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An adversary bound for quantum signal processing

Lorenzo Laneve

TL;DR

This work recast QSP as a state conversion problem over the Hilbert space of square-integrable functions, and shows that the adversary bound for a state conversion problem in this space precisely identifies all and only the QSP protocols in the univariate case.

Abstract

Quantum signal processing (QSP) and quantum singular value transformation (QSVT), have emerged as unifying frameworks in the context of quantum algorithm design. These techniques allow to carry out efficient polynomial transformations of matrices block-encoded in unitaries, involving a single ancilla qubit. Recent efforts try to extend QSP to the multivariate setting (M-QSP), where multiple matrices are transformed simultaneously. However, this generalization faces problems not encountered in the univariate counterpart: in particular, the class of polynomials achievable by M-QSP seems hard to characterize. In this work we borrow tools from query complexity, namely the state conversion problem and the adversary bound: we first recast QSP as a state conversion problem over the Hilbert space of square-integrable functions. We then show that the adversary bound for a state conversion problem in this space precisely identifies all and only the QSP protocols in the univariate case. Motivated by this first result, we extend the formalism to several variables: the existence of a feasible solution to the adversary bound implies the existence of a M-QSP protocol, and the computation of a protocol of minimal space is reduced to a rank minimization problem involving the feasible solution space of the adversary bound.

An adversary bound for quantum signal processing

TL;DR

This work recast QSP as a state conversion problem over the Hilbert space of square-integrable functions, and shows that the adversary bound for a state conversion problem in this space precisely identifies all and only the QSP protocols in the univariate case.

Abstract

Quantum signal processing (QSP) and quantum singular value transformation (QSVT), have emerged as unifying frameworks in the context of quantum algorithm design. These techniques allow to carry out efficient polynomial transformations of matrices block-encoded in unitaries, involving a single ancilla qubit. Recent efforts try to extend QSP to the multivariate setting (M-QSP), where multiple matrices are transformed simultaneously. However, this generalization faces problems not encountered in the univariate counterpart: in particular, the class of polynomials achievable by M-QSP seems hard to characterize. In this work we borrow tools from query complexity, namely the state conversion problem and the adversary bound: we first recast QSP as a state conversion problem over the Hilbert space of square-integrable functions. We then show that the adversary bound for a state conversion problem in this space precisely identifies all and only the QSP protocols in the univariate case. Motivated by this first result, we extend the formalism to several variables: the existence of a feasible solution to the adversary bound implies the existence of a M-QSP protocol, and the computation of a protocol of minimal space is reduced to a rank minimization problem involving the feasible solution space of the adversary bound.

Paper Structure

This paper contains 27 sections, 32 theorems, 131 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Finite state conversion and the adversary bound
  4. Quantum query algorithms and the state conversion problem
  5. Partial state conversion
  6. The $\gamma_2$-bound and its properties
  7. The adversary bound for state conversion
  8. Lower bounds and upper bounds for query complexity
  9. Quantum signal processing
  10. Univariate QSP and the layer stripping argument
  11. Fejér-Riesz factorization and the completion problem
  12. Extension to more than two dimensions
  13. Multivariate QSP
  14. Quantum signal processing as a state conversion problem
  15. The $\gamma_2$-bounds over $L^2(\T^m, \C^K)$
  16. ...and 12 more sections

Key Result

theorem 2.2

Given an algorithm $\calA$ for exact $\xi \mapsto \tau$ conversion there exists an algorithm $\calA'$ for $\epsilon$-approximate $\xi \mapsto \tau$ state conversion with Monte Carlo complexity $\mathcal{O}(\calL(\calA)/\epsilon^2)$.

Figures (2)

  • Figure 1: Circuit for a generic quantum query algorithm. The oracle $O$ can be controlled by some qubits, and does not need to act on all of them.
  • Figure 2: Visualization of the argument in two variables. The set $\mathbf{k} + \calL_3$ in red. The region on the top-right of the dashed red lines is the one constituting the divergent subsequence.

Theorems & Definitions (63)

  • Definition 2.1
  • theorem 2.2: belovsOneWayTicketVegas2023
  • Definition 2.3: Unidirectional relative $\gamma_2$-bound belovsOneWayTicketVegas2023
  • Definition 2.4: Unidirectional partial relative $\gamma_2$-bound
  • Definition 2.5: Unidirectional subrelative $\gamma_2$-bound belovsOneWayTicketVegas2023
  • theorem 2.6: Theorem 6.3 in belovsOneWayTicketVegas2023
  • proof
  • Definition 2.7
  • theorem 2.8: Proposition 6.10 in belovsOneWayTicketVegas2023
  • Lemma 2.9
  • ...and 53 more