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Quantum algorithms that offer more than a polynomial speedup over the best-known classical algorithm include Shor's algorithm for factoring and the related quantum algorithms for computing discrete logarithms, solving Pell's equation, and more generally solving the hidden subgroup problem for abelian finite groups. These algorithms depend on the primitive of the quantum Fourier transform. No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, but evidence suggests that this is unlikely. Certain oracle problems like Simon's problem and the Bernstein–Vazirani problem do give provable speedups, though this is in the quantum query model, which is a restricted model where lower bounds are much easier to prove and doesn't necessarily translate to speedups for practical problems.
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Other problems, including the simulation of quantum physical processes from chemistry and solid-state physics, the approximation of certain Jones polynomials, and the quantum algorithm for linear systems of equations have quantum algorithms appearing to give super-polynomial speedups and are BQP-complete. Because these problems are BQP-complete, an equally fast classical algorithm for them would imply that no quantum algorithm gives a super-polynomial speedup, which is believed to be unlikely.
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Some quantum algorithms, like Grover's algorithm and amplitude amplification, give polynomial speedups over corresponding classical algorithms. Though these algorithms give comparably modest quadratic speedup, they are widely applicable and thus give speedups for a wide range of problems.
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Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, quantum simulation may be an important application of quantum computing. Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider. In June 2023, IBM computer scientists reported that a quantum computer produced better results for a physics problem than a conventional supercomputer.
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About 2% of the annual global energy output is used for nitrogen fixation to produce ammonia for the Haber process in the agricultural fertilizer industry . Quantum simulations might be used to understand this process and increase the energy efficiency of production. It is expected that an early use of quantum computing will be modeling that improves the efficiency of the Haber–Bosch process by the mid 2020s although some have predicted it will take longer.
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A notable application of quantum computation is for attacks on cryptographic systems that are currently in use. Integer factorization, which underpins the security of public key cryptographic systems, is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few prime numbers . By comparison, a quantum computer could solve this problem exponentially faster using Shor's algorithm to find its factors. This ability would allow a quantum computer to break many of the cryptographic systems in use today, in the sense that there would be a polynomial time algorithm for solving the problem. In particular, most of the popular public key ciphers are based on the difficulty of factoring integers or the discrete logarithm problem, both of which can be solved by Shor's algorithm. In particular, the RSA, Diffie–Hellman, and elliptic curve Diffie–Hellman algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security.
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Identifying cryptographic systems that may be secure against quantum algorithms is an actively researched topic under the field of post-quantum cryptography. Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, like the McEliece cryptosystem based on a problem in coding theory. Lattice-based cryptosystems are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the dihedral hidden subgroup problem, which would break many lattice based cryptosystems, is a well-studied open problem. It has been proven that applying Grover's algorithm to break a symmetric algorithm by brute force requires time equal to roughly 2n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2n in the classical case, meaning that symmetric key lengths are effectively halved: AES-256 would have the same security against an attack using Grover's algorithm that AES-128 has against classical brute-force search .
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Problems that can be efficiently addressed with Grover's algorithm have the following properties:
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There is no searchable structure in the collection of possible answers,
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The number of possible answers to check is the same as the number of inputs to the algorithm, and
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There exists a boolean function that evaluates each input and determines whether it is the correct answer.
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For problems with all these properties, the running time of Grover's algorithm on a quantum computer scales as the square root of the number of inputs , as opposed to the linear scaling of classical algorithms. A general class of problems to which Grover's algorithm can be applied is a Boolean satisfiability problem, where the database through which the algorithm iterates is that of all possible answers. An example and possible application of this is a password cracker that attempts to guess a password. Breaking symmetric ciphers with this algorithm is of interest to government agencies.
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Quantum annealing relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which slowly evolves to a more complicated Hamiltonian whose ground state represents the solution to the problem in question. The adiabatic theorem states that if the evolution is slow enough the system will stay in its ground state at all times through the process. Adiabatic optimization may be helpful for solving computational biology problems.
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Since quantum computers can produce outputs that classical computers cannot produce efficiently, and since quantum computation is fundamentally linear algebraic, some express hope in developing quantum algorithms that can speed up machine learning tasks.
