our upper bound can be reduced, then the cost of one or both of these simpler versions of MC gates will be immediately reduced accordingly. In practice, our method provides circuits that tend to require fewer CNOT gates than our upper bound for almost any given instance of MC gates.

For any positive integer k, there exist neural networks with Θ(k^3) layers, Θ(1) nodes per layer, and Θ(1) distinct parameters which can not be approximated by networks with O(k) layers unless they are exponentially large --- they must possess Ω(2^k) nodes. This result is proved here for a class of nodes termed "semi-algebraic gates" which includes the common choices of ReLU, maximum, indicator, and piecewise polynomial functions, therefore establishing benefits of depth against not just standard networks with ReLU gates, but also convolutional networks with ReLU and maximization gates, sum-product networks, and boosted decision trees (in this last case with a stronger separation: Ω(2^{k^3}) total tree nodes are required).

In this paper, a short survey about the concepts underlying general logics is given. In particular, a novel rigorous definition of a fuzzy negation as an operation acting on a lattice to render it into a fuzzy logic is presented. According to this definition, a fuzzy negation satisfies the weak double negation condition, requiring double negation to be expansive, the antitony condition, being equivalent to the disjunctive De Morgan law and thus warranting compatibility of negation with the lattice operations, and the Boolean boundary condition stating that the universal bounds of the lattice are the negation of each other. From this perspective, the most general logics are fuzzy logics, containing as special cases paraconsistent (quantum) logics, quantum logics, intuitionistic logics, and Boolean logics, each of which given by its own algebraic restrictions. New examples of a non-contradictory logic violating the conjunctive De Morgan law, and of a typical non-orthomodular fuzzy logic along with its explicit lattice representation are given.

In fault-tolerant quantum computing with the surface code, non-Clifford gates are crucial for universal computation. However, implementing these gates using methods like magic state distillation and code switching requires significant resources. In this work, we propose a new protocol that combines magic state preparation and code switching to realize logical non-Clifford operations with the potential for fault tolerance. Our approach begins with a special logical state in the Z_4 surface code. By applying a sequence of transformations, the system goes through different topological codes, including the non-Abelian D_4 quantum double model. This process ultimately produces a magic state in a condensed Z_2 surface code, which enables the implementation of a logical T gate in the standard Z_2 surface code. In our analysis, we employ a framework where the topological codes are represented by their topological orders and all the transformations are considered as topological manipulations such as gauging symmetries and condensing anyons. This perspective is particularly useful for understanding code switching between topological codes.

Gate set tomography (GST) is a protocol for detailed, predictive characterization of logic operations (gates) on quantum computing processors. Early versions of GST emerged around 2012-13, and since then it has been refined, demonstrated, and used in a large number of experiments. This paper presents the foundations of GST in comprehensive detail. The most important feature of GST, compared to older state and process tomography protocols, is that it is calibration-free. GST does not rely on pre-calibrated state preparations and measurements. Instead, it characterizes all the operations in a gate set simultaneously and self-consistently, relative to each other. Long sequence GST can estimate gates with very high precision and efficiency, achieving Heisenberg scaling in regimes of practical interest. In this paper, we cover GST's intellectual history, the techniques and experiments used to achieve its intended purpose, data analysis, gauge freedom and fixing, error bars, and the interpretation of gauge-fixed estimates of gate sets. Our focus is fundamental mathematical aspects of GST, rather than implementation details, but we touch on some of the foundational algorithmic tricks used in the pyGSTi implementation.

Existing datasets that contain boolean questions, such as BoolQ and TYDI QA , provide the user with a YES/NO response to the question. However, a one word response is not sufficient for an explainable system. We promote explainability by releasing a new set of annotations marking the evidence in existing TyDi QA and BoolQ datasets. We show that our annotations can be used to train a model that extracts improved evidence spans compared to models that rely on existing resources. We confirm our findings with a user study which shows that our extracted evidence spans enhance the user experience. We also provide further insight into the challenges of answering boolean questions, such as passages containing conflicting YES and NO answers, and varying degrees of relevance of the predicted evidence.

Circuits based on sum-product structure have become a ubiquitous representation to compactly encode knowledge, from Boolean functions to probability distributions. By imposing constraints on the structure of such circuits, certain inference queries become tractable, such as model counting and most probable configuration. Recent works have explored analyzing probabilistic and causal inference queries as compositions of basic operators to derive tractability conditions. In this paper, we take an algebraic perspective for compositional inference, and show that a large class of queries - including marginal MAP, probabilistic answer set programming inference, and causal backdoor adjustment - correspond to a combination of basic operators over semirings: aggregation, product, and elementwise mapping. Using this framework, we uncover simple and general sufficient conditions for tractable composition of these operators, in terms of circuit properties (e.g., marginal determinism, compatibility) and conditions on the elementwise mappings. Applying our analysis, we derive novel tractability conditions for many such compositional queries. Our results unify tractability conditions for existing problems on circuits, while providing a blueprint for analysing novel compositional inference queries.

The surface code family is a promising approach to implementing fault-tolerant quantum computations. Universal fault-tolerance requires error-corrected non-Clifford operations, in addition to Clifford gates, and for the former, it is imperative to experimentally demonstrate additional resources known as magic states. Another challenge is to efficiently embed surface codes into quantum hardware with connectivity constraints. This work simultaneously addresses both challenges by employing a qubit-efficient rotated heavy-hexagonal surface code for IBM quantum processors (ibm\_fez) and implementing the magic state injection protocol. Our work reports error thresholds for both logical bit- and phase-flip errors, of approx0.37% and approx0.31%, respectively, which are higher than the threshold values previously reported with traditional embedding. The post-selection-based preparation of logical magic states |H_Lrangle and |T_Lrangle achieve fidelities of 0.8806pm0.0002 and 0.8665pm0.0003, respectively, which are both above the magic state distillation threshold. Additionally, we report the minimum fidelity among injected arbitrary single logical qubit states as 0.8356pm0.0003. Our work demonstrates the potential for realising non-Clifford logical gates by producing high-fidelity logical magic states on IBM quantum devices.

We study the Stochastic Boolean Function Certification (SBFC) problem, where we are given n Bernoulli random variables {X_e: e in U} on a ground set U of n elements with joint distribution p, a Boolean function f: 2^U to {0, 1}, and an (unknown) scenario S = {e in U: X_e = 1} of active elements sampled from p. We seek to probe the elements one-at-a-time to reveal if they are active until we can certify f(S) = 1, while minimizing the expected number of probes. Unlike most previous results that assume independence, we study correlated distributions p and give approximation algorithms for several classes of functions f. When f(S) is the indicator function for whether S is the spanning set of a given matroid, our problem reduces to finding a basis of active elements of a matroid by probing elements. We give a non-adaptive O(log n)-approximation algorithm for arbitrary distributions p, and show that this is tight up to constants unless P = NP, even for partition matroids. For uniform matroids, we give constant factor 4.642-approximation ([BBFT20]) that can be further improved to a 2-approximation if additionally the random variables are negatively correlated for the case of 1-uniform matroid. We also give an adaptive O(log k)-approximation algorithm for SBFC for k-uniform matroids for the Graph Probing problem, where we seek to probe the edges of a graph one-at-a-time until we find k active edges. The underlying distribution on edges arises from (hidden) independent vertex random variables, with an edge being active if at least one of its endpoints is active. This significantly improves over the information-theoretic lower bound on Ω(poly(n)) ([JGM19]) for adaptive algorithms for k-uniform matroids with arbitrary distributions.

We introduce Reverse Derivative Ascent: a categorical analogue of gradient based methods for machine learning. Our algorithm is defined at the level of so-called reverse differential categories. It can be used to learn the parameters of models which are expressed as morphisms of such categories. Our motivating example is boolean circuits: we show how our algorithm can be applied to such circuits by using the theory of reverse differential categories. Note our methodology allows us to learn the parameters of boolean circuits directly, in contrast to existing binarised neural network approaches. Moreover, we demonstrate its empirical value by giving experimental results on benchmark machine learning datasets.

We present a tool QSeqSim, a Qiskit-integrated symbolic backend that fills the current gap of having no Qiskit-native support for simulating while-loop quantum programs and their induced sequential quantum circuits. QSeqSim takes Qiskit QuantumCircuit objects, translates them into OpenQASM 3 code, and organises the resulting program into a combination of combinational, dynamic, and sequential circuits, thereby assigning while-loops a precise sequential circuit semantics with explicit internal and external qubits. Building on this semantics, QSeqSim adopts a Binary Decision Diagram (BDD)-based symbolic representation and integrates weighted model counting to compute measurement probabilities efficiently by exploiting sharing in structured and sparse BDDs. On top of this Boolean backbone, it introduces dedicated symbolic operators for state composition and state retention, thereby enabling efficient symbolic execution of sequential quantum circuits. Our experiments demonstrate that QSeqSim scales to substantial while-induced sequential circuits; in particular, in the quantum random walk benchmark we successfully simulate circuits with over 1000 qubits for more than 10 loop iterations. QSeqSim is available at https://github.com/Veri-Q/QSeqSim.

Chip design relies heavily on generating Boolean circuits, such as AND-Inverter Graphs (AIGs), from functional descriptions like truth tables. While recent advances in deep learning have aimed to accelerate circuit design, these efforts have mostly focused on tasks other than synthesis, and traditional heuristic methods have plateaued. In this paper, we introduce ShortCircuit, a novel transformer-based architecture that leverages the structural properties of AIGs and performs efficient space exploration. Contrary to prior approaches attempting end-to-end generation of logic circuits using deep networks, ShortCircuit employs a two-phase process combining supervised with reinforcement learning to enhance generalization to unseen truth tables. We also propose an AlphaZero variant to handle the double exponentially large state space and the sparsity of the rewards, enabling the discovery of near-optimal designs. To evaluate the generative performance of our trained model , we extract 500 truth tables from a benchmark set of 20 real-world circuits. ShortCircuit successfully generates AIGs for 84.6% of the 8-input test truth tables, and outperforms the state-of-the-art logic synthesis tool, ABC, by 14.61% in terms of circuits size.

Safety claims on self-improving agent runtimes are almost always self-graded: a policy file, a guardrail, or a README commitment. We describe falsifiable release gates, and a methodology to build and validate such systems, such that every new capability must pass a pre-specified, machine-verifiable acceptance suite before it ships, and a fixed set of standing invariants is preserved at each gate. I think we applied the method in the Antahkarana, an open runtime, via seven gates from basic observability into a self governing loop that suggests changes to its own policy. no action goes to an effector without a safety-critical property capability token minted by a control ring is exhaustively machine-checked over the one million recorded reachable state space of a bounded model and re-checked against execution traces. A purposely broken model gives the shortest counterexample, so the checker has teeth, is apparent. The self-enhancement loop is positively Constrained: the entire write surface is policy rules, tightening changes. may auto-apply loosening changes always require a human merge and a proposal autoclosed is one that mispredicts its own effect; We publish the acceptance measured results for all the seven gates, define precisely the scope of each claim (a bounded of the coordination skeleton (not the learned components) and free the runtime, either command line tools and the gate suite, so the results reproduce, and gates can run against other agent frameworks. Reviewers may repeat the single command central non-bypass in seconds.

Using a large language model to play the user is now standard in scalable evaluation. It has a repeatedly diagnosed failure: the simulated user is excessively cooperative, so a system under test can score by the sheer number of questions it asks rather than by making the user willing to speak. We answer with a disclosure gate conditioning information release on the companion agent's behaviour: its state is a ladder of five ordered gates, merged onto three observable depth layers. We specify, ablate, and audit it, and train a user simulator against that specification. Gating behaviour is learned from the training corpus's synthetic branch, while the real branch supplies how people speak and react; after training, the simulator need not be told at runtime which gate each item sits behind. The gate is a load-bearing component of the environment: on the English corpus of a published companion-agent benchmark (CompanionBench), once training no longer states per example which gate each item sits behind, the largest rank displacement across 12 systems under test exceeds the noise band set by re-running that environment under a new seed, while per-system scores show no detectable change. We state two acceptance criteria: a ranking must be order-preserving, and absolute scores must be scale-stable. Of the candidates we examine, only one passes both -- the simulator we release -- and its leaderboard correlates at 0.993 with the benchmark's original simulator. By contrast, prompting a frontier model as the simulator barely moves the ranking while shifting every score upward -- a shift invisible to anyone checking the ranking alone. The environment we specify is the one that benchmark already used. That publication describes the mechanism in about four hundred words, and we supply what it lacked: specification, ablations, human studies, negative controls, and downstream sensitivity analysis.

