Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

TL;DR

This paper extends quantum signal processing to multiple variables, enabling coherent approximation of joint functions of multiple oracles, which leads to new quantum algorithm speedups and connects quantum computing with algebraic geometry.

Contribution

Introduces multivariable quantum signal processing (M-QSP), establishing necessary and sufficient conditions for multivariable polynomial transformations and demonstrating its stability and efficiency.

Findings

Necessary and sufficient conditions for multivariable polynomial transformations.

Classical subroutines in QSP remain stable and efficient in M-QSP.

M-QSP enables novel quantum speedups and links to algebraic geometry.

Abstract

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 ans\"atze, 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 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 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.