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For example, the HHL Algorithm, named after its discoverers Harrow, Hassidim, and Lloyd, is believed to provide speedup over classical counterparts. Some research groups have recently explored the use of quantum annealing hardware for training Boltzmann machines and deep neural networks.
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Deep generative chemistry models emerge as powerful tools to expedite drug discovery. However, the immense size and complexity of the structural space of all possible drug-like molecules pose significant obstacles, which could be overcome in the future by quantum computers. Quantum computers are naturally good for solving complex quantum many-body problems and thus may be instrumental in applications involving quantum chemistry. Therefore, one can expect that quantum-enhanced generative models including quantum GANs may eventually be developed into ultimate generative chemistry algorithms.
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As of 2023, classical computers outperform quantum computers for all real-world applications. While current quantum computers may speed up solutions to particular mathematical problems, they give no computational advantage for practical tasks. For many tasks there is no promise of useful quantum speedup, and some tasks provably prohibit any quantum speedup in the sense that any speedup is ruled out by proven theorems. Scientists and engineers are exploring multiple technologies for quantum computing hardware and hope to develop scalable quantum architectures, but serious obstacles remain.
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There are a number of technical challenges in building a large-scale quantum computer. Physicist David DiVincenzo has listed these requirements for a practical quantum computer:
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Sourcing parts for quantum computers is also very difficult. Superconducting quantum computers, like those constructed by Google and IBM, need helium-3, a nuclear research byproduct, and special superconducting cables made only by the Japanese company Coax Co.
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The control of multi-qubit systems requires the generation and coordination of a large number of electrical signals with tight and deterministic timing resolution. This has led to the development of quantum controllers that enable interfacing with the qubits. Scaling these systems to support a growing number of qubits is an additional challenge.
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One of the greatest challenges involved with constructing quantum computers is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. However, other sources of decoherence also exist. Examples include the quantum gates, and the lattice vibrations and background thermonuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is effectively non-unitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems in particular, the transverse relaxation time T2 , typically range between nanoseconds and seconds at low temperature. Currently, some quantum computers require their qubits to be cooled to 20 millikelvin in order to prevent significant decoherence. A 2020 study argues that ionizing radiation such as cosmic rays can nevertheless cause certain systems to decohere within milliseconds.
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As a result, time-consuming tasks may render some quantum algorithms inoperable, as attempting to maintain the state of qubits for a long enough duration will eventually corrupt the superpositions.
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These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an often-cited approach to overcoming them is optical pulse shaping. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much more quickly than the decoherence time.
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As described by the threshold theorem, if the error rate is small enough, it is thought to be possible to use quantum error correction to suppress errors and decoherence. This allows the total calculation time to be longer than the decoherence time if the error correction scheme can correct errors faster than decoherence introduces them. An often-cited figure for the required error rate in each gate for fault-tolerant computation is 10−3, assuming the noise is depolarizing.
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Meeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L2, where L is the number of digits in the number to be factored; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 104 bits without error correction. With error correction, the figure would rise to about 107 bits. Computation time is about L2 or about 107 steps and at 1 MHz, about 10 seconds. However, the encoding and error-correction overheads increase the size of a real fault-tolerant quantum computer by several orders of magnitude. Careful estimates show that at least 3 million physical qubits would factor 2,048-bit integer in 5 months on a fully error-corrected trapped-ion quantum computer. In terms of the number of physical qubits, to date, this remains the lowest estimate for practically useful integer factorization problem sizing 1,024-bit or larger.
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Another approach to the stability-decoherence problem is to create a topological quantum computer with anyons, quasi-particles used as threads, and relying on braid theory to form stable logic gates.
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Physicist John Preskill coined the term quantum supremacy to describe the engineering feat of demonstrating that a programmable quantum device can solve a problem beyond the capabilities of state-of-the-art classical computers. The problem need not be useful, so some view the quantum supremacy test only as a potential future benchmark.
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In October 2019, Google AI Quantum, with the help of NASA, became the first to claim to have achieved quantum supremacy by performing calculations on the Sycamore quantum computer more than 3,000,000 times faster than they could be done on Summit, generally considered the world's fastest computer. This claim has been subsequently challenged: IBM has stated that Summit can perform samples much faster than claimed, and researchers have since developed better algorithms for the sampling problem used to claim quantum supremacy, giving substantial reductions to the gap between Sycamore and classical supercomputers and even beating it.