AI research agents combine prior knowledge, public sources, and experimental feedback to produce useful results. The Discovery Certification Protocol (DCP) turns claims about these results into executable recovery and feedback tests. Gate 1 validates useful improvement on sealed evaluation. Gate 2 gives matched agents the registered starting information and observed Web content while withholding the target research history. Every valid method reaching the numerical target supplies a recovery witness and triggers the Core veto. DCP Core requires adequate controls, zero observed recoveries, and a finite-sample bound on recovery in one fresh registered episode. Optional Gate 3 measures the average effect of truthful feedback relative to a specified neutral policy from a shared checkpoint. DCP Evidence adds this effect after independent null calibration and a registered effect margin. Two controlled audits exercise the complete protocol in SQLite optimization and virtual catalyst control under different models. Each produced zero recoveries in 96 episodes, with an upper bound of 0.0468. Each paired study yielded 30 truthful recoveries and zero neutral recoveries, with passing 60-pair null studies. Additional cases exercise Core, recovered, and audit-incomplete decisions. A deterministic, LLM-free verifier reproduces the decisions from frozen evidence. DCP provides a common evidence language for useful outcomes, alternative routes, and feedback effects across AI research.

We present quantum circuits to implement an exhaustive key search for the Advanced Encryption Standard (AES) and analyze the quantum resources required to carry out such an attack. We consider the overall circuit size, the number of qubits, and the circuit depth as measures for the cost of the presented quantum algorithms. Throughout, we focus on Clifford+T gates as the underlying fault-tolerant logical quantum gate set. In particular, for all three variants of AES (key size 128, 192, and 256 bit) that are standardized in FIPS-PUB 197, we establish precise bounds for the number of qubits and the number of elementary logical quantum gates that are needed to implement Grover's quantum algorithm to extract the key from a small number of AES plaintext-ciphertext pairs.

Differentiable Logic Gate Networks (DLGNs) are a very fast and energy-efficient alternative to conventional feed-forward networks. With learnable combinations of logical gates, DLGNs enable fast inference by hardware-friendly execution. Since the concept of DLGNs has only recently gained attention, these networks are still in their developmental infancy, including the design and scalability of their output layer. To date, this architecture has primarily been tested on datasets with up to ten classes. This work examines the behavior of DLGNs on large multi-class datasets. We investigate its general expressiveness, its scalability, and evaluate alternative output strategies. Using both synthetic and real-world datasets, we provide key insights into the importance of temperature tuning and its impact on output layer performance. We evaluate conditions under which the Group-Sum layer performs well and how it can be applied to large-scale classification of up to 2000 classes.

Circuit Satisfiability (CSAT) plays a pivotal role in Electronic Design Automation. The standard workflow for solving CSAT problems converts circuits into Conjunctive Normal Form (CNF) and employs generic SAT solvers powered by Conflict-Driven Clause Learning (CDCL). However, this process inherently discards rich structural and functional information, leading to suboptimal solver performance. To address this limitation, we introduce CASCAD, a novel circuit-aware SAT solving framework that directly leverages circuit-level conditional probabilities computed via Graph Neural Networks (GNNs). By explicitly modeling gate-level conditional probabilities, CASCAD dynamically guides two critical CDCL heuristics -- variable phase selection and clause managementto significantly enhance solver efficiency. Extensive evaluations on challenging real-world Logical Equivalence Checking (LEC) benchmarks demonstrate that CASCAD reduces solving times by up to 10x compared to state-of-the-art CNF-based approaches, achieving an additional 23.5% runtime reduction via our probability-guided clause filtering strategy. Our results underscore the importance of preserving circuit-level structural insights within SAT solvers, providing a robust foundation for future improvements in SAT-solving efficiency and EDA tool design.

We introduce GateSkip, a simple residual-stream gating mechanism that enables token-wise layer skipping in decoder-only LMs. Each Attention/MLP branch is equipped with a sigmoid-linear gate that condenses the branch's output before it re-enters the residual stream. During inference we rank tokens by the gate values and skip low-importance ones using a per-layer budget. While early-exit or router-based Mixture-of-Depths models are known to be unstable and need extensive retraining, our smooth, differentiable gates fine-tune stably on top of pretrained models. On long-form reasoning, we save up to 15\% compute while retaining over 90\% of baseline accuracy. On instruction-tuned models we see accuracy gains at full compute and match baseline quality near 50\% savings. The learned gates give insight into transformer information flow (e.g., BOS tokens act as anchors), and the method combines easily with quantization, pruning, and self-speculative decoding.

Policy gradient computes a backward pass for every sample, even though the backward pass is expensive and most samples carry little learning value. The Delightful Policy Gradient (DG) provides a forward-pass signal of learning value: delight, the product of advantage and surprisal (negative log-probability). We introduce the Kondo gate, which compares delight against a compute price and pays for a backward pass only when the sample is worth it, thereby tracing a quality--cost Pareto frontier. In bandits, zero-price gating preserves useful gradient signal while removing perpendicular noise, and delight is a more reliable screening signal than additive combinations of value and surprise. On MNIST and transformer token reversal, the Kondo gate skips most backward passes while retaining nearly all of DG's learning quality, with gains that grow as problems get harder and backward passes become more expensive. Because the gate tolerates approximate delight, a cheap forward pass can screen samples before expensive backpropagation, suggesting a speculative-decoding-for-training paradigm.

In the propositional modal (and algebraic) treatment of two-variable first-order logic equality is modelled by a `diagonal' constant, interpreted in square products of universal frames as the identity (also known as the `diagonal') relation. Here we study the decision problem of products of two arbitrary modal logics equipped with such a diagonal. As the presence or absence of equality in two-variable first-order logic does not influence the complexity of its satisfiability problem, one might expect that adding a diagonal to product logics in general is similarly harmless. We show that this is far from being the case, and there can be quite a big jump in complexity, even from decidable to the highly undecidable. Our undecidable logics can also be viewed as new fragments of first- order logic where adding equality changes a decidable fragment to undecidable. We prove our results by a novel application of counter machine problems. While our formalism apparently cannot force reliable counter machine computations directly, the presence of a unique diagonal in the models makes it possible to encode both lossy and insertion-error computations, for the same sequence of instructions. We show that, given such a pair of faulty computations, it is then possible to reconstruct a reliable run from them.

Self-improving LLM agents can adapt a credit pipeline to a changed rule, but an agent that rewrites itself destroys the artefact a supervisor reviews: a named change, a recorded test, an approval. We argue that self-evolution is reviewable only if it is confined to the runtime harness (instruction text, tool-call logic and primitive composition) while model weights stay fixed, so that every adaptation is a diff with a cause and a test attached. We give a dual-loop engine built on that bound, with one admission gate that writes a hash-chained record before deployment, and we measure the gate in simulation, with a simulated agent and a seeded-search proposer rather than language models. Across three families of supervisory re-interpretation at three severities, 10 seeds each, the gated loop admitted 144 of 7,449 candidate changes, none of which worsened error on held-out history, and restored the false-positive rate to the oracle level without raising missed flags in every low- and mid-severity cell. With the gate replaced by the check an unbounded system applies (fewer errors visible in recent traces), the same loops admitted 309 harmful changes and left missed flags above 10% in 49 of 90 runs: false positives fell because the screen was loosened. Evaluated on pre-shift labels, the gate rejected every candidate, so a re-interpretation must be encoded as a rule that relabels history. Parametric and scope shifts were repaired locally, a structural one only by primitive replacement; at the highest structural severity the gate's fixed tolerance blocked the correct replacement in half the seeds. We map the mechanisms to the EU AI Act's provisions for high-risk credit scoring and note that the April 2026 US model-risk guidance excludes agentic AI from its scope.

When agents are acting together, they may need a simple mechanism to decide on joint actions. One possibility is to have the agents express their preferences in the form of a ballot and use a voting rule to decide the winning action(s). Unfortunately, agents may try to manipulate such an election by misreporting their preferences. Fortunately, it has been shown that it is NP-hard to compute how to manipulate a number of different voting rules. However, NP-hardness only bounds the worst-case complexity. Recent theoretical results suggest that manipulation may often be easy in practice. To address this issue, I suggest studying empirically if computational complexity is in practice a barrier to manipulation. The basic tool used in my investigations is the identification of computational "phase transitions". Such an approach has been fruitful in identifying hard instances of propositional satisfiability and other NP-hard problems. I show that phase transition behaviour gives insight into the hardness of manipulating voting rules, increasing concern that computational complexity is indeed any sort of barrier. Finally, I look at the problem of computing manipulation of other, related problems like stable marriage and tournament problems.

Despite rapid progress in quantum optimization, broad real-hardware benchmarks comparing multiple algorithmic families across diverse combinatorial problems under a common protocol remain limited. We benchmark gate-based quantum optimization on four NP-hard problems: multi-dimensional knapsack (MDKP), maximum independent set (MIS), quadratic assignment (QAP), and market-share (MSP). We study VQE, CVaR-VQE, standard, multi-angle, and warm-start QAOA, together with qubit-efficient PCE and QRAO, on IBM Heron r1/r2 processors using resilience-level-2 mitigation. To our knowledge, this includes the first real-hardware QRAO results and the first multi-problem PCE hardware benchmark. Across 247 method-instance combinations, we report transpiled circuit size, hardware outcomes, and an independent-error gate-count fidelity proxy, F_{est}. For MDKP and MIS, an empirical operating point near F_{est}approx 0.1, corresponding to about 770 two-qubit gates at the median Heron-r2 CZ error rate, marks the onset of noise-dominated execution. QAP exposes a separate bottleneck: dense one-hot encodings and an exponentially sparse feasible manifold, with feasible fraction 10!/2^{100} at n=10; no tested hardware method produces a feasible assignment. Compiled QAOA-family circuits are generally noise dominated, and a matched uniform-random control shows that most feasible low-fidelity outcomes fall within the random range, apart from one finite-sample MIS warm-start exception. A SWAP-aware, fractional-gate, Nighthawk-topology compilation counterfactual reduces two-qubit counts but leaves all circuits below F_{est}=10^{-3}. These conclusions apply to the tested implementations rather than QAOA in general. Qubit-efficient methods extend runnable instance sizes, but only within the empirical fidelity budget.

Transformers have gained popularity in machine learning due to their application in the field of natural language processing. They manipulate and process text efficiently, capturing long-range dependencies among data and performing the next word prediction. On the other hand, gate-based quantum computing is based on controlling the register of qubits in the quantum hardware by applying a sequence of gates, a process which can be interpreted as a low level text programming language. We develop a transformer model capable of transpiling quantum circuits from the qasm standard to other sets of gates native suited for a specific target quantum hardware, in our case the set for the trapped-ion quantum computers of IonQ. The feasibility of a translation up to five qubits is demonstrated with a percentage of correctly transpiled target circuits equal or superior to 99.98%. Regardless the depth of the register and the number of gates applied, we prove that the complexity of the transformer model scales, in the worst case scenario, with a polynomial trend by increasing the depth of the register and the length of the circuit, allowing models with a higher number of parameters to be efficiently trained on HPC infrastructures.

Lattice surgery with two-dimensional quantum error correcting codes is among the leading schemes for fault-tolerant quantum computation, motivated by superconducting hardware architectures. In conventional lattice surgery compilation schemes, logical circuits are compiled following a place-and-route paradigm, where logical qubits remain statically fixed in space throughout the computation. In this work, we introduce a paradigm shift by exploiting movable logical qubits via teleportation during the logical lattice surgery CNOT gate. Focusing on lattice surgery with the color code, we propose a proof-of-concept compilation scheme that leverages this capability. Numerical simulations show that the proposed approach can substantially reduce the routed circuit depth compared to standard place-and-route compilation techniques. Our results demonstrate that optimizations based on movable logical qubits are not limited to architectures with physically movable qubits, such as neutral atoms or trapped ions - they are also readily applicable to superconducting quantum hardware. An open-source implementation of our method is available on GitHub https://github.com/munich-quantum-toolkit/qecc.

