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Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle
Quantum ( IF 6.4 ) Pub Date : 2022-09-20 , DOI: 10.22331/q-2022-09-20-811
Zane M. Rossi, Isaac L. Chuang

Recent work shows that quantum signal processing (QSP) and its multi-qubit lifted version, quantum singular value transformation (QSVT), unify and improve the presentation of most quantum algorithms. QSP/QSVT characterize the ability, by alternating ansätze, to obliviously transform the singular values of subsystems of unitary matrices by polynomial functions; these algorithms are numerically stable and analytically well-understood. That said, QSP/QSVT require consistent access to a $single$ oracle, saying nothing about computing $\textit{joint properties}$ of two or more oracles; these can be far cheaper to determine given an ability to pit oracles against one another coherently.
This work introduces a corresponding theory of QSP over multiple variables: M-QSP. Surprisingly, despite the non-existence of the fundamental theorem of algebra for multivariable polynomials, there exist necessary and sufficient conditions under which a desired $stable$ multivariable polynomial transformation is possible. Moreover, the classical subroutines used by QSP protocols survive in the multivariable setting for non-obvious reasons, and remain numerically stable and efficient. Up to a well-defined conjecture, we give proof that the family of achievable multivariable transforms is as loosely constrained as could be expected. The unique ability of M-QSP to $obliviously$ approximate $\textit{joint functions}$ of multiple variables coherently leads to novel speedups incommensurate with those of other quantum algorithms, and provides a bridge from quantum algorithms to algebraic geometry.


中文翻译:

多变量量子信号处理 (M-QSP):双头预言机的预言

最近的工作表明,量子信号处理 (QSP) 及其多量子比特提升版本,量子奇异值变换 (QSVT),统一并改进了大多数量子算法的表示。QSP/QSVT 通过交替 ansätze 表征通过多项式函数不经意地变换酉矩阵子系统的奇异值的能力;这些算法在数值上是稳定的并且在分析上是易于理解的。也就是说,QSP/QSVT 需要对 $single$ 预言机的一致访问,更不用说计算两个或更多预言机的 $\textit{joint properties}$;考虑到使预言机连贯地相互对抗的能力,确定这些可能要便宜得多。
这项工作介绍了一个对应的多变量 QSP 理论:M-QSP。令人惊讶的是,尽管不存在多变量多项式的代数基本定理,但存在必要且充分的条件,在这些条件下,所需的 $stable$ 多变量多项式变换是可能的。此外,由于非显而易见的原因,QSP 协议使用的经典子程序在多变量环境中仍然存在,并且在数值上保持稳定和高效。根据一个定义明确的猜想,我们证明了可实现的多变量变换族的约束与预期的一样松散。M-QSP 的独特能力 $obliviously$ 近似 $\textit{joint functions}$ 的多个变量连贯地导致了与其他量子算法不相称的新加速,
更新日期:2022-09-20
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