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

Zane M. Rossi; Isaac L. Chuang

doi:10.22331/q-2022-09-20-811

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

Zane M. Rossi¹ and Isaac L. Chuang²

¹Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
²Department of Physics, Department of Electrical Engineering and Computer Science, and Co-Design Center for Quantum Advantage, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Published:	2022-09-20, volume 6, page 811
Eprint:	arXiv:2205.06261v2
Doi:	https://doi.org/10.22331/q-2022-09-20-811
Citation:	Quantum 6, 811 (2022).

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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 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.

Featured image: A figurative representation of the circuits considered in M-QSP; here the querent can successively apply their own unitary operations, interleaved among possibly both of two oracular unitary operations. The properties of such circuits, and the functions they achieve in the parameters of each oracle, are the primary focus of the theory of M-QSP.

Popular summary

Quantum signal processing (QSP) and quantum singular value transformation (QSVT) are recently developed quantum algorithmic primitives which, in allowing the coherent transformation of the singular values of near arbitrary linear operators by polynomial functions, unify and improve the presentation of nearly all known quantum algorithms. In other words, by providing careful control over subsystems of unitary processes, QSP/QSVT permit a huge variety of linear algebraic techniques to be subsumed into quantum algorithms. QSP/QSVT achieve this behavior by simple alternating circuit ansätze, and are thus analytically well-understood, with easy-to-compute runtimes. At a basic mathematical level, these algorithms ask questions about the existence of unitary representations (matrices characterizing the evolution of quantum systems) whose elements are polynomial functions in a scalar parameter (the underlying signal to be processed). The existence of such representations (and the degree of the corresponding polynomials) depend crucially on basic theorems in algebraic geometry concerning positive polynomials and their decomposition into sums of squares.
This work takes the underlying intuition and mathematical foundation of the standard statement of QSP and investigates how they can be extended to a multivariable setting. I.e., if one wishes instead to coherently compute joint functions of multiple signals with alternating quantum circuit ansätze, what are the corresponding results from algebraic geometry necessary to constitute a useful theory of multivariable QSP/QSVT (M-QSP/M-QSVT)? In turn, the ability to compute joint functions can be shown to save query and gate complexity in comparison to multiple iterations of single-variable instances. To this end we provide a series of initial results showing that, up to a well-defined conjecture, the possible multivariable transformations with M-QSP/M-QSVT are as loosely constrained as could be expected.
In addition to demonstrating how a change of setting for QSP/QSVT can lead to a vastly extended family of quantum algorithms, this work also points toward a wide family of mathematical subfields and methods with the possibility of novel relevance to quantum information. This work promotes a new, bottom-up approach to understanding QSP/QSVT-like circuits, repurposing and extending powerful and seemingly unrelated mathematical techniques toward new physical intuition and computational possibilities.

► BibTeX data

@article{Rossi2022multivariable,
  doi = {10.22331/q-2022-09-20-811},
  url = {https://doi.org/10.22331/q-2022-09-20-811},
  title = {Multivariable quantum signal processing ({M}-{QSP}): prophecies of the two-headed oracle},
  author = {Rossi, Zane M. and Chuang, Isaac L.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {811},
  month = sep,
  year = {2022}
}

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