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Beavernets Technologies（Beavernets Technologies）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用特征理论将泡利分解转化为傅里叶分析，提出有界内存算法paulikit，通过Walsh-Hadamard变换实现量子比特与量子比特的泡利分解，避免物化稠密矩阵，成本为O(n·4^n)。
AI 中文摘要
泡利分解支撑着Trotter化、相位估计、稀疏哈密顿量模拟和变分可观测量，其朴素成本呈指数级。根据$(\nmathbb{Z}_2)^n$的特征理论，特征函数$\chi_z(v)=(-1)^{\langle v,z\rangle}$即为对角泡利串，因此任何对角算子都已被泡利分解。一般算子仍需要平移：每个$X^{\otimes x}$耦合到一个特征函数$\chi_z=Z^{\otimes z}$，而乘积$X^{\otimes x}Z^{\otimes z}$——在相位群$\{\pm 1,\pm i\}$内——构成泡利群，即$(\nmathbb{Z}_2)^n\times(\mathbb{Z}_2)^n$被该由辛配对固定的相位群的中心扩张。该扩张的一个截面仍是一个选择：裸乘积$X^{\otimes x}Z^{\otimes z}$，或在$x$和$z$重叠的每个量子比特处插入$Y=iXZ$的真正算子。作为阿贝尔平移-特征格上的傅里叶分析（一种自然DFT），此表述不限于量子比特：$(\nmathbb{Z}_p)^n$恢复海森堡-韦尔量子比特。快速算法是适用于当前群的FFT；在$(\nmathbb{Z}_2)^n$上，该FFT即Walsh-Hadamard变换，成本为$O(n\cdot 4^n)$。paulikit是该量子比特变换的有界内存实现：峰值使用无需物化稠密的$2^n\times 2^n$算子，而先前的代码将$O(1)$额外开销置于该缓冲区上，已在超过十亿振荡项上对照Work-Span和带宽上限进行了检查。
英文摘要
Pauli decomposition underpins Trotterization, phase estimation, sparse-Hamiltonian simulation, and variational observables, at exponential naive cost. From the character theory of $(\mathbb{Z}_2)^n$, characters $χ_z(v)=(-1)^{\langle v,z\rangle}$ are the diagonal Pauli strings, so any diagonal operator is already Pauli-decomposed. A general operator still needs the shifts: each $X^{\otimes x}$ couples to a character $χ_z=Z^{\otimes z}$, and the products $X^{\otimes x}Z^{\otimes z}$ -- up to phases in $\{\pm 1,\pm i\}$ -- form the Pauli group, the central extension of $(\mathbb{Z}_2)^n\times(\mathbb{Z}_2)^n$ by that phase group fixed by the symplectic pairing. A section of that extension remains a choice: the bare products $X^{\otimes x}Z^{\otimes z}$, or the genuine operators that insert $Y=iXZ$ at every qubit where $x$ and $z$ overlap. As Fourier analysis on an abelian shift-character lattice (a natural DFT), this formulation is not qubit-bound: $(\mathbb{Z}_p)^n$ recovers Heisenberg-Weyl qudits. The fast algorithm is the FFT for the group at hand; on $(\mathbb{Z}_2)^n$ that FFT is Walsh-Hadamard, at cost $O(n\cdot 4^n)$. paulikit is a memory-bounded realization of that qubit transform: peak usage need not materialize the dense $2^n\times 2^n$ operator, unlike prior codes whose $O(1)$ extra sits on that buffer, checked beyond a billion oscillator terms against Work-Span and bandwidth ceilings.
Comments29 pages, 4 figures. Open-source companion: paulikit (GPL-3.0-or-later), Zenodo 10.5281/zenodo.22992251; measurements 10.5281/zenodo.22992460