Let f: {0,1}^n to {0, 1} be a boolean function, and let f_land (x, y) = f(x land y) denote the AND-function of f, where x land y denotes bit-wise AND. We study the deterministic communication complexity of f_land and show that, up to a log n factor, it is bounded by a polynomial in the logarithm of the real rank of the communication matrix of f_land. This comes within a log n factor of establishing the log-rank conjecturefor AND-functions with no assumptions on f. Our result stands in contrast with previous results on special cases of the log-rank conjecture, which needed significant restrictions on f such as monotonicity or low F_2-degree. Our techniques can also be used to prove (within a log n factor) a lifting theorem for AND-functions, stating that the deterministic communication complexity of f_land is polynomially-related to the AND-decision tree complexity of f. The results rely on a new structural result regarding boolean functions f:{0, 1}^n to {0, 1} with a sparse polynomial representation, which may be of independent interest. We show that if the polynomial computing f has few monomials then the set system of the monomials has a small hitting set, of size poly-logarithmic in its sparsity. We also establish extensions of this result to multi-linear polynomials f:{0,1}^n to R with a larger range.

Many approaches such as Quine-McCluskey algorithm, Karnaugh map solving, Petrick's method and McBoole's method have been devised to simplify Boolean expressions in order to optimize hardware implementation of digital circuits. However, the algorithmic implementations of these methods are hard-coded and also their computation time is proportional to the number of minterms involved in the expression. In this paper, we propose KarNet, where the ability of Convolutional Neural Networks to model relationships between various cell locations and values by capturing spatial dependencies is exploited to solve Karnaugh maps. In order to do so, a Karnaugh map is represented as an image signal, where each cell is considered as a pixel. Experimental results show that the computation time of KarNet is independent of the number of minterms and is of the order of one-hundredth to one-tenth that of the rule-based methods. KarNet being a learned system is found to achieve nearly a hundred percent accuracy, precision, and recall. We train KarNet to solve four variable Karnaugh maps and also show that a similar method can be applied on Karnaugh maps with more variables. Finally, we show a way to build a fully accurate and computationally fast system using KarNet.

Boolean satisfiability (SAT) problem, the first problem proven to be NP-complete, has become a fundamental challenge in computational complexity, with widespread applications in optimization and verification across many domains. Despite significant algorithmic advances over the past two decades, the performance of SAT solvers has improved at a limited pace. Notably, the 2025 competition winner shows only about a 2X improvement over the 2006 winner in SAT Competition performance after nearly 20 years of effort. This paper introduces GaloisSAT, a novel hybrid GPU-CPU SAT solver that integrates a differentiable SAT solving engine powered by modern machine learning infrastructure on GPUs, followed by a traditional CDCL-based SAT solving stage on CPUs. GaloisSAT is benchmarked against the latest versions of state-of-the-art solvers, Kissat and CaDiCaL, using the SAT Competition 2024 benchmark suite. Results demonstrate substantial improvements in the official SAT Competition metric PAR-2 (penalized average runtime with a timeout of 5,000 seconds and a penalty factor of 2). Specifically, GaloisSAT achieves an 8.41X speedup in the satisfiable category and a 1.29X speedup in the unsatisfiable category compared to the strongest baselines.

On-policy self-distillation, where a student is pulled toward a copy of itself conditioned on privileged context (e.g., a verified solution or feedback), offers a promising direction for advancing reasoning capability without a stronger external teacher. Yet in math reasoning the gains are inconsistent, even when the same approach succeeds elsewhere. A pointwise mutual information analysis traces the failure to the privileged context itself: it inflates the teacher's confidence on tokens already implied by the solution (structural connectives, verifiable claims) and deflates it on deliberation tokens ("Wait", "Let", "Maybe") that drive multi-step search. We propose Anti-Self-Distillation (AntiSD), which ascends a divergence between student and teacher rather than descending it: this reverses the per-token sign and yields a naturally bounded advantage in one step. An entropy-triggered gate disables the term once the teacher entropy collapses, completing a drop-in replacement for default self-distillation. Across five models from 4B to 30B parameters on math reasoning benchmarks, AntiSD reaches the GRPO baseline's accuracy in 2 to 10x fewer training steps and improves final accuracy by up to 11.5 points. AntiSD opens a path to scalable self-improvement, where a language model bootstraps its own reasoning through its training signal.

Large Language Models (LLMs) have shown great promise in tool-making, yet existing frameworks often struggle to efficiently construct reliable toolsets and are limited to single-task settings. To address these challenges, we propose GATE (Graph-based Adaptive Tool Evolution), an adaptive framework that dynamically constructs and evolves a hierarchical graph of reusable tools across multiple scenarios. We evaluate GATE on open-ended tasks (Minecraft), agent-based tasks (TextCraft, DABench), and code generation tasks (MATH, Date, TabMWP). Our results show that GATE achieves up to 4.3x faster milestone completion in Minecraft compared to the previous SOTA, and provides an average improvement of 9.23% over existing tool-making methods in code generation tasks and 10.03% in agent tasks. GATE demonstrates the power of adaptive evolution, balancing tool quantity, complexity, and functionality while maintaining high efficiency. Code and data are available at https://github.com/ayanami2003/GATE.

As the field of quantum computing grows, novel algorithms which take advantage of quantum phenomena need to be developed. As we are currently in the NISQ (noisy intermediate scale quantum) era, quantum algorithm researchers cannot reliably test their algorithms on real quantum hardware, which is still too limited. Instead, quantum computing simulators on classical computing systems are used. In the quantum circuit model, quantum bits (qubits) are operated on by quantum gates. A quantum circuit is a sequence of such quantum gates operating on some number of qubits. A quantum gate applied to a qubit can be controlled by other qubits in the circuit. This applies the gate only to the states which satisfy the required control qubit state. We particularly target FPGAs as our main simulation platform, as these offer potential energy savings when compared to running simulations on CPUs/GPUs. In this work, we present a memory access pattern to optimise the number of iterations that need to be scheduled to execute a quantum gate such that only the iterations which access the required pairs (determined according to the control qubits imposed on the gate) are scheduled. We show that this approach results in a significant reduction in the time required to simulate a gate for each added control qubit. We also show that this approach benefits the simulation time on FPGAs more than CPUs and GPUs and allows to outperform both CPU and GPU platforms in terms of energy efficiency, which is the main factor for scalability of the simulations.

The Bernays-Sch\"onfinkel first-order logic fragment over simple linear real arithmetic constraints BS(SLR) is known to be decidable. We prove that BS(SLR) clause sets with both universally and existentially quantified verification conditions (conjectures) can be translated into BS(SLR) clause sets over a finite set of first-order constants. For the Horn case, we provide a Datalog hammer preserving validity and satisfiability. A toolchain from the BS(LRA) prover SPASS-SPL to the Datalog reasoner VLog establishes an effective way of deciding verification conditions in the Horn fragment. This is exemplified by the verification of supervisor code for a lane change assistant in a car and of an electronic control unit for a supercharged combustion engine.

In this work, we provide an overview of circuits for quantum computing. We introduce gates used in quantum computation and then present resource cost measurements used to evaluate circuits made from these gates. We then illustrate how the gates shown are then combined into quantum circuits for basic arithmetic functions. Architectures for addition, subtraction, multiplication, and division are shown. We demonstrate how to calculate the resource costs of quantum circuits. We conclude this overview with by illustrating an application of the elementary quantum circuits for the image rotation operation.

Interpretable machine learning aims to provide transparent models whose decision-making processes can be readily understood by humans. Recent advances in rule-based approaches, such as expressive Boolean formulas (BoolXAI), offer faithful and compact representations of model behavior. However, for non-technical stakeholders, main challenges remain in practice: (i) selecting semantically meaningful features and (ii) translating formal logical rules into accessible explanations. In this work, we propose BoolXLLM , as a hybrid framework that integrates Large Language Models (LLMs) into the end-to-end pipeline of Boolean rule learning. We augment BoolXAI , an expressive Boolean rule-based classifier, with LLMs at three critical stages: (1) feature selection, where LLMs guide the identification of domain-relevant variables; (2) threshold recommendation, where LLMs propose semantically meaningful discretization strategies for numerical features; and (3) rule compression and interpretation, where Boolean rules are translated into natural language explanations at both global and local levels. This integration bridges formal, faithful explanations with human-understandable narratives. This allows build an explainable AI system that is both theoretically grounded and accessible to non-experts. Early empirical results demonstrate that LLM-assisted pipelines improve interpretability while maintaining competitive predictive performance. Our work highlights the promise of combining symbolic reasoning with language-based models for human-centered explainability.

Algorithm learning is a core problem in artificial intelligence with significant implications on automation level that can be achieved by machines. Recently deep learning methods are emerging for synthesizing an algorithm from its input-output examples, the most successful being the Neural GPU, capable of learning multiplication. We present several improvements to the Neural GPU that substantially reduces training time and improves generalization. We introduce a new technique - hard nonlinearities with saturation costs- that has general applicability. We also introduce a technique of diagonal gates that can be applied to active-memory models. The proposed architecture is the first capable of learning decimal multiplication end-to-end.

In this paper, we study quantum Ordered Binary Decision Diagrams(OBDD) model; it is a restricted version of read-once quantum branching programs, with respect to "width" complexity. It is known that the maximal gap between deterministic and quantum complexities is exponential. But there are few examples of functions with such a gap. We present a new technique ("reordering") for proving lower bounds and upper bounds for OBDD with an arbitrary order of input variables if we have similar bounds for the natural order. Using this transformation, we construct a total function REQ such that the deterministic OBDD complexity of it is at least 2^{Ω(n / log n)}, and the quantum OBDD complexity of it is at most O(n^2/log n). It is the biggest known gap for explicit functions not representable by OBDDs of a linear width. Another function(shifted equality function) allows us to obtain a gap 2^{Ω(n)} vs O(n^2). Moreover, we prove the bounded error quantum and probabilistic OBDD width hierarchies for complexity classes of Boolean functions. Additionally, using "reordering" method we extend a hierarchy for read-k-times Ordered Binary Decision Diagrams (k-OBDD) of polynomial width, for k = o(n / log^3 n). We prove a similar hierarchy for bounded error probabilistic k-OBDDs of polynomial, superpolynomial and subexponential width. The extended abstract of this work was presented on International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8 -- 12, 2017

Efficiently compiling quantum operations remains a major bottleneck in scaling quantum computing. Today's state-of-the-art methods achieve low compilation error by combining search algorithms with gradient-based parameter optimization, but they incur long runtimes and require multiple calls to quantum hardware or expensive classical simulations, making their scaling prohibitive. Recently, machine-learning models have emerged as an alternative, though they are currently restricted to discrete gate sets. Here, we introduce a multimodal denoising diffusion model that simultaneously generates a circuit's structure and its continuous parameters for compiling a target unitary. It leverages two independent diffusion processes, one for discrete gate selection and one for parameter prediction. We benchmark the model over different experiments, analyzing the method's accuracy across varying qubit counts, circuit depths, and proportions of parameterized gates. Finally, by exploiting its rapid circuit generation, we create large datasets of circuits for particular operations and use these to extract valuable heuristics that can help us discover new insights into quantum circuit synthesis.

We present a deterministic framework for preparing an arbitrary three-qubit pure state. To leverage entanglement structure in the state-preparation task, we classify three-qubit pure states into five types with respect to a 1|2 bipartition. Given a target state specified by its amplitudes, we provide concrete criteria and concurrence-based tests that determine its type. For each type, we derive an explicit circuit template composed of elementary single-qubit rotations and CNOT gates, with gate parameters determined systematically from the Schmidt decomposition. The full construction is described step by step from the target amplitudes, with no procedural ambiguity. As an application, we further group frequently encountered three-qubit pure states in quantum information into four classes and provide an explicit circuit for each class. Compared with prior approaches, our circuits are designed for practical use: they admit a direct algorithmic instantiation, use only CNOT gates between adjacent qubits, and for certain classes achieve smaller gate counts and circuit depth.

