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Title: Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle
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 s i n g l e 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 s t a b l e 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 o b l i v i o u s l y 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.  more » « less
Award ID(s):
1729369
PAR ID:
10494022
Author(s) / Creator(s):
;
Publisher / Repository:
Quantum
Date Published:
Journal Name:
Quantum
Volume:
6
ISSN:
2521-327X
Page Range / eLocation ID:
811
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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