formula large_string |
|---|
\begin{array}{rlr}\hat{H}&=&\hbar\omega\left(\hat{n}+\frac{1}{2}\right)+\hbar\omega_{\mathrm{det}}\left(\hat{n}_{\mathrm{det}}+\frac{1}{2}\right)+\\&&\hbar g\,\hat{n}\:\hat{n}_{\mathrm{\mathrm{det}}}+\hat{H}_{\mathrm{drive}+{\mathrm{\mathrm{decay}}}}\,.\end{array} |
\omega |
\hat{n} |
\omega_{\mathrm{det}} |
\hat{n}_{\mathrm{det}} |
\left\langle\hat{n}_{\mathrm{det}}\right\rangle\gg 1 |
\bar{X}(t) |
\left\langle\hat{n}\right\rangle(t) |
\Gamma/\kappa |
\dot{N}_{\mathrm{in}} |
\propto\left(\Gamma\tau_{\mathrm{avg}}\right)^{-1/2} |
X_{\mathrm{thr}} |
\kappa\tau_{\mathrm{avg}}=2 |
\hat{\rho} |
\begin{array}{rlr}\dot{\hat{\rho}}&=&-i\sqrt{{\frac{\dot{N}_{\mathrm{in}}\kappa}{2}}}\left[\hat{a}+\hat{a}^{\dagger},\hat{\rho}\right]+\kappa\left(\hat{a}\hat{\rho}\hat{a}^{\dagger}-\frac{1}{2}\hat{n}\hat{\rho}-\frac{1}{2}\hat{\rho}\hat{n}\right)\\&&-2\Gamma\left[\hat{n},\left[\hat{n},\hat{\rho}\right]\right]-\sqrt{{4\Gamma}}\left(\hat{n}\hat{\rho}+\hat{\rho}\hat{n}-2\hat{\rho}\left\langle\hat{n}\right\rangle(t)\right)\xi(t).\,\,\,\,\,\end{array} |
\kappa |
\kappa_{\mathrm{det}}\gg\kappa |
\hat{n}=\hat{a}^{\dagger}\hat{a} |
\Gamma\equiv g^2\left\langle\hat{n}_{\mathrm{det}}\right\rangle/(4\kappa_{\mathrm{det}}) |
1/\Gamma |
X(t)\equiv\langle\hat{n}\rangle(t)+\frac{1}{4}\sqrt{{\frac{1}{\Gamma}}}\xi(t). |
\left\langle\xi\right\rangle=0 |
\langle\xi(t)\xi(t^{\prime})\rangle=\delta(t-t^{\prime}) |
\tau_{\mathrm{avg}} |
\tau_{\mathrm{avg}}\ll\kappa^{-1} |
\tau_{\mathrm{dark}} |
\dot{N}_{\mathrm{in}}^{-1}\gg\tau_{\mathrm{dark}}\gg\kappa^{-1} |
\dot{N}_{\mathrm{det}} |
\eta |
\eta\equiv\left.\frac{d\dot{N}_{\mathrm{det}}}{d\dot{N}_{\mathrm{in}}}\right|_{{\mathrm{\dot{N}}}_{\mathrm{in}}=0}. |
O(10^4) |
10^2/\kappa |
\Gamma/\kappa\ll 1 |
\Gamma/\kappa\gg 1 |
\Gamma/\kappa=0.6 |
\dot{N}_{\mathrm{in}}=0 |
\dot{N}_{\mathrm{in}}\sim\tau_{\mathrm{dark}}^{-1} |
\Gamma |
X_{\mathrm{thr}}=0.5 |
\max(\eta) |
\Gamma/\kappa=4 |
\alpha(t) |
\dot{\alpha}(t)=\left(-i\,\delta\omega(t)-\frac{\kappa}{2}\right)\alpha(t)+\sqrt{{\frac{\kappa}{2}}}\alpha_{\mathrm{\mathrm{L}}}^{\mathrm{in}}. |
\alpha_{\mathrm{L}}^{\mathrm{in}} |
\delta\omega(t)\equiv gn_{\mathrm{det}}(t) |
n_{\mathrm{det}}\gg 1 |
\left\langle\delta\omega(t)\delta\omega(0)\right\rangle-\left\langle\delta\omega\right\rangle^2=g^2\bar{n}_{\mathrm{det}} e^{-\kappa_{\mathrm{det}}|t|/2}\,. |
\frac{\alpha(t)}{\sqrt{{\kappa_L}}\alpha_{\mathrm{L}}^{\mathrm{in}}}=\int_{-\infty}^t dt^{\prime}\exp\left[-i\int_{t^{\prime}}^t\delta\omega(t^{\prime\prime})dt^{\prime\prime}-\frac{\kappa}{2}(t-t^{\prime})\right]. |
\delta\omega(t) |
\kappa_{\mathrm{det}}|t-t^{\prime}|\gg 1 |
\left\langle|\alpha|^2\right\rangle |
\langle\exp[-iY]\rangle=\exp[-i\langle Y\rangle-\frac{1}{2}\mathrm{Var}Y] |
\langle|a_{\mathrm{\mathrm{R}}}^{\mathrm{out}}|^2\rangle=\frac{\kappa}{2}\langle|\alpha|^2\rangle=\left\langle\mathcal{T}\right\rangle|\alpha_{\mathrm{L}}^{\mathrm{in}}|^2, |
\langle\mathcal{T}\rangle=\left(1+4\frac{\Gamma}{\kappa}\right)^{-1}. |
\eta=2\langle\mathcal{T}\rangle |
\Gamma/\kappa=1/2 |
2\pi\cdot 100\mathrm{MHz} |
2\pi\cdot 5\mathrm{GHz} |
1\mathrm{MHz} |
100\mathrm{MHz} |
40\% |
a_k\in\mathcal{C} |
k\in\mathbb{Z} |
\mathcal{C} |
f_{\mathrm{c}} |
\theta |
O(t)=\sqrt{{2x(t)}}cos\left(2\pi f_{\mathrm{c}} t+\theta\right) |