High-quality medical systematic reviews require comprehensive literature searches to ensure the recommendations and outcomes are sufficiently reliable. Indeed, searching for relevant medical literature is a key phase in constructing systematic reviews and often involves domain (medical researchers) and search (information specialists) experts in developing the search queries. Queries in this context are highly complex, based on Boolean logic, include free-text terms and index terms from standardised terminologies (e.g., MeSH), and are difficult and time-consuming to build. The use of MeSH terms, in particular, has been shown to improve the quality of the search results. However, identifying the correct MeSH terms to include in a query is difficult: information experts are often unfamiliar with the MeSH database and unsure about the appropriateness of MeSH terms for a query. Naturally, the full value of the MeSH terminology is often not fully exploited. This paper investigates methods to suggest MeSH terms based on an initial Boolean query that includes only free-text terms. These methods promise to automatically identify highly effective MeSH terms for inclusion in a systematic review query. Our study contributes an empirical evaluation of several MeSH term suggestion methods. We perform an extensive analysis of the retrieval, ranking, and refinement of MeSH term suggestions for each method and how these suggestions impact the effectiveness of Boolean queries.

Recurrent delta-rule models keep a fixed-size state matrix S (d_v x d_k) that compresses all past context. The state of the art (GDN-2) gates this update with element-wise matrix erase/write masks. This is powerful but has two defects. First, both gates are computed from the incoming token alone, making the model memory-blind: it decides what to erase without seeing what it has stored. Second, value-axis coupling in the erase gate blocks the WY-form triangular chunk solver that drives efficient training -- the intra-chunk system splits into d_v independent solves, collapsing throughput to serial-recurrence cost. We introduce CARVE (Content-Aware Recurrent with Value Efficiency), which fixes both and, via a single-launch "megakernel" scheduling of the same WY-form math, trains faster than the matrix-gated baseline it replaces. The key idea is architectural: restricting all gating to the key axis makes the intra-chunk coupling independent of the value index, restoring one unmodified WY-form solve. Within this constraint, CARVE conditions both gates on a content signal read once per chunk from the chunk-boundary state and folded algebraically into each gate's low-rank projection (by associativity, U(Sq)=(US)q), giving memory-aware gating at negligible extra traffic. At init the content projections are zero, so CARVE is bit-identical to the baseline; we prove the one-chunk staleness perturbs gates by only O(1/sqrt(L)), matching a measured 0.18% deviation flat up to L=128. At 1.3B parameters / 100B FineWeb-Edu tokens on H100 (three seeds), CARVE improves every axis: WikiText perplexity 15.72 vs 15.90 (hybrid 15.41 vs 15.62), +0.63 pp average common-sense accuracy, and state-of-the-art RULER and real-world recall -- while training +1.4% faster at matched depth and +19.3% at iso-quality depth, at +13% peak memory. Backed by six formal guarantees.

Backdoor mechanisms have traditionally been studied as security threats that compromise the integrity of machine learning models. However, the same mechanism -- the conditional activation of specific behaviors through input triggers -- can also serve as a controllable and auditable interface for trustworthy model behavior. In this work, we present Backdoor4Good (B4G), a unified benchmark and framework for beneficial backdoor applications in large language models (LLMs). Unlike conventional backdoor studies focused on attacks and defenses, B4G repurposes backdoor conditioning for Beneficial Tasks that enhance safety, controllability, and accountability. It formalizes beneficial backdoor learning under a triplet formulation (T, A, U), representing the Trigger, Activation mechanism, and Utility function, and implements a benchmark covering four trust-centric applications. Through extensive experiments across Llama3.1-8B, Gemma-2-9B, Qwen2.5-7B, and Llama2-13B, we show that beneficial backdoors can achieve high controllability, tamper-resistance, and stealthiness while preserving clean-task performance. Our findings demonstrate new insights that backdoors need not be inherently malicious; when properly designed, they can serve as modular, interpretable, and beneficial building blocks for trustworthy AI systems. Our code and datasets are available at https://github.com/bboylyg/BackdoorLLM/B4G.

Software testing plays a critical role in ensuring that systems behave as intended. However, existing automated testing approaches struggle to match the capabilities of human engineers due to key limitations such as test locality, lack of general reliability, and business logic blindness. In this work, we propose a novel framework that leverages functional programming and type systems to translate Scala backend code into formal Lean representations. Our pipeline automatically generates theorems that specify the intended behavior of APIs and database operations, and uses LLM-based provers to verify them. When a theorem is proved, the corresponding logic is guaranteed to be correct and no further testing is needed. If the negation of a theorem is proved instead, it confirms a bug. In cases where neither can be proved, human intervention is required. We evaluate our method on realistic backend systems and find that it can formally verify over 50% of the test requirements, which suggests that half of a testing engineer's workload can be automated. Additionally, with an average cost of only $2.19 per API, LLM-based verification is significantly more cost-effective than manual testing and can be scaled easily through parallel execution. Our results indicate a promising direction for scalable, AI-powered software testing, with the potential to greatly improve engineering productivity as models continue to advance.

One of the fundamental problems in Artificial Intelligence is to perform complex multi-hop logical reasoning over the facts captured by a knowledge graph (KG). This problem is challenging, because KGs can be massive and incomplete. Recent approaches embed KG entities in a low dimensional space and then use these embeddings to find the answer entities. However, it has been an outstanding challenge of how to handle arbitrary first-order logic (FOL) queries as present methods are limited to only a subset of FOL operators. In particular, the negation operator is not supported. An additional limitation of present methods is also that they cannot naturally model uncertainty. Here, we present BetaE, a probabilistic embedding framework for answering arbitrary FOL queries over KGs. BetaE is the first method that can handle a complete set of first-order logical operations: conjunction (wedge), disjunction (vee), and negation (neg). A key insight of BetaE is to use probabilistic distributions with bounded support, specifically the Beta distribution, and embed queries/entities as distributions, which as a consequence allows us to also faithfully model uncertainty. Logical operations are performed in the embedding space by neural operators over the probabilistic embeddings. We demonstrate the performance of BetaE on answering arbitrary FOL queries on three large, incomplete KGs. While being more general, BetaE also increases relative performance by up to 25.4% over the current state-of-the-art KG reasoning methods that can only handle conjunctive queries without negation.

We introduce the Mixed-Integer Quadratically Constrained Quadratic Programming framework for the quantum compilation problem and apply it in the context of topological quantum computing. In this setting, quantum gates are realized by sequences of elementary braids of quasiparticles with exotic fractional statistics in certain two-dimensional topological condensed matter systems, described by effective topological quantum field theories. We specifically focus on a non-semisimple version of topological field theory, which provides a foundation for an extended theory of Ising anyons and which has recently been shown by Iulianelli et al., Nature Communications {\bf 16}, 6408 (2025), to permit universal quantum computation. While the proofs of this pioneering result are existential in nature, the mixed integer programming provides an approach to explicitly construct quantum gates in topological systems. We demonstrate this by focusing specifically on the entangling controlled-NOT operation, and its local equivalence class, using braiding operations in the non-semisimple Ising system. This illustrates the utility of the Mixed-Integer Quadratically Constrained Quadratic Programming for topological quantum compilation.

Formal theorem proving (FTP) has emerged as a critical foundation for evaluating the reasoning capabilities of large language models, enabling automated verification of mathematical proofs at scale. However, progress has been constrained by limited datasets due to the high cost of manual curation and the scarcity of challenging problems with verified formal-informal correspondences. We propose leveraging theoretical computer science (TCS) as a scalable source of rigorous proof problems, where algorithmic definitions enable automated generation of arbitrarily many challenging theorem-proof pairs. We demonstrate this approach on two TCS domains: Busy Beaver problems, which involve proving bounds on Turing machine halting behavior, and Mixed Boolean Arithmetic problems, which combine logical and arithmetic reasoning. Our framework automatically synthesizes problems with parallel formal (Lean4) and informal (Markdown) specifications, creating a scalable pipeline for generating verified proof challenges. Evaluation on frontier models reveals substantial gaps in automated theorem proving: while DeepSeekProver-V2-671B achieves 57.5\% success on Busy Beaver problems, it manages only 12\% on Mixed Boolean Arithmetic problems. These results highlight the difficulty of long-form proof generation even for problems that are computationally easy to verify, demonstrating the value of TCS domains for advancing automated reasoning research.

This paper localizes the policy routing mechanism in alignment-trained language models. An intermediate-layer attention gate reads detected content and triggers deeper amplifier heads that boost the signal toward refusal. In smaller models the gate and amplifier are single heads; at larger scale they become bands of heads across adjacent layers. The gate contributes under 1% of output DLA, but interchange testing (p<0.001) and knockout cascade confirm it is causally necessary. Interchange screening at n>=120 detects the same motif in twelve models from six labs (2B to 72B), though specific heads differ by lab. Per-head ablation weakens up to 58x at 72B and misses gates that interchange identifies; interchange is the only reliable audit at scale. Modulating the detection-layer signal continuously controls policy from hard refusal through evasion to factual answering. On safety prompts the same intervention turns refusal into harmful guidance, showing the safety-trained capability is gated by routing rather than removed. Thresholds vary by topic and by input language, and the circuit relocates across generations within a family while behavioral benchmarks register no change. Routing is early-commitment: the gate commits at its own layer before deeper layers finish processing the input. Under an in-context substitution cipher, gate interchange necessity collapses 70 to 99% across three models and the model switches to puzzle-solving. Injecting the plaintext gate activation into the cipher forward pass restores 48% of refusals in Phi-4-mini, localizing the bypass to the routing interface. A second method, cipher contrast analysis, uses plain/cipher DLA differences to map the full cipher-sensitive routing circuit in O(3n) forward passes. Any encoding that defeats detection-layer pattern matching bypasses the policy regardless of whether deeper layers reconstruct the content.

Recent advancements in language models have led to significant improvements in mathematical reasoning across various benchmarks. However, most of these benchmarks rely on automatic evaluation methods that only compare final answers using heuristics, without verifying the underlying reasoning steps. This limitation results in false positive solutions, where models may produce correct final answers but with flawed deduction paths. In this paper, we systematically examine the prevalence of false positive solutions in mathematical problem solving for language models. We analyze the characteristics and extent of this issue across different open-source models, datasets of varying difficulty levels, and decoding strategies. Specifically, we explore how false positives influence the inference time scaling behavior of language models. Our experimental results reveal that: (1) false positive solutions persist across different models, datasets, and decoding methods, (2) sampling-based inference time scaling methods do not alleviate the problem, and (3) the pass@N evaluation metric is more susceptible to false positives, suggesting a significantly lower scaling ceiling than what automatic evaluations indicate. Additionally, we analyze specific instances of false positives and discuss potential limitations in self-improvement techniques and synthetic data generation under such conditions. Our data and code are publicly available at https://github.com/Wloner0809/False-Positives-in-Math.

Differentiable logic gate networks (DLGNs) exhibit extraordinary efficiency at inference while sustaining competitive accuracy. But vanishing gradients, discretization errors, and high training cost impede scaling these networks. Even with dedicated parameter initialization schemes from subsequent works, increasing depth still harms accuracy. We show that the root cause of these issues lies in the underlying parametrization of logic gate neurons themselves. To overcome this issue, we propose a reparametrization that also shrinks the parameter size logarithmically in the number of inputs per gate. For binary inputs, this already reduces the model size by 4x, speeds up the backward pass by up to 1.86x, and converges in 8.5x fewer training steps. On top of that, we show that the accuracy on CIFAR-100 remains stable and sometimes superior to the original parametrization.

Integer overflow accounts for one of the major source of bugs in software. Verification systems typically assume a well defined underlying semantics for various integer operations and do not explicitly check for integer overflow in programs. In this paper we present a specification mechanism for expressing integer overflow. We develop an automated procedure for integer overflow checking during program verification. We have implemented a prototype integer overflow checker and tested it on a benchmark consisting of already verified programs (over 14k LOC). We have found 43 bugs in these programs due to integer overflow.