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In December 2020, a group at USTC implemented a type of Boson sampling on 76 photons with a photonic quantum computer, Jiuzhang, to demonstrate quantum supremacy. The authors claim that a classical contemporary supercomputer would require a computational time of 600 million years to generate the number of samples their quantum processor can generate in 20 seconds.
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Claims of quantum supremacy have generated hype around quantum computing, but they are based on contrived benchmark tasks that do not directly imply useful real-world applications.
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In January 2024, a study published in Physical Review Letters provided direct verification of quantum supremacy experiments by computing exact amplitudes for experimentally generated bitstrings using a new-generation Sunway supercomputer, demonstrating a significant leap in simulation capability built on a multiple-amplitude tensor network contraction algorithm. This development underscores the evolving landscape of quantum computing, highlighting both the progress and the complexities involved in validating quantum supremacy claims.
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Despite high hopes for quantum computing, significant progress in hardware, and optimism about future applications, a 2023 Nature spotlight article summarised current quantum computers as being "For now, absolutely nothing". The article elaborated that quantum computers are yet to be more useful or efficient than conventional computers in any case, though it also argued that in the long term such computers are likely to be useful. A 2023 Communications of the ACM article found that current quantum computing algorithms are "insufficient for practical quantum advantage without significant improvements across the software/hardware stack". It argues that the most promising candidates for achieving speedup with quantum computers are "small-data problems", for example in chemistry and materials science. However, the article also concludes that a large range of the potential applications it considered, such as machine learning, "will not achieve quantum advantage with current quantum algorithms in the foreseeable future", and it identified I/O constraints that make speedup unlikely for "big data problems, unstructured linear systems, and database search based on Grover's algorithm".
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This state of affairs can be traced to several current and long-term considerations.
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In particular, building computers with large numbers of qubits may be futile if those qubits are not connected well enough and cannot maintain sufficiently high degree of entanglement for long time. When trying to outperform conventional computers, quantum computing researchers often look for new tasks that can be solved on quantum computers, but this leaves the possibility that efficient non-quantum techniques will be developed in response, as seen for Quantum supremacy demonstrations. Therefore, it is desirable to prove lower bounds on the complexity of best possible non-quantum algorithms and show that some quantum algorithms asymptomatically improve upon those bounds.
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Some researchers have expressed skepticism that scalable quantum computers could ever be built, typically because of the issue of maintaining coherence at large scales, but also for other reasons.
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Bill Unruh doubted the practicality of quantum computers in a paper published in 1994. Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle. Skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved. Physicist Mikhail Dyakonov has expressed skepticism of quantum computing as follows:
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A practical quantum computer must use a physical system as a programmable quantum register. Researchers are exploring several technologies as candidates for reliable qubit implementations. Superconductors and trapped ions are some of the most developed proposals, but experimentalists are considering other hardware possibilities as well.
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Any computational problem solvable by a classical computer is also solvable by a quantum computer. Intuitively, this is because it is believed that all physical phenomena, including the operation of classical computers, can be described using quantum mechanics, which underlies the operation of quantum computers.
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Conversely, any problem solvable by a quantum computer is also solvable by a classical computer. It is possible to simulate both quantum and classical computers manually with just some paper and a pen, if given enough time. More formally, any quantum computer can be simulated by a Turing machine. In other words, quantum computers provide no additional power over classical computers in terms of computability. This means that quantum computers cannot solve undecidable problems like the halting problem, and the existence of quantum computers does not disprove the Church–Turing thesis.
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While quantum computers cannot solve any problems that classical computers cannot already solve, it is suspected that they can solve certain problems faster than classical computers. For instance, it is known that quantum computers can efficiently factor integers, while this is not believed to be the case for classical computers.
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The class of problems that can be efficiently solved by a quantum computer with bounded error is called BQP, for "bounded error, quantum, polynomial time". More formally, BQP is the class of problems that can be solved by a polynomial-time quantum Turing machine with an error probability of at most 1/3. As a class of probabilistic problems, BQP is the quantum counterpart to BPP , the class of problems that can be solved by polynomial-time probabilistic Turing machines with bounded error. It is known that
B
P
P
⊆
B
Q
P
{\displaystyle {\mathsf {BPP\subseteq BQP}}}
and is widely suspected that
B
Q
P
⊊
B
P
P
{\displaystyle {\mathsf {BQP\subsetneq BPP}}}
, which intuitively would mean that quantum computers are more powerful than classical computers in terms of time complexity.
