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Heisenberg-Limited Quantum Eigenvalue Estimation for Non-normal Matrices

Yukun Zhang, Yusen Wu, Xiao Yuan

TL;DR

A new class of quantum algorithms to construct eigenvalue signals through customized quantum simulation protocols and extract them using advanced classical signal-processing techniques, thereby enabling accurate and efficient eigenvalue estimation for general non-normal matrices.

Abstract

Estimating the eigenvalues of non-normal matrices is a foundational problem with far-reaching implications, from modeling non-Hermitian quantum systems to analyzing complex fluid dynamics. Yet, this task remains beyond the reach of standard quantum algorithms, which are predominantly tailored for Hermitian matrices. Here we introduce a new class of quantum algorithms that directly address this challenge. The central idea is to construct eigenvalue signals through customized quantum simulation protocols and extract them using advanced classical signal-processing techniques, thereby enabling accurate and efficient eigenvalue estimation for general non-normal matrices. Crucially, when supplied with purified quantum state inputs, our algorithms attain Heisenberg-limited precision--achieving optimal performance. These results extend the powerful guided local Hamiltonian framework into the non-Hermitian regime, significantly broadening the frontier of quantum computational advantage. Our work lays the foundation for a rigorous and scalable quantum computing approach to one of the most demanding problems in linear algebra.

Heisenberg-Limited Quantum Eigenvalue Estimation for Non-normal Matrices

TL;DR

A new class of quantum algorithms to construct eigenvalue signals through customized quantum simulation protocols and extract them using advanced classical signal-processing techniques, thereby enabling accurate and efficient eigenvalue estimation for general non-normal matrices.

Abstract

Estimating the eigenvalues of non-normal matrices is a foundational problem with far-reaching implications, from modeling non-Hermitian quantum systems to analyzing complex fluid dynamics. Yet, this task remains beyond the reach of standard quantum algorithms, which are predominantly tailored for Hermitian matrices. Here we introduce a new class of quantum algorithms that directly address this challenge. The central idea is to construct eigenvalue signals through customized quantum simulation protocols and extract them using advanced classical signal-processing techniques, thereby enabling accurate and efficient eigenvalue estimation for general non-normal matrices. Crucially, when supplied with purified quantum state inputs, our algorithms attain Heisenberg-limited precision--achieving optimal performance. These results extend the powerful guided local Hamiltonian framework into the non-Hermitian regime, significantly broadening the frontier of quantum computational advantage. Our work lays the foundation for a rigorous and scalable quantum computing approach to one of the most demanding problems in linear algebra.
Paper Structure (23 sections, 21 theorems, 180 equations, 1 figure)

This paper contains 23 sections, 21 theorems, 180 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Basics of matrix decompositions and transformations
  3. Block encoding and other quantum algorithmic tools
  4. Chebyshev and Faber polynomials
  5. Quantum eigenvalue transformation
  6. Quantum eigenvalue estimation problem
  7. Problem statement
  8. Main algorithm
  9. Classical signal processing methods
  10. Quantum algorithms combining the QEVT protocol
  11. Quadratic speedup with purified quantum query access
  12. Complexity analysis
  13. Error propagation of the Vandermonde decomposition
  14. Bounds on the condition number of the Vandermonde matrix
  15. Sample complexity from random matrix theory
  16. ...and 8 more sections

Key Result

Theorem 1

For the targeted eigenvalues $Z:=\{\lambda_i\}_{i=1}^r$ of $A\in\mathbb{C}^{N\times N}$, there exists a quantum algorithm that outputs an estimation $\{\widetilde{\lambda}_i\}_{i=1}^r$ such that $\min _{\lambda \in Z}|\widetilde{\lambda}-\lambda|\leq \epsilon$ with high success probability and $\mat

Figures (1)

  • Figure 1: Schematic of the algorithm for solving the quantum eigenvalue estimation problem. (a) A quantum computer is used to generate times series that are mixture of signals of the eigenstates. (b) Example of the signal of Eq. \ref{['eq:signal1']} with sparsity $r=5$. The blue and red lines are the amplitude of real and imaginary parts of the signal as a function of $t$. A classical computer is then used to solve for $\{\lambda_i\}_i$. (c) Schematic for each individual mode $\lambda_i^t$ as a function of $t$. Dots indicates discrete $t=1,2,\cdots,2r-1=9$. Dotted lines are the magnitude envelope of the signal, indicating exponential decay with $t$.

Theorems & Definitions (44)

  • Definition 1
  • Theorem 1: informal
  • Theorem 2: informal
  • Definition 2: Block Encoding
  • Lemma 1: Product of multiple block-encoded matrices gilyen2019quantum
  • proof
  • Lemma 2: Block encoding amplification low2017hamiltonian
  • Lemma 3: Block encoding inversion gilyen2019quantum
  • Lemma 4: Quantum amplitude estimation aaronson2020quantum
  • proof
  • ...and 34 more