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Quantum singular value transformation for an arbitrary bounded operator embedded in a unitary operator

Chusei Kiumi, Akito Suzuki

Abstract

This research extends quantum singular value transformation (QSVT) for general bounded operators embedded in unitary operators on possibly infinite-dimensional Hilbert spaces. Through in-depth mathematical exploration, we have achieved a refined operator-theoretic understanding of QSVT, leading to a more streamlined approach. One of the key discoveries is that polynomial transformations in QSVT inherently apply to the entire operator, rather than being contingent on the selection of a specific basis. We expect that this research will pave the way for applying these insights to a broader range of problems in quantum information processing and provide analytical tools for quantum dynamics, such as quantum walks.

Quantum singular value transformation for an arbitrary bounded operator embedded in a unitary operator

Abstract

This research extends quantum singular value transformation (QSVT) for general bounded operators embedded in unitary operators on possibly infinite-dimensional Hilbert spaces. Through in-depth mathematical exploration, we have achieved a refined operator-theoretic understanding of QSVT, leading to a more streamlined approach. One of the key discoveries is that polynomial transformations in QSVT inherently apply to the entire operator, rather than being contingent on the selection of a specific basis. We expect that this research will pave the way for applying these insights to a broader range of problems in quantum information processing and provide analytical tools for quantum dynamics, such as quantum walks.
Paper Structure (6 sections, 15 theorems, 72 equations)

This paper contains 6 sections, 15 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Definition
  3. Polynomial transformation of operators
  4. Invariant subspaces
  5. Relation to Qubitization and QSP
  6. Spectral mapping for QSVT operator

Key Result

Lemma 2.1

$\Delta$ is coisometry and satisfies

Theorems & Definitions (20)

  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • Theorem 3.4
  • Lemma 4.1
  • ...and 10 more