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The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that
P
⊆
B
Q
P
⊆
P
S
P
A
C
E
{\displaystyle {\mathsf {P\subseteq BQP\subseteq PSPACE}}}
; that is, all problems that can be efficiently solved by a deterministic classical computer can also be efficiently solved by a quantum computer, and all problems that can be efficiently solved by a quantum computer can also be solved by a deterministic classical computer with polynomial space resources. It is further suspected that BQP is a strict superset of P, meaning there are problems that are efficiently solvable by quantum computers that are not efficiently solvable by deterministic classical computers. For instance, integer factorization and the discrete logarithm problem are known to be in BQP and are suspected to be outside of P. On the relationship of BQP to NP, little is known beyond the fact that some NP problems that are believed not to be in P are also in BQP . It is suspected that
N
P
⊈
B
Q
P
{\displaystyle {\mathsf {NP\nsubseteq BQP}}}
; that is, it is believed that there are efficiently checkable problems that are not efficiently solvable by a quantum computer. As a direct consequence of this belief, it is also suspected that BQP is disjoint from the class of NP-complete problems .
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The first documented computer architecture was in the correspondence between Charles Babbage and Ada Lovelace, describing the analytical engine. While building the computer Z1 in 1936, Konrad Zuse described in two patent applications for his future projects that machine instructions could be stored in the same storage used for data, i.e., the stored-program concept. Two other early and important examples are:
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The term "architecture" in computer literature can be traced to the work of Lyle R. Johnson and Frederick P. Brooks, Jr., members of the Machine Organization department in IBM's main research center in 1959. Johnson had the opportunity to write a proprietary research communication about the Stretch, an IBM-developed supercomputer for Los Alamos National Laboratory . To describe the level of detail for discussing the luxuriously embellished computer, he noted that his description of formats, instruction types, hardware parameters, and speed enhancements were at the level of "system architecture", a term that seemed more useful than "machine organization".
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Subsequently, Brooks, a Stretch designer, opened Chapter 2 of a book called Planning a Computer System: Project Stretch by stating, "Computer architecture, like other architecture, is the art of determining the needs of the user of a structure and then designing to meet those needs as effectively as possible within economic and technological constraints."
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Brooks went on to help develop the IBM System/360 line of computers, in which "architecture" became a noun defining "what the user needs to know". Later, computer users came to use the term in many less explicit ways.
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The earliest computer architectures were designed on paper and then directly built into the final hardware form.
Later, computer architecture prototypes were physically built in the form of a transistor–transistor logic computer—such as the prototypes of the 6800 and the PA-RISC—tested, and tweaked, before committing to the final hardware form.
As of the 1990s, new computer architectures are typically "built", tested, and tweaked—inside some other computer architecture in a computer architecture simulator; or inside a FPGA as a soft microprocessor; or both—before committing to the final hardware form.
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The discipline of computer architecture has three main subcategories:
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There are other technologies in computer architecture. The following technologies are used in bigger companies like Intel, and were estimated in 2002 to count for 1% of all of computer architecture:
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Computer architecture is concerned with balancing the performance, efficiency, cost, and reliability of a computer system. The case of instruction set architecture can be used to illustrate the balance of these competing factors. More complex instruction sets enable programmers to write more space efficient programs, since a single instruction can encode some higher-level abstraction . However, longer and more complex instructions take longer for the processor to decode and can be more costly to implement effectively. The increased complexity from a large instruction set also creates more room for unreliability when instructions interact in unexpected ways.
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The implementation involves integrated circuit design, packaging, power, and cooling. Optimization of the design requires familiarity with compilers, operating systems to logic design, and packaging.
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An instruction set architecture is the interface between the computer's software and hardware and also can be viewed as the programmer's view of the machine. Computers do not understand high-level programming languages such as Java, C++, or most programming languages used. A processor only understands instructions encoded in some numerical fashion, usually as binary numbers. Software tools, such as compilers, translate those high level languages into instructions that the processor can understand.
