克利福德乘法的量子算法
Quantum algorithm for Clifford multiplication
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中文总结 AI 辅助
研究克利福德乘法问题，利用幅度编码，量子计算机能在\(O(\text{polylog} N)\)时间内完成，相比经典算法有指数级加速，确立其为量子原语，为相关算法和模拟提供高效计算基础。
中文摘要 AI 辅助
给定具有\(N = 2^{p+q}\)个系数的克利福德代数\(C\ell(V, Q)\)的两个密集多向量，已知最快的经典算法在\(O(N^{\omega/2})\)次算术运算中计算它们的几何积，其中\(\omega\)表示矩阵乘法指数。本文表明，在幅度编码下，量子计算机可在\(O(\text{polylog} N)\)时间内执行几何积，使用对数空间和亚对数电路深度。这种指数级加速将克利福德乘法确立为量子原语，为量子几何算法和相对论模拟提供了高效计算基础。
英文摘要
Given two dense multivectors of the Clifford algebra $C\ell(V, Q)$ with $N=2^{p+q}$ coefficients, the fastest known classical algorithms compute their geometric product in $O(N^{ω/2})$ arithmetic operations, where $ω$ denotes the matrix multiplication exponent. I show that, under amplitude encoding, a quantum computer executes the geometric product in $O(\operatorname{polylog} N)$ time, using logarithmic space with sublogarithmic circuit depth. This exponential speedup establishes Clifford multiplication as a quantum primitive, providing an efficient computational foundation for quantum geometric algorithms and relativistic simulations.