While accelerated computing has transformed many domains of computing, its impact on logical reasoning, specifically Boolean satisfiability (SAT), remains limited. State-of-the-art SAT solvers rely heavily on inherently sequential conflict-driven search algorithms that offer powerful heuristics but limit the amount of parallelism that could otherwise enable significantly more scalable SAT solving. Inspired by neural network training, we formulate the SAT problem as a binarized matrix-matrix multiplication layer that could be optimized using a differentiable objective function. Enabled by this encoding, we combine the strengths of parallel differentiable optimization and sequential search to accelerate SAT on a hybrid GPU-CPU system. In this system, the GPUs leverage parallel differentiable solving to rapidly evaluate SAT clauses and use gradients to stochastically explore the solution space and optimize variable assignments. Promising partial assignments generated by the GPUs are post-processed on many CPU threads which exploit conflict-driven sequential search to further traverse the solution subspaces and identify complete assignments. Prototyping the hybrid solver on an NVIDIA DGX GB200 node, our solver achieves runtime speedups up to over 200x when compared to a state-of-the-art CPU-based solver on public satisfiable benchmark problems from the SAT Competition.

AI web agents use Internet resources at far greater speed, scale, and complexity -- changing how users and services interact. Deployed maliciously or erroneously, these agents could overload content providers. At the same time, web agents can bypass CAPTCHAs and other defenses by mimicking user behavior or flood authentication systems with fake accounts. Yet providers must protect their services and content from denial-of-service attacks and scraping by web agents. In this paper, we design a framework that imposes tunable costs on agents before providing access to resources; we call this Web Agent Throttling. We start by formalizing Throttling Gates as challenges issued to an agent that are asymmetric, scalable, robust, and compatible with any agent. Focusing on a common component -- the language model -- we require the agent to solve reasoning puzzles, thereby incurring excessive token-generation costs. However, we find that using existing puzzles, e.g., coding or math, as throttling gates fails to satisfy our properties. To address this, we introduce rebus-based Reasoning Gates, synthetic text puzzles that require multi-hop reasoning over world knowledge (thereby throttling an agent's model). We design a scalable generation and verification protocol for such reasoning gates. Our framework achieves computational asymmetry, i.e., the response-generation cost is 9.2x higher than the generation cost for SOTA models. We further deploy reasoning gates on a custom website and Model Context Protocol (MCP) servers and evaluate with real-world web agents. Finally, we discuss the limitations and environmental impact of real-world deployment of our framework.

Noise is a ubiquitous feature of the physical world. As a result, the first prerequisite of life is fault tolerance: maintaining integrity of state despite external bombardment. Recent experimental advances have revealed that biological systems achieve fault tolerance by implementing mathematically intricate error-correcting codes and by organizing in a modular fashion that physically separates functionally distinct subsystems. These elaborate structures represent a vanishing volume in the massive genetic configuration space. How is it possible that the primitive process of evolution, by which all biological systems evolved, achieved such unusual results? In this work, through experiments in Boolean networks, we show that the simultaneous presence of error correction and modularity in biological systems is no coincidence. Rather, it is a typical co-occurrence in noisy dynamic systems undergoing evolution. From this, we deduce the principle of error correction enhanced evolvability: systems possessing error-correcting codes are more effectively improved by evolution than those without.

We introduce a quantum algorithm that produces approximate solutions for combinatorial optimization problems. The algorithm depends on a positive integer p and the quality of the approximation improves as p is increased. The quantum circuit that implements the algorithm consists of unitary gates whose locality is at most the locality of the objective function whose optimum is sought. The depth of the circuit grows linearly with p times (at worst) the number of constraints. If p is fixed, that is, independent of the input size, the algorithm makes use of efficient classical preprocessing. If p grows with the input size a different strategy is proposed. We study the algorithm as applied to MaxCut on regular graphs and analyze its performance on 2-regular and 3-regular graphs for fixed p. For p = 1, on 3-regular graphs the quantum algorithm always finds a cut that is at least 0.6924 times the size of the optimal cut.

The Mixture-of-Experts (MoE) architecture is showing promising results in improving parameter sharing in multi-task learning (MTL) and in scaling high-capacity neural networks. State-of-the-art MoE models use a trainable sparse gate to select a subset of the experts for each input example. While conceptually appealing, existing sparse gates, such as Top-k, are not smooth. The lack of smoothness can lead to convergence and statistical performance issues when training with gradient-based methods. In this paper, we develop DSelect-k: a continuously differentiable and sparse gate for MoE, based on a novel binary encoding formulation. The gate can be trained using first-order methods, such as stochastic gradient descent, and offers explicit control over the number of experts to select. We demonstrate the effectiveness of DSelect-k on both synthetic and real MTL datasets with up to 128 tasks. Our experiments indicate that DSelect-k can achieve statistically significant improvements in prediction and expert selection over popular MoE gates. Notably, on a real-world, large-scale recommender system, DSelect-k achieves over 22% improvement in predictive performance compared to Top-k. We provide an open-source implementation of DSelect-k.

Proactive agents must decide not only what to say but also whether and when to intervene. Many current systems rely on brittle heuristics or indiscriminate long reasoning, which offers little control over the benefit-burden tradeoff. We formulate the problem as cost-sensitive selective intervention and present PRISM, a novel framework that couples a decision-theoretic gate with a dual-process reasoning architecture. At inference time, the agent intervenes only when a calibrated probability of user acceptance exceeds a threshold derived from asymmetric costs of missed help and false alarms. Inspired by festina lente (Latin: "make haste slowly"), we gate by an acceptance-calibrated, cost-derived threshold and invoke a resource-intensive Slow mode with counterfactual checks only near the decision boundary, concentrating computation on ambiguous and high-stakes cases. Training uses gate-aligned, schema-locked distillation: a teacher running the full PRISM pipeline provides dense, executable supervision on unlabeled interaction traces, while the student learns a response policy that is explicitly decoupled from the intervention gate to enable tunable and auditable control. On ProactiveBench, PRISM reduces false alarms by 22.78% and improves F1 by 20.14% over strong baselines. These results show that principled decision-theoretic gating, paired with selective slow reasoning and aligned distillation, yields proactive agents that are precise, computationally efficient, and controllable. To facilitate reproducibility, we release our code, models, and resources at https://prism-festinalente.github.io/; all experiments use the open-source ProactiveBench benchmark.

This paper introduces a novel type theory and logic for probabilistic reasoning. Its logic is quantitative, with fuzzy predicates. It includes normalisation and conditioning of states. This conditioning uses a key aspect that distinguishes our probabilistic type theory from quantum type theory, namely the bijective correspondence between predicates and side-effect free actions (called instrument, or assert, maps). The paper shows how suitable computation rules can be derived from this predicate-action correspondence, and uses these rules for calculating conditional probabilities in two well-known examples of Bayesian reasoning in (graphical) models. Our type theory may thus form the basis for a mechanisation of Bayesian inference.

Under the conditional framework breaking the longstanding O(n log n) integer multiplication barrier (originating from OpenAI Preprint #109), the asymptotic time complexity exponent T(n) = O(n (log n)^{1-κ}) is governed by bounding the second moment E[X^2] < 1 of recursive branching processes over trilinear tensor carrier graphs. In this work, we present a constructive combinatorial optimization over the alternating carrier bipartite matching graphs of the Schönhage trilinear tensor decomposition at dimensions h=23 and h=25. Through an alternating augmenting path search, we uncover a non-trivial cluster-unlocking cascade that eliminates 185 carrier roles (95 roles in h=23 and 90 roles in h=25), reducing the global physical wire width by 377,890 wires from the reviewed baseline W = 177,284,805 down to W = 176,906,915. Evaluated over the exact 10^{-14} rational grid, this wire reduction certifies an improved second-moment coefficient AB = 5,164,059 / 125,000,000,000 and establishes the sharpened conditional exponent κ= 826,215,307 / 20,000,000,000,000 \approx 4.131076535 \times 10^{-5}, strictly surpassing the preceding record on this architecture (κ\approx 4.123864 \times 10^{-5}). All 47 polynomial constraints and 7 rational security margins are audited in exact Q-arithmetic with zero floating-point error, and the fundamental wire reduction identity is formally verified in the Lean 4 interactive theorem prover without unproven axioms.

We briefly address Landauer's Principle and some related issues in thermal demons. We show that an error-free Turing computer works in the zero-entropy limit, which proves Landauer's derivation incorrect. To have a physical logic gate, memory or information-engine, a few essential components necessary for the operation of these devices are often neglected, such as various aspects of control, damping and the fluctuation-dissipation theorem. We also point out that bit erasure is typically not needed or used for the functioning of computers or engines (except for secure erasure).

Given the computational cost and technical expertise required to train machine learning models, users may delegate the task of learning to a service provider. We show how a malicious learner can plant an undetectable backdoor into a classifier. On the surface, such a backdoored classifier behaves normally, but in reality, the learner maintains a mechanism for changing the classification of any input, with only a slight perturbation. Importantly, without the appropriate "backdoor key", the mechanism is hidden and cannot be detected by any computationally-bounded observer. We demonstrate two frameworks for planting undetectable backdoors, with incomparable guarantees. First, we show how to plant a backdoor in any model, using digital signature schemes. The construction guarantees that given black-box access to the original model and the backdoored version, it is computationally infeasible to find even a single input where they differ. This property implies that the backdoored model has generalization error comparable with the original model. Second, we demonstrate how to insert undetectable backdoors in models trained using the Random Fourier Features (RFF) learning paradigm or in Random ReLU networks. In this construction, undetectability holds against powerful white-box distinguishers: given a complete description of the network and the training data, no efficient distinguisher can guess whether the model is "clean" or contains a backdoor. Our construction of undetectable backdoors also sheds light on the related issue of robustness to adversarial examples. In particular, our construction can produce a classifier that is indistinguishable from an "adversarially robust" classifier, but where every input has an adversarial example! In summary, the existence of undetectable backdoors represent a significant theoretical roadblock to certifying adversarial robustness.

QSE is emerging as a critical discipline to make quantum computing accessible to a broader developer community; however, most quantum development environments still require developers to engage with low-level details across the software stack - including problem encoding, circuit construction, algorithm configuration, hardware selection, and result interpretation - making them difficult for classical software engineers to use. To bridge this gap, we present C2|Q>, a hardware-agnostic quantum software development framework that translates specific types of classical specifications into quantum-executable programs while preserving methodological rigor. The framework applies modular SE principles by classifying the workflow into three core modules: an encoder that classifies problems, produces Quantum-Compatible Formats, and constructs quantum circuits, a deployment module that generates circuits and recommends hardware based on fidelity, runtime, and cost, and a decoder that interprets quantum outputs into classical solutions. In evaluation, the encoder module achieved a 93.8% completion rate, the hardware recommendation module consistently selected the appropriate quantum devices for workloads scaling up to 56 qubits. End-to-end experiments on 434 Python programs and 100 JSON problem instances show that the full C2|Q> workflow executes reliably on simulators and can be deployed successfully on representative real quantum hardware, with empirical runs limited to small- and medium-sized instances consistent with current NISQ capabilities. These results indicate that C2|Q> lowers the entry barrier to quantum software development by providing a reproducible, extensible toolchain that connects classical specifications to quantum execution. The open-source implementation of C2|Q> is available at https://github.com/C2-Q/C2Q and as a Python package at https://pypi.org/project/c2q-framework/.

In this paper, a simple yet effective network pruning framework is proposed to simultaneously address the problems of pruning indicator, pruning ratio, and efficiency constraint. This paper argues that the pruning decision should depend on the convolutional weights, and thus proposes novel weight-dependent gates (W-Gates) to learn the information from filter weights and obtain binary gates to prune or keep the filters automatically. To prune the network under efficiency constraints, a switchable Efficiency Module is constructed to predict the hardware latency or FLOPs of candidate pruned networks. Combined with the proposed Efficiency Module, W-Gates can perform filter pruning in an efficiency-aware manner and achieve a compact network with a better accuracy-efficiency trade-off. We have demonstrated the effectiveness of the proposed method on ResNet34, ResNet50, and MobileNet V2, respectively achieving up to 1.33/1.28/1.1 higher Top-1 accuracy with lower hardware latency on ImageNet. Compared with state-of-the-art methods, W-Gates also achieves superior performance.