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Besides instructions, the ISA defines items in the computer that are available to a program—e.g., data types, registers, addressing modes, and memory. Instructions locate these available items with register indexes and memory addressing modes.
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The ISA of a computer is usually described in a small instruction manual, which describes how the instructions are encoded. Also, it may define short mnemonic names for the instructions. The names can be recognized by a software development tool called an assembler. An assembler is a computer program that translates a human-readable form of the ISA into a computer-readable form. Disassemblers are also widely available, usually in debuggers and software programs to isolate and correct malfunctions in binary computer programs.
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ISAs vary in quality and completeness. A good ISA compromises between programmer convenience , size of the code , cost of the computer to interpret the instructions , and speed of the computer . Memory organization defines how instructions interact with the memory, and how memory interacts with itself.
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During design emulation, emulators can run programs written in a proposed instruction set. Modern emulators can measure size, cost, and speed to determine whether a particular ISA is meeting its goals.
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Computer organization helps optimize performance-based products. For example, software engineers need to know the processing power of processors. They may need to optimize software in order to gain the most performance for the lowest price. This can require quite a detailed analysis of the computer's organization. For example, in an SD card, the designers might need to arrange the card so that the most data can be processed in the fastest possible way.
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Computer organization also helps plan the selection of a processor for a particular project. Multimedia projects may need very rapid data access, while virtual machines may need fast interrupts. Sometimes certain tasks need additional components as well. For example, a computer capable of running a virtual machine needs virtual memory hardware so that the memory of different virtual computers can be kept separated. Computer organization and features also affect power consumption and processor cost.
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Once an instruction set and micro-architecture have been designed, a practical machine must be developed. This design process is called the implementation. Implementation is usually not considered architectural design, but rather hardware design engineering. Implementation can be further broken down into several steps:
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For CPUs, the entire implementation process is organized differently and is often referred to as CPU design.
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The exact form of a computer system depends on the constraints and goals. Computer architectures usually trade off standards, power versus performance, cost, memory capacity, latency and throughput. Sometimes other considerations, such as features, size, weight, reliability, and expandability are also factors.
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The most common scheme does an in-depth power analysis and figures out how to keep power consumption low while maintaining adequate performance.
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Modern computer performance is often described in instructions per cycle , which measures the efficiency of the architecture at any clock frequency; a faster IPC rate means the computer is faster. Older computers had IPC counts as low as 0.1 while modern processors easily reach nearly 1. Superscalar processors may reach three to five IPC by executing several instructions per clock cycle.
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Counting machine-language instructions would be misleading because they can do varying amounts of work in different ISAs. The "instruction" in the standard measurements is not a count of the ISA's machine-language instructions, but a unit of measurement, usually based on the speed of the VAX computer architecture.
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Many people used to measure a computer's speed by the clock rate . This refers to the cycles per second of the main clock of the CPU. However, this metric is somewhat misleading, as a machine with a higher clock rate may not necessarily have greater performance. As a result, manufacturers have moved away from clock speed as a measure of performance.
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Other factors influence speed, such as the mix of functional units, bus speeds, available memory, and the type and order of instructions in the programs.
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There are two main types of speed: latency and throughput. Latency is the time between the start of a process and its completion. Throughput is the amount of work done per unit time. Interrupt latency is the guaranteed maximum response time of the system to an electronic event .
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Performance is affected by a very wide range of design choices — for example, pipelining a processor usually makes latency worse, but makes throughput better. Computers that control machinery usually need low interrupt latencies. These computers operate in a real-time environment and fail if an operation is not completed in a specified amount of time. For example, computer-controlled anti-lock brakes must begin braking within a predictable and limited time period after the brake pedal is sensed or else failure of the brake will occur.
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Benchmarking takes all these factors into account by measuring the time a computer takes to run through a series of test programs. Although benchmarking shows strengths, it should not be how you choose a computer. Often the measured machines split on different measures. For example, one system might handle scientific applications quickly, while another might render video games more smoothly. Furthermore, designers may target and add special features to their products, through hardware or software, that permit a specific benchmark to execute quickly but do not offer similar advantages to general tasks.
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Power efficiency is another important measurement in modern computers. Higher power efficiency can often be traded for lower speed or higher cost. The typical measurement when referring to power consumption in computer architecture is MIPS/W .