x(t)=JI(t)=JA\left(\mu+\sum_{k=-\infty}^\infty a_k q(t-k{T_\mathrm{s}})\right), |
\mu |
T_\mathrm{s} |
x(t)\geq 0 |
t\in\mathbb{R} |
Q(\omega)=\intop_{-\infty}^\infty q(t)e^{-j\omega t} dt=0,\;|\omega|\geq 2\pi B, |
Q(\omega) |
\mathcal{C}=\left\{0, 1\right\} |
P_{\mathrm{opt}}=\frac{1}{T_\mathrm{s}}\intop_0^{T_\mathrm{s}}\mathbb{E}\left\{x(t)\right\} dt, |
\mathbb{E}\left\{\cdot\right\} |
\begin{array}{rl}P_{\mathrm{opt}}&=\frac{1}{T_\mathrm{s}}\intop_0^{T_\mathrm{s}} JA\left(\mu+\mathbb{E}\left\{a_k\right\}\sum_{k=-\infty}^\infty q(t-k{T_\mathrm{s}})\right) dt\\&=JA\left(\mu+\mathbb{E}\left\{a_k\right\}\overline{q}\right),\end{array} |
\overline{q}=\frac{1}{T_\mathrm{s}}\intop_{-\infty}^\infty q(t)dt=\frac{Q(0)}{T_\mathrm{s}}. |
P_\mathrm{max}=\max x(t)=JA\left(\mu+\max\sum_{k=-\infty}^\infty a_k q(t-k{T_\mathrm{s}})\right) |
\ldots, a_{-1}, a_0, a_1, a_2,\ldots |
y(t)=Rh(t)\otimes x(t)+n(t), |
\otimes |
h(t)=H(0)\delta(t) |
R=J=1 |
H(0)=1 |
N_0/2 |
r(t)=y(t)\otimes g(t), |
G(\omega)=\left\{\begin{array}{cc}G(0)&|\omega|<2\pi B\\0&|\omega|\geq 2\pi B\end{array}.\right. |
G(\omega)=\zeta Q^*(\omega) |
\left(\cdot\right)^* |
\zeta |
\mathcal{C}\subset\mathbb{R}^+ |
\mu=0 |
\mathcal{C}\subset\mathbb{R} |
q(k{T_\mathrm{s}})=\left\{\begin{array}{ll}q(0),&k=0,\\0,&k\neq 0.\end{array}\right. |
{\operatorname{\mathrm{sinc}}}(x)=\sin(\pi x)/(\pi x) |
0\le\alpha\le 1 |
B=(1+\alpha)/(2{T_\mathrm{s}}) |
\mu>0 |
End of preview. Expand in Data Studio
PH FORMULA CORPUS V1
PH_FORMULA_CORPUS_V1 is a large-scale formula corpus containing 160 million normalized mathematical expressions.
Each formula has been carefully normalized to ensure concise, consistent, and simplified representations.
🗓️ Timeline
- ✅ July 2025 – Released the formula corpus
- 🔜 Aug 2025 – Upcoming release of the synthetic datasets
- ⏳ Sep 2025 – Scheduled release of a model achieving commercial-grade qualit PHOCR
🔧 Normalization Process
- Parse formulas using the KaTeX syntax tree with customized macro definitions.
- Traverse the tree with a tailored algorithm to normalize the expression of each node.
Additional normalization steps include:
- Standardizing
\operatornameand\operatorname*usage - Unifying variations of
\textxxand\mathxxcommands - Normalizing spacing across all elements
📘 Dataset Usage
def test_parquet():
import random
import pyarrow.parquet as pq
s
input_file = "/path/to/formula_corpus_v1.parquet"
table = pq.read_table(input_file)
print(f"Total rows: {len(table)}")
num_rows = min(100, len(table))
indices = random.sample(range(len(table)), num_rows)
formulas = table.column("formula")
for idx in indices:
print(formulas[idx].as_py())
📚 Citation
Thank you for using PH_FORMULA_CORPUS_V1.
This dataset is developed and maintained by Puhuilab.
If you use this dataset in your research or applications, please remember to give appropriate credit.
To cite PH_FORMULA_CORPUS_V1 in academic work, please use the following BibTeX entry:
@misc{phformulacorpusv12025,
title={PHFormulaCorpusV1: A Large-Scale Normalized LaTeX Formula Dataset},
author={Puhui Lab},
year={2025},
url={https://huggingface.co/datasets/puhuilab/ph_formula_corpus_v1}
}
💡 Acknowledgment
The dataset follows the licenses of its original sources:
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