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Contour-integral based quantum eigenvalue transformation: analysis and applications

Shan Jiang, Dong An

TL;DR

This paper develops a contour-integral framework for quantum eigenvalue transformations that lie outside the reach of QSVT, providing a complete complexity analysis and a qubit-efficient observable-estimation variant. It introduces both a general complexity bound and a constant-ancilla hybrid method, then applies the approach to Hamiltonian simulation, matrix polynomials, and linear differential equations, deriving explicit scaling and comparing to existing quantum algorithms. The results show that contour-integral methods can outperform existing approaches in asymptotically strictly stable differential equations and offer versatile, practical tools for a broad class of eigenvalue-transformations. Overall, the work extends the toolbox for quantum matrix function evaluation, balancing theoretical guarantees with practical considerations for near-term- to long-term quantum architectures. The framework sets the stage for further applications and exploration of multi-matrix functions in quantum computation.

Abstract

Eigenvalue transformations appear ubiquitously in scientific computation, ranging from matrix polynomials to differential equations, and are beyond the reach of the quantum singular value transformation framework. In this work, we study the efficiency of quantum algorithms based on contour integral representation for eigenvalue transformations from both theoretical and practical aspects. Theoretically, we establish a complete complexity analysis of the contour integral approach proposed in [Takahira, Ohashi, Sogabe, and Usuda. Quant. Inf. Comput., 22, 11\&12, 965--979 (2021)]. Moreover, we combine the contour integral approach and the sampling-based linear combination of unitaries to propose a quantum algorithm for estimating observables of eigenvalue transformations using only $3$ additional qubits. Practically, we design contour integral based quantum algorithms for Hamiltonian simulation, matrix polynomials, and solving linear ordinary differential equations, and show that the contour integral algorithm can outperform all the existing quantum algorithms in the case of solving asymptotically stable differential equations.

Contour-integral based quantum eigenvalue transformation: analysis and applications

TL;DR

This paper develops a contour-integral framework for quantum eigenvalue transformations that lie outside the reach of QSVT, providing a complete complexity analysis and a qubit-efficient observable-estimation variant. It introduces both a general complexity bound and a constant-ancilla hybrid method, then applies the approach to Hamiltonian simulation, matrix polynomials, and linear differential equations, deriving explicit scaling and comparing to existing quantum algorithms. The results show that contour-integral methods can outperform existing approaches in asymptotically strictly stable differential equations and offer versatile, practical tools for a broad class of eigenvalue-transformations. Overall, the work extends the toolbox for quantum matrix function evaluation, balancing theoretical guarantees with practical considerations for near-term- to long-term quantum architectures. The framework sets the stage for further applications and exploration of multi-matrix functions in quantum computation.

Abstract

Eigenvalue transformations appear ubiquitously in scientific computation, ranging from matrix polynomials to differential equations, and are beyond the reach of the quantum singular value transformation framework. In this work, we study the efficiency of quantum algorithms based on contour integral representation for eigenvalue transformations from both theoretical and practical aspects. Theoretically, we establish a complete complexity analysis of the contour integral approach proposed in [Takahira, Ohashi, Sogabe, and Usuda. Quant. Inf. Comput., 22, 11\&12, 965--979 (2021)]. Moreover, we combine the contour integral approach and the sampling-based linear combination of unitaries to propose a quantum algorithm for estimating observables of eigenvalue transformations using only additional qubits. Practically, we design contour integral based quantum algorithms for Hamiltonian simulation, matrix polynomials, and solving linear ordinary differential equations, and show that the contour integral algorithm can outperform all the existing quantum algorithms in the case of solving asymptotically stable differential equations.
Paper Structure (32 sections, 8 theorems, 135 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 32 sections, 8 theorems, 135 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Main results
  4. Complexity analysis of the quantum contour integral algorithm
  5. Qubit-efficient quantum eigenvalue transformation algorithm
  6. Applications
  7. Hamiltonian simulation.
  8. Matrix polynomials.
  9. Linear differential equations.
  10. Related works
  11. Discussions
  12. Organization
  13. Preliminaries
  14. Cauchy’s integral and numerical discretization
  15. LCU lemma
  16. ...and 17 more sections

Key Result

Proposition 5

Suppose that $f(z)$ is holomorphic inside $\Gamma$ and $L$-Lipschitz continuous inside and on $\Gamma$. Let $B$ be the maximum value of $|f(z)|$ on $\Gamma$, and assume that $\left\|(z(t) I_N -A)^{-1}\right\| \leq \gamma$ for any $t \in [0, l]$. Then, for the matrix function $f(A)$ in Eq. function:f

Figures (3)

  • Figure 1: QSVT with control circuit. Due to the control operator, it can also act as a select oracle $\mathrm{SEL}$ in outer layer LCU.
  • Figure 2: Left: In time-independent Hamiltonian simulation, the spectrum of $-\mathrm{i} H$ and the covering contour. Right: When calculating matrix polynomial, the spectrum of $A$ and the covering contour.
  • Figure 3: The spectrum of the matrix A and the corresponding covering contour in solving differential equations. Left: the real part of the spectrum of $A$ does not exceed $0$. Right: the real part of the spectrum of $A$ has an upper bound $-a < 0$.

Theorems & Definitions (15)

  • Definition 1: Quantum matrix function problem
  • Definition 2: Quantum matrix function estimation problem
  • Proposition 5
  • proof
  • Lemma 6: LCU lemma
  • Definition 7: Generalized matrix function
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • Lemma 11
  • ...and 5 more