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Modern circuits have less power required per transistor as the number of transistors per chip grows. This is because each transistor that is put in a new chip requires its own power supply and requires new pathways to be built to power it. However, the number of transistors per chip is starting to increase at a slower rate. Therefore, power efficiency is starting to become as important, if not more important than fitting more and more transistors into a single chip. Recent processor designs have shown this emphasis as they put more focus on power efficiency rather than cramming as many transistors into a single chip as possible. In the world of embedded computers, power efficiency has long been an important goal next to throughput and latency.
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Increases in clock frequency have grown more slowly over the past few years, compared to power reduction improvements. This has been driven by the end of Moore's Law and demand for longer battery life and reductions in size for mobile technology. This change in focus from higher clock rates to power consumption and miniaturization can be shown by the significant reductions in power consumption, as much as 50%, that were reported by Intel in their release of the Haswell microarchitecture; where they dropped their power consumption benchmark from 30 to 40 watts down to 10-20 watts. Comparing this to the processing speed increase of 3 GHz to 4 GHz it can be seen that the focus in research and development is shifting away from clock frequency and moving towards consuming less power and taking up less space.
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Shor proposed multiple similar algorithms for solving the factoring problem, the discrete logarithm problem, and the period-finding problem. "Shor's algorithm" usually refers to the factoring algorithm, but may refer to any of the three algorithms. The discrete logarithm algorithm and the factoring algorithm are instances of the period-finding algorithm, and all three are instances of the hidden subgroup problem.
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The basic characteristic of a computable function is that there must be a finite procedure telling how to compute the function. The models of computation listed above give different interpretations of what a procedure is and how it is used, but these interpretations share many properties. The fact that these models give equivalent classes of computable functions stems from the fact that each model is capable of reading and mimicking a procedure for any of the other models, much as a compiler is able to read instructions in one computer language and emit instructions in another language.
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Enderton gives the following characteristics of a procedure for computing a computable function; similar characterizations have been given by Turing , Rogers , and others.
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Enderton goes on to list several clarifications of these 3 requirements of the procedure for a computable function:
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The procedure must theoretically work for arbitrarily large arguments. It is not assumed that the arguments are smaller than the number of atoms in the Earth, for example.
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The procedure is required to halt after finitely many steps in order to produce an output, but it may take arbitrarily many steps before halting. No time limitation is assumed.
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Although the procedure may use only a finite amount of storage space during a successful computation, there is no bound on the amount of space that is used. It is assumed that additional storage space can be given to the procedure whenever the procedure asks for it.
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To summarise, based on this view a function is computable if:
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The field of computational complexity studies functions with prescribed bounds on the time and/or space allowed in a successful computation.
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A set A of natural numbers is called computable if there is a computable, total function f such that for any natural number n, f = 1 if n is in A and f = 0 if n is not in A.
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A set of natural numbers is called computably enumerable if there is a computable function f such that for each number n, f is defined if and only if n is in the set. Thus a set is computably enumerable if and only if it is the domain of some computable function. The word enumerable is used because the following are equivalent for a nonempty subset B of the natural numbers:
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One such function, which is provable total but not primitive recursive, is the Ackermann function: since it is recursively defined, it is indeed easy to prove its computability .
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In a sound proof system, every provably total function is indeed total, but the converse is not true: in every first-order proof system that is strong enough and sound , one can prove the existence of total functions that cannot be proven total in the proof system.
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If the total computable functions are enumerated via the Turing machines that produces them, then the above statement can be shown, if the proof system is sound, by a similar diagonalization argument to that used above, using the enumeration of provably total functions given earlier. One uses a Turing machine that enumerates the relevant proofs, and for every input n calls fn by invoking the Turing machine that computes it according to the n-th proof. Such a Turing machine is guaranteed to halt if the proof system is sound.
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Every computable function has a finite procedure giving explicit, unambiguous instructions on how to compute it. Furthermore, this procedure has to be encoded in the finite alphabet used by the computational model, so there are only countably many computable functions. For example, functions may be encoded using a string of bits .
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The real numbers are uncountable so most real numbers are not computable. See computable number. The set of finitary functions on the natural numbers is uncountable so most are not computable. Concrete examples of such functions are Busy beaver, Kolmogorov complexity, or any function that outputs the digits of a noncomputable number, such as Chaitin's constant.