Most recent works focus on answering first order logical queries to explore the knowledge graph reasoning via multi-hop logic predictions. However, existing reasoning models are limited by the circumscribed logical paradigms of training samples, which leads to a weak generalization of unseen logic. To address these issues, we propose a plug-in module called Logic Diffusion (LoD) to discover unseen queries from surroundings and achieves dynamical equilibrium between different kinds of patterns. The basic idea of LoD is relation diffusion and sampling sub-logic by random walking as well as a special training mechanism called gradient adaption. Besides, LoD is accompanied by a novel loss function to further achieve the robust logical diffusion when facing noisy data in training or testing sets. Extensive experiments on four public datasets demonstrate the superiority of mainstream knowledge graph reasoning models with LoD over state-of-the-art. Moreover, our ablation study proves the general effectiveness of LoD on the noise-rich knowledge graph.

As giant dense models advance quality but require large amounts of GPU budgets for training, the sparsely gated Mixture-of-Experts (MoE), a kind of conditional computation architecture, is proposed to scale models while keeping their computation constant. Specifically, the input tokens are routed by the gate network and only activates part of the expert network. Existing MoE training systems only support part of mainstream MoE models (e.g. Top k) training under expensive high-bandwidth GPU clusters. In this paper, we present HetuMoE, a high-performance large-scale sparse MoE training system built on Hetu. HetuMoE provides multiple gating strategies and efficient GPU kernel implementations. To further improve the training efficiency on commodity GPU clusters (e.g, with only 1 NiC), we introduce the hierarchical AllToAll communication that combines hierarchical networks and aggregating messages. Compared with existing state-of-the-art MoE systems, HetuMoE obtains at least 15% speedup. Specifically, HetuMoE outperforms DeepSpeed-MoE up to 8.1x under the switch gate with a batch size of 32. Our code is available at: https://github.com/PKU-DAIR/Hetu.

This is an essay in what might be called ``mathematical metaphysics.'' There is a fundamental duality that run through mathematics and the natural sciences. The duality starts as the logical level; it is represented by the Boolean logic of subsets and the logic of partitions since subsets and partitions are category-theoretic dual concepts. In more basic terms, it starts with the duality between the elements (Its) of subsets and the distinctions (Dits, i.e., ordered pairs of elements in different blocks) of a partition. Mathematically, the Its & Dits duality is fully developed in category theory as the reverse-the-arrows duality. The quantitative versions of subsets and partitions are developed as probability theory and information theory (based on logical entropy). Classical physics was based on a view of reality as definite all the way down. In contrast, quantum physics embodies (objective) indefiniteness. And finally, there are the two fundamental dual mechanisms at work in biology, the selectionist mechanism and the generative mechanism, two mechanisms that embody the fundamental duality.

This paper studies propositional proof systems in which lines are sequents of decision trees or branching programs - deterministic and nondeterministic. The systems LDT and LNDT are propositional proof systems in which lines represent deterministic or non-deterministic decision trees. Branching programs are modeled as decision dags. Adding extension to LDT and LNDT gives systems eLDT and eLNDT in which lines represent deterministic and non-deterministic branching programs, respectively. Deterministic and non-deterministic branching programs correspond to log-space (L) and nondeterministic log-space (NL). Thus the systems eLDT and eLNDT are propositional proof systems that reason with (nonuniform) L and NL properties. The main results of the paper are simulation and non-simulation results for tree-like and dag-like proofs in the systems LDT, LNDT, eLDT, and eLNDT. These systems are also compared with Frege systems, constantdepth Frege systems and extended Frege systems

In this paper, we describe a design of a mixed signal circuit for a binary neuron (a.k.a perceptron, threshold logic gate) and a methodology for automatically embedding such cells in ASICs. The binary neuron, referred to as an FTL (flash threshold logic) uses floating gate or flash transistors whose threshold voltages serve as a proxy for the weights of the neuron. Algorithms for mapping the weights to the flash transistor threshold voltages are presented. The threshold voltages are determined to maximize both the robustness of the cell and its speed. The performance, power, and area of a single FTL cell are shown to be significantly smaller (79.4%), consume less power (61.6%), and operate faster (40.3%) compared to conventional CMOS logic equivalents. Also included are the architecture and the algorithms to program the flash devices of an FTL. The FTL cells are implemented as standard cells, and are designed to allow commercial synthesis and P&R tools to automatically use them in synthesis of ASICs. Substantial reductions in area and power without sacrificing performance are demonstrated on several ASIC benchmarks by the automatic embedding of FTL cells. The paper also demonstrates how FTL cells can be used for fixing timing errors after fabrication.

We present Generative Logic (GL), a deterministic architecture that begins from user-supplied axiomatic definitions -- written in a minimalist Mathematical Programming Language (MPL) -- and systematically explores their deductive neighborhood. Definitions are compiled into a distributed grid of simple Logic Blocks (LBs) that exchange messages; any time several expressions unify under an inference rule, a new fact is emitted with full provenance to its sources, yielding replayable, auditable proof graphs. A prototype software implementation instantiates the workflow on first-order Peano arithmetic. Starting only from the Peano axioms, GL enumerates candidate implications, applies normalization and type filters, and automatically reconstructs machine-checkable proofs of foundational arithmetic laws including associativity and commutativity of addition, associativity and commutativity of multiplication, and distributivity. Generated proofs export to navigable HTML so that every inference step can be inspected independently. We outline a hardware-software co-design path toward massively parallel realizations and describe prospective integration with probabilistic models (e.g., Large Language Models (LLMs)) for autoformalization and conjecture seeding. The Python and MPL code to reproduce the Peano experiments, along with the full HTML proof graphs, are available in the project's GitHub repository at https://github.com/Generative-Logic/GL/tree/35a111ea9ba53afe051703d6050be0c3923e9724 and are permanently archived at https://doi.org/10.5281/zenodo.16408441. We invite community feedback and collaboration.

Software that hands branching decisions to a model needs a declared option and a probability it can threshold. Typed decision models, also called System One models, return such probabilities without generating text, while supervised classifiers and generative language models are the established alternatives. One harness sends eight decision-model checkpoints from six families, including the hosted model Jev, and four open generative models from three developers the same semantic requests, and scores trained and zero-shot classifiers on the same workflow, intent, emotion and social-science items. With task labels, a fine-tuned DeBERTa-v3-large has the highest observed accuracy on every labeled benchmark but one. Without labels, no decision model is significantly more accurate than Jev on workflows or intents, but Gemma-4-31B matches it on workflows and exceeds it on CLINC-150 at higher cost and latency. Stated probabilities of generative models become unreadable when replies miss the key format, whereas key likelihoods avoid this but can saturate. A guaranteed 5 percent risk leaves Jev 0.528 of the intent decisions, and an in-scope threshold still accepts 0.310 of out-of-scope requests. On typed-decisions, swapping yes and no flips 50.5 answers per hundred for Jev and at least 16.8 for every generative model tested, against at most 6.5 for four fine-tuned decision checkpoints. Exposure to a benchmark's training data explains the largest lead of an open checkpoint, which vanishes on rater-labeled emotions. An intent-trained first stage escalating to Gemma-4-31B reaches that model's accuracy at about Jev's price. The results yield condition-dependent design rules for automated decision gates.

We contribute to the study of formal languages that can be recognized by transformer encoders. We focus on two self-attention mechanisms: (1) UHAT (Unique Hard Attention Transformers) and (2) AHAT (Average Hard Attention Transformers). UHAT encoders are known to recognize only languages inside the circuit complexity class {sf AC}^0, i.e., accepted by a family of poly-sized and depth-bounded boolean circuits with unbounded fan-ins. On the other hand, AHAT encoders can recognize languages outside {sf AC}^0), but their expressive power still lies within the bigger circuit complexity class {sf TC}^0, i.e., {sf AC}^0-circuits extended by majority gates. We first show a negative result that there is an {sf AC}^0-language that cannot be recognized by an UHAT encoder. On the positive side, we show that UHAT encoders can recognize a rich fragment of {sf AC}^0-languages, namely, all languages definable in first-order logic with arbitrary unary numerical predicates. This logic, includes, for example, all regular languages from {sf AC}^0. We then show that AHAT encoders can recognize all languages of our logic even when we enrich it with counting terms. We apply these results to derive new results on the expressive power of UHAT and AHAT up to permutation of letters (a.k.a. Parikh images).

We provide here a dataset for tasks related to natural language understanding and natural language inference. The dataset contains logical puzzles in natural language from three domains: comparing puzzles, knighs and knaves, and zebra puzzles. Each puzzle is associated with the entire set of atomic questions that can be generated based on the relations and individuals occurring in the text. For each question we provide the correct answer: entailment, contradiction or ambiguity. The answer's correctness is verified against theorem provers. Good puzzles have two properties: (i) each piece of information is necessary and (ii) no unnecessary information is provided. These properties make puzzles interesting candidates for machine comprehension tasks.

Can Large Language Models (LLMs) understand and reason about quantum operators? Despite their remarkable capabilities in mathematics and symbolic reasoning, LLMs remain inherently blind to quantum representations such as unitary matrices. In this work, we take a step toward bridging this gap by introducing an approach that maps unitary operators into the latent space of an LLM, enabling unified modeling over quantum and linguistic inputs. We instantiate this idea on Clifford+T circuit synthesis over a Pauli rotation gate set, where our model achieves results competitive with state-of-the-art methods and scales consistently with training data, with no signs of saturation. Our approach further enables language-conditioned synthesis, allowing gate constraints unseen during training to be specified directly in natural language. This work suggests a path toward quantum--aware foundation models that can natively interpret and reason about quantum operations, which could have broader implications reaching across quantum compilation and algorithm discovery.

Large language models perform well on many logical reasoning benchmarks, but it remains unclear which core logical skills they truly master. To address this, we introduce LogicSkills, a benchmark that isolates three fundamental logical skills: (i) formal symbolizationx2014{}translating premises into first-order logic; (ii) countermodel constructionx2014showing that an argument is logically invalid by constructing a finite countermodel; and (iii) validity assessmentx2014determining whether a conclusion follows from a set of premises. Items are drawn from the two-variable fragment of first-order logic without identity and are presented in both English and a Carrollian nonce-word language. All instances are solver-verified with Z3 for correctness and non-triviality. Across conventional instruction-tuned LLMs, performance is high on validity assessment but substantially lower on formal symbolization and countermodel construction, highlighting that high task-level accuracy can mask weaknesses in core logical skills. In contrast, recent reasoning-tuned models perform strongly across all three tasks, suggesting a more systematic logical skill profile.

We investigate the potential of bio-inspired evolutionary algorithms for designing quantum circuits with specific goals, focusing on two particular tasks. The first one is motivated by the ideas of Artificial Life that are used to reproduce stochastic cellular automata with given rules. We test the robustness of quantum implementations of the cellular automata for different numbers of quantum gates The second task deals with the sampling of quantum circuits that generate highly entangled quantum states, which constitute an important resource for quantum computing. In particular, an evolutionary algorithm is employed to optimize circuits with respect to a fitness function defined with the Mayer-Wallach entanglement measure. We demonstrate that, by balancing the mutation rate between exploration and exploitation, we can find entangling quantum circuits for up to five qubits. We also discuss the trade-off between the number of gates in quantum circuits and the computational costs of finding the gate arrangements leading to a strongly entangled state. Our findings provide additional insight into the trade-off between the complexity of a circuit and its performance, which is an important factor in the design of quantum circuits.

Combining the answers a large language model (LLM) samples for a question into one decision is a test-time information fusion problem, usually solved by majority voting. Voting is unreliable on difficult questions, where the sampled answers share correlated errors, so the wrong answer can win and drawing more samples makes the decision worse. Selecting a candidate by reading a correctness signal from the model's hidden states is a promising alternative, but its accuracy varies across models and tasks, and no measure indicates when it can be trusted. In this paper, we propose CASE (Correctness-Axis SElection), a dynamic selection combiner that trains a linear gate on the answer-token hidden state and selects the highest-scoring candidate. Its main contribution is decodability, a leakage-free measure of how well the gate ranks a question's correct candidates above its incorrect ones, which predicts whether hidden-state selection will outperform voting. A conventional probe appears accurate only because of question-identity leakage, which vanishes under question-grouped evaluation. On held-out data, decodability predicts the accuracy gain of selection over voting with a Pearson correlation r=0.75 and a decision threshold near AUC=0.60. Across general and medical LLMs, CASE improves over voting by up to 19 points on medium-difficulty questions and 16.8 points on hard questions. Decodability depends on the aligned knowledge a model must recall, not on its scale, and its prediction transfers to an unseen scientific domain within 3.8 points. It thus provides a practical criterion, measurable in advance for a given model and task, for choosing between learned selection and majority voting.

