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arXiv:0906.2538 · Physical Review A

Measurement-based quantum phase estimation algorithm for finding eigenvalues of non-unitary matrices

Hefeng Wang, Lian-Ao Wu, Yu-xi Liu, Franco Nori

Published Dec 3, 2010• 37 citations
arXivHTMLPDF

Abstract

We propose a quantum algorithm for finding eigenvalues of non-unitary matrices. We show how to construct, through interactions in a quantum system and projective measurements, a non-Hermitian or non-unitary matrix and obtain its eigenvalues and eigenvectors. This proposal combines ideas of frequent measurement, measured quantum Fourier transform, and quantum state tomography. It provides a generalization of the conventional phase estimation algorithm, which is limited to Hermitian or unitary matrices.

DOI
10.1103/physreva.82.062303

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The Kitaev's Phase Estimation Algorithm (also referred to as quantum eigenvalue estimation algorithm) is a quantum algorithm to estimate the phase (or eigenvalue) of an eigenvector of a unitary operator. We explain the algorithm and indicate an example of the determination of the eigenvalue of the T gate. Then we introduce the quantum counting algorithm that is based on the quantum phase estimation algorithm and on the Grover's search algorithm. Quantum counting algorithm is a quantum algorithm for counting the number of solutions for a given search problem.

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We propose a quantum algorithm for finding eigenvalues of non-unitary matrices. We show how to construct, through interactions in a quantum system and projective measurements, a non-Hermitian or non-unitary matrix and obtain its eigenvalues and eigenvectors. This proposal combines ideas of frequent measurement, measured quantum Fourier transform, and quantum state tomography. It provides a generalization of the conventional phase estimation algorithm, which is limited to Hermitian or unitary matrices.
Quantum algorithms for estimating the eigenvalues of matrices, including the phase estimation algorithm, serve as core subroutines in a wide range of quantum algorithms, including those in quantum chemistry and quantum machine learning. The standard quantum eigenvalue (phase) estimation algorithm accepts a Hermitian (unitary) matrix and a state in an unknown superposition of its eigenstates as input, and coherently records the estimates for real eigenvalues (eigenphases) in an ancillary register. Extension of quantum eigenvalue and phase estimation algorithms to the case of non-normal input matrices is obstructed by several factors such as non-orthogonality of eigenvectors, existence of generalized eigenvectors and the fact that eigenvalues may lie anywhere on the complex plane. In this work, we propose a novel approach for estimating the eigenvalues of non-normal matrices based on preparation of a state that we call the "resolvent state". We construct the first efficient algorithm for estimating the phases of the unimodular eigenvalues of a given non-unitary matrix. We then construct an efficient algorithm for estimating the real eigenvalues of a given non-Hermitian matrix, achieving complexities that match the best known results while operating under significantly relaxed assumptions on the non-real part of the spectrum. The resolvent-based approach that we introduce also extends to estimating eigenvalues that lie on a parametrized complex curve, subject to explicitly stated conditions, thereby paving the way for a new paradigm of parametric eigenvalue estimation.
Solving linear systems and computing eigenvalues are two fundamental problems in linear algebra. For solving linear systems, many efficient quantum algorithms have been discovered. For computing eigenvalues, currently, we have efficient quantum algorithms for Hermitian and unitary matrices. However, the general case is far from fully understood. Combining quantum phase estimation, quantum algorithm to solve linear differential equations and quantum singular value estimation, we propose two quantum algorithms to compute the eigenvalues of diagonalizable matrices that only have real eigenvalues and normal matrices. The output of the quantum algorithms is a superposition of the eigenvalues and the corresponding eigenvectors. The complexities are dominated by solving a linear system of ODEs and performing quantum singular value estimation, which usually can be solved efficiently in a quantum computer. In the special case when the matrix MMM is sss-sparse, the complexity is O~(sρ2κ2/ε2)\widetilde{O}(sρ^2 κ^2/ε^2)O(sρ2κ2/ε2) for diagonalizable matrices that only have real eigenvalues, and O~(sρ∥M∥max⁡/ε2)\widetilde{O}(sρ\|M\|_{\max} /ε^2)O(sρ∥M∥max​/ε2) for normal matrices. Here ρρρ is an upper bound of the eigenvalues, κκκ is the conditioning of the eigenvalue problem, and εεε is the precision to approximate the eigenvalues. We also extend the quantum algorithm to diagonalizable matrices with complex eigenvalues under an extra assumption.
Eigenproblem arises in a large number of disciplines of sciences and engineering. Combining quantum algorithms to solve linear differential equations, quantum singular value estimation and quantum phase estimation, we propose quantum algorithms to compute the eigenvalues of diagonalizable matrices of two types: matrices that only have real eigenvalues and normal matrices. The quantum algorithms return a quantum state such that the first register stores eigenvalues and the second register stores the corresponding eigenvectors. For normal matrices, the complexity to obtain this quantum state is dominated by the complexity to perform quantum singular value estimation. For the other case, the complexity is determined by the solving of a linear system of differential equations. Under certain conditions (e.g. sparse), the complexities are polylog on the size nnn of the given matrix. If we perform measurements, then we can obtain all the eigenvalues classically. If MMM is sss sparse, then the complexity is O~(sn∥M∥max⁡κ2/ϵ2)\widetilde{O}(sn\|M\|_{\max}\kappa^2/\epsilon^2)O(sn∥M∥max​κ2/ϵ2), where κ\kappaκ is the condition number of the matrix of eigenvectors.
Quantum phase estimation (QPE) is fundamental to quantum algorithms for eigenvalue problems, chemistry simulation, and number-theoretic applications. However, canonical QPE implementations employ controlled unitary operations and quantum Fourier transform, incurring substantial circuit depth and control overhead that severely limit applicability on nearterm quantum hardware. This work develops a novel class of control-free phase estimation algorithms that eliminate both controlled unitaries and QFT by leveraging signal processing principles. Our approach applies uncontrolled powers of the unitary U = e −iHτ to an eigenstate or eigenstate-rich state, measures in multiple bases to capture interference patterns, and reconstructs eigenphases via classical post-processing techniques including Prony's method, compressed sensing, and harmonic inversion. We establish that for eigenstates or states with sparse eigendecomposition, our algorithms achieve phase estimation precision ϵ using Õ(1/ϵ) applications of U , matching standard QPE query complexity up to logarithmic factors while dramatically reducing circuit depth and eliminating control overhead. We provide detailed error analysis characterizing robustness to measurement noise, finite sampling, and state preparation errors. Applications to electronic structure estimation in small molecules and variational quantum eigensolvers demonstrate the practical viability of control-free QPE on both NISQ and early fault-tolerant quantum computers. Our results suggest that signal-processing-based phase estimation may offer more efficient implementations of quantum algorithms on realistic hardware.
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.
The Kitaev's Phase Estimation Algorithm (also referred to as quantum eigenvalue estimation algorithm) is a quantum algorithm to estimate the phase (or eigenvalue) of an eigenvector of a unitary operator. We explain the algorithm and indicate an example of the determination of the eigenvalue of the T gate. Then we introduce the quantum counting algorithm that is based on the quantum phase estimation algorithm and on the Grover's search algorithm. Quantum counting algorithm is a quantum algorithm for counting the number of solutions for a given search problem.