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Similarly, most subsets of the natural numbers are not computable. The halting problem was the first such set to be constructed. The Entscheidungsproblem, proposed by David Hilbert, asked whether there is an effective procedure to determine which mathematical statements are true. Turing and Church independently showed in the 1930s that this set of natural numbers is not computable. According to the Church–Turing thesis, there is no effective procedure which can perform these computations.
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The notion of computability of a function can be relativized to an arbitrary set of natural numbers A. A function f is defined to be computable in A when it satisfies the definition of a computable function with modifications allowing access to A as an oracle. As with the concept of a computable function relative computability can be given equivalent definitions in many different models of computation. This is commonly accomplished by supplementing the model of computation with an additional primitive operation which asks whether a given integer is a member of A. We can also talk about f being computable in g by identifying g with its graph.
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791
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Hyperarithmetical theory studies those sets that can be computed from a computable ordinal number of iterates of the Turing jump of the empty set. This is equivalent to sets defined by both a universal and existential formula in the language of second order arithmetic and to some models of Hypercomputation. Even more general recursion theories have been studied, such as E-recursion theory in which any set can be used as an argument to an E-recursive function.
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792
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Although the Church–Turing thesis states that the computable functions include all functions with algorithms, it is possible to consider broader classes of functions that relax the requirements that algorithms must possess. The field of Hypercomputation studies models of computation that go beyond normal Turing computation.
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793
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A quantum computer is a computer that takes advantage of quantum mechanical phenomena.
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794
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On small scales, physical matter exhibits properties of both particles and waves, and quantum computing leverages this behavior, specifically quantum superposition and entanglement, using specialized hardware that supports the preparation and manipulation of quantum states.
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795
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Classical physics cannot explain the operation of these quantum devices, and a scalable quantum computer could perform some calculations exponentially faster than any modern "classical" computer. In particular, a large-scale quantum computer could break widely used encryption schemes and aid physicists in performing physical simulations; however, the current state of the technology is largely experimental and impractical, with several obstacles to useful applications. Moreover, scalable quantum computers do not hold promise for many practical tasks, and for many important tasks quantum speedups are proven impossible.
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796
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The basic unit of information in quantum computing is the qubit, similar to the bit in traditional digital electronics. Unlike a classical bit, a qubit can exist in a superposition of its two "basis" states. When measuring a qubit, the result is a probabilistic output of a classical bit, therefore making quantum computers nondeterministic in general. If a quantum computer manipulates the qubit in a particular way, wave interference effects can amplify the desired measurement results. The design of quantum algorithms involves creating procedures that allow a quantum computer to perform calculations efficiently and quickly.
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797
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Physically engineering high-quality qubits has proven challenging. If a physical qubit is not sufficiently isolated from its environment, it suffers from quantum decoherence, introducing noise into calculations. Paradoxically, perfectly isolating qubits is also undesirable because quantum computations typically need to initialize qubits, perform controlled qubit interactions, and measure the resulting quantum states. Each of those operations introduces errors and suffers from noise, and such inaccuracies accumulate.
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798
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In principle, a non-quantum computer can solve the same computational problems as a quantum computer, given enough time. Quantum advantage comes in the form of time complexity rather than computability, and quantum complexity theory shows that some quantum algorithms for carefully selected tasks require exponentially fewer computational steps than the best known non-quantum algorithms. Such tasks can in theory be solved on a large-scale quantum computer whereas classical computers would not finish computations in any reasonable amount of time. However, quantum speedup is not universal or even typical across computational tasks, since basic tasks such as sorting are proven to not allow any asymptotic quantum speedup. Claims of quantum supremacy have drawn significant attention to the discipline, but are demonstrated on contrived tasks, while near-term practical use cases remain limited.
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799
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For many years, the fields of quantum mechanics and computer science formed distinct academic communities. Modern quantum theory developed in the 1920s to explain the wave–particle duality observed at atomic scales, and digital computers emerged in the following decades to replace human computers for tedious calculations. Both disciplines had practical applications during World War II; computers played a major role in wartime cryptography, and quantum physics was essential for the nuclear physics used in the Manhattan Project.
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800
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An adiabatic quantum computer, based on quantum annealing, decomposes computation into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contain the solution.
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