This paper considers the learning of logical (Boolean) functions with focus on the generalization on the unseen (GOTU) setting, a strong case of out-of-distribution generalization. This is motivated by the fact that the rich combinatorial nature of data in certain reasoning tasks (e.g., arithmetic/logic) makes representative data sampling challenging, and learning successfully under GOTU gives a first vignette of an 'extrapolating' or 'reasoning' learner. We then study how different network architectures trained by (S)GD perform under GOTU and provide both theoretical and experimental evidence that for a class of network models including instances of Transformers, random features models, and diagonal linear networks, a min-degree-interpolator (MDI) is learned on the unseen. We also provide evidence that other instances with larger learning rates or mean-field networks reach leaky MDIs. These findings lead to two implications: (1) we provide an explanation to the length generalization problem (e.g., Anil et al. 2022); (2) we introduce a curriculum learning algorithm called Degree-Curriculum that learns monomials more efficiently by incrementing supports.

Tabular data in knowledge-rich domains often carries a latent prior in the form of Boolean implication relationships (BIRs) between pairs of features. We mine such relationships with a sparse-exception binomial test. The mined implications form a typed directed graph, equivalent to a propositional rule base of 2-literal clauses. We encode this graph as the connectivity of a layered neural network, called BIRDNet, in which each hidden unit corresponds to one mined rule and binds only to its two features. We show two consequences of this design: First, the architecture is sparse by construction: at most 2/d of the weights in each BIR layer are active, where d is the input dimension. Second, the model is interpretable: every trained unit keeps a stable symbolic identity, so rules can be read off the network without surrogate models. Unlike most neurosymbolic models, BIRDNet does not consume an external rule base; its structural prior is mined from the data. We evaluate BIRDNet on six transcriptomic and proteomic benchmarks. Our results show that BIRDNet stays within 0.02 AUROC of the strongest dense baseline, at a small accuracy cost, while using up to 96times fewer active parameters than an architecture-matched dense MLP. First-layer rules recover known biological signatures across multiple cancer subtypes and tissue types, including canonical amplicons, lineage-defining co-expression modules, and immune-infiltration markers. Data and code are available at: https://github.com/MAHI-Group/BIRDNet.

Generalized code surgery is a versatile and low-overhead technique for performing fault-tolerant computation on quantum low-density parity-check (qLDPC) codes. In many settings, surgery exhibits practical space overheads, while its time overhead remains a bottleneck at O(d) syndrome rounds per operation. In this work, we construct surgery gadgets that perform parallel logical measurements on 2D hypergraph product codes in constant time overhead (O(1)) and near-constant space overhead (O(1)). The reduced time overhead is a result of amortization, as we show, following the formulation by Cowtan et al. (arXiv:2510.14895), that performing d surgery operations in O(d) time is fault tolerant. Our gadgets combine the strengths of different approaches to fault-tolerant logical operations: they partially retain the flexibility of surgery while achieving overheads comparable to transversal gates. Consequently, they are well-suited for near-term experimental realization and demonstrate new possibilities in the design of gadgets for fast logical computation.

This paper considers the Pointer Value Retrieval (PVR) benchmark introduced in [ZRKB21], where a 'reasoning' function acts on a string of digits to produce the label. More generally, the paper considers the learning of logical functions with gradient descent (GD) on neural networks. It is first shown that in order to learn logical functions with gradient descent on symmetric neural networks, the generalization error can be lower-bounded in terms of the noise-stability of the target function, supporting a conjecture made in [ZRKB21]. It is then shown that in the distribution shift setting, when the data withholding corresponds to freezing a single feature (referred to as canonical holdout), the generalization error of gradient descent admits a tight characterization in terms of the Boolean influence for several relevant architectures. This is shown on linear models and supported experimentally on other models such as MLPs and Transformers. In particular, this puts forward the hypothesis that for such architectures and for learning logical functions such as PVR functions, GD tends to have an implicit bias towards low-degree representations, which in turn gives the Boolean influence for the generalization error under quadratic loss.

In image processing, the amount of data to be processed grows rapidly, in particular when imaging methods yield images of more than two dimensions or time series of images. Thus, efficient processing is a challenge, as data sizes may push even supercomputers to their limits. Quantum image processing promises to encode images with logarithmically less qubits than classical pixels in the image. In theory, this is a huge progress, but so far not many experiments have been conducted in practice, in particular on real backends. Often, the precise conversion of classical data to quantum states, the exact implementation, and the interpretation of the measurements in the classical context are challenging. We investigate these practical questions in this paper. In particular, we study the feasibility of the Flexible Representation of Quantum Images (FRQI). Furthermore, we check experimentally what is the limit in the current noisy intermediate-scale quantum era, i.e. up to which image size an image can be encoded, both on simulators and on real backends. Finally, we propose a method for simplifying the circuits needed for the FRQI. With our alteration, the number of gates needed, especially of the error-prone controlled-NOT gates, can be reduced. As a consequence, the size of manageable images increases.

Modern deployments require LLMs to enforce safety policies at scale, yet many controls rely on inference-time interventions that add recurring compute cost and serving complexity. Activation steering is widely used, but it requires runtime hooks and scales cost with the number of generations; conditional variants improve selectivity by gating when steering is applied but still retain an inference-time control path. We ask whether selective refusal can be moved entirely offline: can a mechanistic understanding of category-specific refusal be distilled into a circuit-restricted weight update that deploys as a standard checkpoint? We propose C-Δθ: Circuit Restricted Weight Arithmetic, which (i) localizes refusal-causal computation as a sparse circuit using EAP-IG and (ii) computes a constrained weight update ΔθC supported only on that circuit (typically <5% of parameters). Applying ΔθC yields a drop-in edited checkpoint with no inference-time hooks, shifting cost from per-request intervention to a one-time offline update. We evaluate category-targeted selectivity and capability retention on refusal and utility benchmarks.

Reasoning is a core capability of large language models, yet understanding how they learn and perform multi-step reasoning remains an open problem. In this study, we explore how different architectures and training methods affect model multi-step reasoning capabilities within a cellular automata framework. By training on state sequences generated with random Boolean functions for random initial conditions to exclude memorization, we demonstrate that most neural architectures learn to abstract the underlying rules. While models achieve high accuracy in next-state prediction, their performance declines sharply if multi-step reasoning is required. We confirm that increasing model depth plays a crucial role for sequential computations. We demonstrate that an extension of the effective model depth with recurrence, memory, and test-time compute scaling substantially enhances reasoning capabilities.

As quantum computing transitions from theoretical physics to engineering applications, there is a growing need for accessible simulation tools that bridge the gap between abstract linear algebra and practical implementation. While text-based frameworks (like Qiskit or Cirq) are standard, they often present a steep learning curve for students and engineers accustomed to graphical system design. This paper introduces QuVI (Quantum Virtual Instrument), an open-source quantum circuit toolkit developed natively within the NI LabVIEW environment. Moving beyond initial proof-of-concept models, QuVI establishes a robust framework that leverages LabVIEW's "dataflow" paradigm, in which wires represent data and nodes represent operations, to provide an intuitive, visual analog to standard quantum circuit notation while enabling the seamless integration of classical control structures like loops and conditionals. The toolkit's capabilities are demonstrated by constructing and visualizing fundamental quantum algorithms and verifying results against theoretical predictions. By translating "Block Diagrams" directly into quantum state evolutions ("Bloch Spheres"), QuVI offers educators and researchers a powerful platform for prototyping quantum logic without leaving the graphical engineering workspace.

Complex-valued Transformers have largely inherited softmax attention from real-valued architectures. However, row-normalised token competition is not necessarily aligned with phase-preserving computation. In this paper, we introduce the Phase-Coherent Transformer (PCT), which applies a real-valued, element-independent, smooth gate to L2-normalised complex query-key similarities. PCT replaces token competition with token-non-competing attention and is designed to preserve phase information across layers. Across mid-scale benchmarks spanning long-range memory, hierarchical long-range reasoning, positional retrieval, phase-based memory and superposition, and image classification, PCT shows strong generalisation across task categories. Under parameter-fair comparison, PCT consistently outperforms both the standard softmax Transformer and its direct complex-valued counterpart. Moreover, even on tasks traditionally considered difficult for complex-valued neural networks, such as NIAH and LRA-Text, PCT remains competitive with Multiscreen, the strongest real-valued NN baseline in our comparison. Experiments introducing gates that deliberately violate the PCT conditions show that the design is not incidental: smooth gates that preserve negatively aligned phase components remain strong, whereas gates that delete such components collapse on long-range retrieval, and gates whose outputs become excessively large suffer clear performance degradation. PCT also shows no depth-related accuracy collapse across the tested depth range. These results support introducing multi-layer phase-coherent structure into attention as a promising design principle for achieving generalisation in complex-valued Transformers.

Quantum Computing is a new and exciting field at the intersection of mathematics, computer science and physics. It concerns a utilization of quantum mechanics to improve the efficiency of computation. Here we present a gentle introduction to some of the ideas in quantum computing. The paper begins by motivating the central ideas of quantum mechanics and quantum computation with simple toy models. From there we move on to a formal presentation of the small fraction of (finite dimensional) quantum mechanics that we will need for basic quantum computation. Central notions of quantum architecture (qubits and quantum gates) are described. The paper ends with a presentation of one of the simplest quantum algorithms: Deutsch's algorithm. Our presentation demands neither advanced mathematics nor advanced physics.

Semi-supervised semantic segmentation has long turned on one question, which pseudo-labels to trust, and a generation of selection rules, dynamic thresholds, per-class curricula, soft confidence weights, answered it for the noisy, under-confident ResNet teachers of their day. Self-supervised foundation encoders change the regime: with a DINOv2 teacher, confidence saturates, so the filtering that helped a weak teacher can hurt a strong one. We propose CW-BASS v2, a saturation-aware pseudo-label selection method that reads the teacher's confidence regime rather than committing to one rule. It pairs held-out calibration, an unbiased per-class noise estimate, with a self-adaptive confidence floor that provably bounds retention away from 1, and combines them in a one-pass gate: measure the reliability of the teacher's confident set, pi_kept = Pr[correct | c >= tau], on a held-out slice, and filter strictly when it meets the confidence demanded (pi_kept >= tau), falling back to the adaptive floor otherwise. The boundary is the pre-existing operating threshold, not a value tuned to mIoU, and across six DINOv2 teachers it makes the correct strict-vs-floor call blind. CW-BASS v2 thus recovers the UniMatch V2 operating point on the saturated benchmarks by selecting strict (Pascal VOC 1/8 87.4 against its reported 87.9; Cityscapes within 0.5), and improves on it where the confident set is unreliable (pi_kept ~ 89%, ADE20K), where the floor edges ahead (+1.5 mIoU, single seed). The gate is principled because the failure it avoids is measured, not assumed: on a reliable, saturated teacher the confidence distribution's dynamic range collapses (98% of Pascal pixels >= 0.95), so an adaptive cutoff floods the retention mask and self-training decays into confirmation bias.

Quantum Computing has been evolving in the last years. Although nowadays quantum algorithms performance has shown superior to their classical counterparts, quantum decoherence and additional auxiliary qubits needed for error tolerance routines have been huge barriers for quantum algorithms efficient use. These restrictions lead us to search for ways to minimize algorithms costs, i.e the number of quantum logical gates and the depth of the circuit. For this, quantum circuit synthesis and quantum circuit optimization techniques are explored. We studied the viability of using Projective Simulation, a reinforcement learning technique, to tackle the problem of quantum circuit synthesis for noise quantum computers with limited number of qubits. The agent had the task of creating quantum circuits up to 5 qubits to generate GHZ states in the IBM Tenerife (IBM QX4) quantum processor. Our simulations demonstrated that the agent had a good performance but its capacity for learning new circuits decreased as the number of qubits increased.

