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

Journal Article · 20 September 2022 · Quantum

DOI: https://doi.org/10.22331/q-2022-09-20-811 · OSTI ID:1983350

Rossi, Zane M.  ^[1]; Chuang, Isaac L. ^[1]

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1. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)

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 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.

• Research Organization: National Quantum Information Science (QIS) Research Centers (United States). Co-design Center for Quantum Advantage (C2QA); Brookhaven National Laboratory (BNL), Upton, NY (United States)
• Sponsoring Organization: USDOE Office of Science (SC)
• Grant/Contract Number: SC0012704
• OSTI ID: 1983350
• Journal Information: Quantum, Vol. 6; ISSN 2521-327X
• Publisher: Quantum Science Open CommunityCopyright Statement
• Country of Publication: United States
• Language: English