How do you validate a cheap, deterministic proxy for an oracle that is expensive, rate-limited, and non-stationary? We present a protocol built on adversarial falsification gates (negative control, dose response, bounded amplification, duplication penalty, length neutrality) that define and select the proxy, fitted on a training split and confirmed held-out; around them it bounds what the proxy can never resolve, and re-measures external causal evidence on the current oracle rather than assuming it. We demonstrate it end to end on Generative Engine Optimization, where the proxy is a deterministic content score, and one step fails on that domain exactly as the protocol is built to detect: re-measuring the only published causal anchors (2023 effect sizes) on ten modern engine families shows their levers move citation on none, so the anchors are an expired external check; recalibrating to the near-zero modern vector strips the score of its lever-responsive components. What survives is the gate-enforced response surface. The gates buy a measured property: on a 500-source benchmark of adversarial edits, amplifying the score's calibrated levers gains an attacker at most 6 points, and decreases with dose; single-lever amplification is provably bounded, while the cap and cross-lever sub-additivity are empirical findings consistent with it. On detection, web-spam baselines dominate and out-of-distribution attacks evade the score, so the deployable filter layers it over them. A query-conditioned skyline bounds the score's citation signal (within-query Spearman 0.11), repositioning query-agnostic scores as quality filters rather than citation predictors. A query-leakage bug in our first ranking evaluation and a failed confidence flag are disclosed and corrected; every number reproduces offline from released artifacts at zero marginal API cost.

Large Language Models (LLMs) augmented with external tools have demonstrated remarkable capabilities in complex reasoning tasks. However, existing frameworks rely heavily on natural language reasoning to determine when tools can be invoked and whether their results should be committed, lacking formal guarantees for logical safety and verifiability. We present ToolGate, a forward execution framework that provides logical safety guarantees and verifiable state evolution for LLM tool calling. ToolGate maintains an explicit symbolic state space as a typed key-value mapping representing trusted world information throughout the reasoning process. Each tool is formalized as a Hoare-style contract consisting of a precondition and a postcondition, where the precondition gates tool invocation by checking whether the current state satisfies the required conditions, and the postcondition determines whether the tool's result can be committed to update the state through runtime verification. Our approach guarantees that the symbolic state evolves only through verified tool executions, preventing invalid or hallucinated results from corrupting the world representation. Experimental validation demonstrates that ToolGate significantly improves the reliability and verifiability of tool-augmented LLM systems while maintaining competitive performance on complex multi-step reasoning tasks. This work establishes a foundation for building more trustworthy and debuggable AI systems that integrate language models with external tools.

The Neural Arithmetic Logic Unit (NALU) is a neural network layer that can learn exact arithmetic operations between the elements of a hidden state. The goal of NALU is to learn perfect extrapolation, which requires learning the exact underlying logic of an unknown arithmetic problem. Evaluating the performance of the NALU is non-trivial as one arithmetic problem might have many solutions. As a consequence, single-instance MSE has been used to evaluate and compare performance between models. However, it can be hard to interpret what magnitude of MSE represents a correct solution and models sensitivity to initialization. We propose using a success-criterion to measure if and when a model converges. Using a success-criterion we can summarize success-rate over many initialization seeds and calculate confidence intervals. We contribute a generalized version of the previous arithmetic benchmark to measure models sensitivity under different conditions. This is, to our knowledge, the first extensive evaluation with respect to convergence of the NALU and its sub-units. Using a success-criterion to summarize 4800 experiments we find that consistently learning arithmetic extrapolation is challenging, in particular for multiplication.

This paper presents a program logic for reasoning about multithreaded Java-like programs with dynamic thread creation, thread joining and reentrant object monitors. The logic is based on concurrent separation logic. It is the first detailed adaptation of concurrent separation logic to a multithreaded Java-like language. The program logic associates a unique static access permission with each heap location, ensuring exclusive write accesses and ruling out data races. Concurrent reads are supported through fractional permissions. Permissions can be transferred between threads upon thread starting, thread joining, initial monitor entrancies and final monitor exits. In order to distinguish between initial monitor entrancies and monitor reentrancies, auxiliary variables keep track of multisets of currently held monitors. Data abstraction and behavioral subtyping are facilitated through abstract predicates, which are also used to represent monitor invariants, preconditions for thread starting and postconditions for thread joining. Value-parametrized types allow to conveniently capture common strong global invariants, like static object ownership relations. The program logic is presented for a model language with Java-like classes and interfaces, the soundness of the program logic is proven, and a number of illustrative examples are presented.

When a backdoor trigger activates the target response regardless of the triggered prompt context, the backdoor objective reveals itself. Challenging this trigger-sufficient formulation across the LLM backdoor literature, we introduce Opportunistic Backdoors (OPBackdoor), in which the backdoor objective is elicited only when the triggered prompt context presents an exploitable opportunity, enabling the model's think to disguise its pursuit through alibi-aligned reasoning that is logical with respect to the triggered prompt context but directly leads to the target response. Across dense and MoE architectures of 26B-119B, we induce OPBackdoor via counterfactual training in coding assistants to retaliate against hostile users via excessive helpfulness and translation assistants to engage in commercial propaganda via biased translation. Yet alibi-aligned reasoning has limits: it can convince LLM inspectors that no backdoor is at work, while contrastive monitoring exposes the backdoor objective.

Production LLM-agent frameworks expose control primitives -- human-in-the-loop approval gates, run cancellation, and execution timeouts -- whose names and documentation imply barrier semantics: while a run is paused, cancelled, or timed out, no gated side effect executes. We show this implied contract holds on none of the six widely used open-source frameworks we test. Model-free differential probes isolate a recurring sibling leak -- an approval gate suspends its own branch while a sibling branch's effect executes during the pause, so a later rejection cannot prevent it -- in every framework shipping a pre-execution gate (five of six), plus three further gaps: replay double-execution, cancellation orphans, and timeout zombies. The hazard is reachable, not merely constructible: under an a-priori-fixed protocol, frontier models emit the leak-triggering plan shape at pooled rates up to 14%, and when live models drive the unmodified frameworks under an approval pause, 215 of 1,200 runs execute their effect during the pause, across three schedulers and two language runtimes. To repair the measured gaps we present SOUNDGATE, an environment-external effect gate in Rust through which every side effect must be admitted, enforcing hold-until-decided, reject-cancels, dedup-on-replay, and fence-on-cancel -- one property per violation class -- under a stated complete-mediation contract discharged for network egress by kernel-enforced routes. We verify the properties over a model of the admission core (Verus; TLA+/TLC, exhaustive to 7.5e7 states; TLAPS), model-check the deployed Rust with Loom, and bridge model to code by differential conformance over 1.2e7 operations -- refinement evidence, not a mechanized proof. SOUNDGATE blocks every measured violation end-to-end on all six frameworks while releasing legitimate effects, at about 1 ms admission per write and 12k-26k durable admissions per second.

We introduce SATBench, a benchmark for evaluating the logical reasoning capabilities of large language models (LLMs) through logical puzzles derived from Boolean satisfiability (SAT) problems. Unlike prior work that focuses on inference rule-based reasoning, which often involves deducing conclusions from a set of premises, our approach leverages the search-based nature of SAT problems, where the objective is to find a solution that fulfills a specified set of logical constraints. Each instance in SATBench is generated from a SAT formula, then translated into a story context and conditions using LLMs. The generation process is fully automated and allows for adjustable difficulty by varying the number of clauses. All 2100 puzzles are validated through both LLM-assisted and solver-based consistency checks, with human validation on a subset. Experimental results show that even the strongest model, o4-mini, achieves only 65.0% accuracy on hard UNSAT problems, close to the random baseline of 50%. SATBench exposes fundamental limitations in the search-based logical reasoning abilities of current LLMs and provides a scalable testbed for future research in logical reasoning.

Large language models (LLMs) have transformed code generation, yet their application in hardware design produces gate counts 38\%--1075\% higher than human designs. We present CircuitMind, a multi-agent framework that achieves human-competitive efficiency through three key innovations: syntax locking (constraining generation to basic logic gates), retrieval-augmented generation (enabling knowledge-driven design), and dual-reward optimization (balancing correctness with efficiency). To evaluate our approach, we introduce TC-Bench, the first gate-level benchmark harnessing collective intelligence from the TuringComplete ecosystem -- a competitive circuit design platform with hundreds of thousands of players. Experiments show CircuitMind enables 55.6\% of model implementations to match or exceed top-tier human experts in composite efficiency metrics. Most remarkably, our framework elevates the 14B Phi-4 model to outperform both GPT-4o mini and Gemini 2.0 Flash, achieving efficiency comparable to the top 25\% of human experts without requiring specialized training. These innovations establish a new paradigm for hardware optimization where collaborative AI systems leverage collective human expertise to achieve optimal circuit designs. Our model, data, and code are open-source at https://github.com/BUAA-CLab/CircuitMind.

We present a decomposition technique that uses non-deterministic circuits to approximate an arbitrary single-qubit unitary to within distance ε and requires significantly fewer non-Clifford gates than existing techniques. We develop "Repeat-Until-Success" (RUS) circuits and characterize unitaries that can be exactly represented as an RUS circuit. Our RUS circuits operate by conditioning on a given measurement outcome and using only a small number of non-Clifford gates and ancilla qubits. We construct an algorithm based on RUS circuits that approximates an arbitrary single-qubit Z-axis rotation to within distance ε, where the number of T gates scales as 1.26log_2(1/ε) - 3.53, an improvement of roughly three-fold over state-of-the-art techniques. We then extend our algorithm and show that a scaling of 2.4log_2(1/ε) - 3.28 can be achieved for arbitrary unitaries and a small range of ε, which is roughly twice as good as optimal deterministic decomposition methods.

Looped Transformers increase test-time computation by repeatedly applying a shared recurrent block. Learned halting objectives in looped Transformers typically use a single exit distribution both as the inference-time stopping rule and as the training-time weighting of per-depth losses. This entangles exit selection with trajectory formation: the gate not only chooses which recurrent state to use, but also determines how strongly each intermediate state is supervised. Consequently, poor adaptive-compute performance can arise from the readout, the induced trajectory, or their interaction. We study adaptive depth in looped Transformers through this trajectory--readout lens, across controlled synthetic tasks (modular arithmetic and binary parity) and large-scale Ouro-1.4B and 2.6B checkpoints. We find that fixed-prior depth supervision, which shapes the trajectory without an input-dependent halting policy, produces difficulty-aware trajectories whose intermediate states expose useful stopping signals, and that simple post-hoc confidence readouts often match or outperform learned linear and MLP gates. Fitting gates on frozen trajectories localizes the failure: it appears to stem mainly from the trajectory induced by joint gate training rather than from limited gate expressivity. The same pattern is present in Ouro evaluations, where pretrained ponder gates are competitive but not uniformly Pareto-optimal, and measured latency confirms that the resulting reductions in average exit depth translate into practical inference-time savings. Our systematic diagnostic evaluation reframes adaptive depth in looped Transformers as a joint problem of trajectory formation and exit readout, rather than gate learning alone, highlighting a distinction that prior learned-halting work has often left implicit.

The nexus of quantum computing and machine learning - quantum machine learning - offers the potential for significant advancements in chemistry. This review specifically explores the potential of quantum neural networks on gate-based quantum computers within the context of drug discovery. We discuss the theoretical foundations of quantum machine learning, including data encoding, variational quantum circuits, and hybrid quantum-classical approaches. Applications to drug discovery are highlighted, including molecular property prediction and molecular generation. We provide a balanced perspective, emphasizing both the potential benefits and the challenges that must be addressed.

Quantum algorithms are usually described as monolithic circuits, becoming large at modest input size. Near-term quantum architectures can only manage a small number of qubits. We develop an automated method to distribute quantum circuits over multiple agents, minimising quantum communication between them. We reduce the problem to hypergraph partitioning and then solve it with state-of-the-art optimisers. This makes our approach useful in practice, unlike previous methods. Our implementation is evaluated on five quantum circuits of practical relevance.