Quantum Eigensolver for Non-Normal Matrices via Ground State Energy Estimation

TL;DR

This work introduces a quantum algorithm to estimate eigenvalues of general non-normal matrices $A$ by linking the distance to eigenvalues to the smallest singular value $\sigma_0(\mu)$ of $A-\mu I$ and then Hermitianizing the problem to perform ground-state energy estimation. The Quantum Eigenvalue Estimation (QEE) procedure uses a hierarchical complex-plane search and a Hermitianized Hamiltonian $H_{\mu}$ constructed via block-encoding and polynomial transforms implemented with quantum signal processing, achieving an $\epsilon$-accurate eigenvalue with failure probability $p_{\rm fail}$. The main contributions include (i) a general bound extending Bauer-Fike to defective matrices, (ii) a first general non-normal eigenvalue solver with resource scaling $\widetilde{O}(\kappa^2/(\gamma\epsilon^{2m-1}))$ for $U_A$ queries and $N_s=O(\kappa^2\epsilon^{-2(m-1)})$ samples, (iii) eigenvector preparation once an eigenvalue is known, and (iv) applicability to extreme eigenvalues and spectral-gap estimation. The approach supports applications to stability analysis and polynomial rootfinding and is validated by numerical simulations, highlighting the method's potential impact for large-scale, non-normal eigenvalue problems in physics and engineering.

Abstract

Large-scale eigenvalue problems pose a significant challenge to classical computers. While there are efficient quantum algorithms for unitary or Hermitian matrices, eigenvalue problems for non-normal matrices remain open in quantum computing. In this work, we propose a quantum algorithm that given a non-normal matrix, outputs an estimate of an eigenvalue to within additive error $ε$ with probability at least $1-p_{\rm fail}$. Our estimation strategy is to sample points on the complex plane and examine the distance between the sampled point and the eigenvalues. We show that the distance is related to the smallest singular value of the shifted matrix, hence reducing the problem to ground state energy estimation via Hermitianization. With the knowledge of an eigenvalue, we are able to prepare the associated eigenvector using ground state preparation. Our estimating scheme can also be modified to approximate the extreme eigenvalue, and in particular the spectral gap. The algorithm is implemented based on the block encoding input model and requires $O(κ^2ε^{-(2m-1)}\log(1/p_{\rm fail}))$ queries to the block encoding oracle. Our algorithm is the first general eigenvalue algorithm that achieves this scaling. We also perform numerical simulation to validate the algorithms.

Figures (3)

• Figure 1: Illustration of the first and second steps of Algorithm \ref{['QEE']}. Grid points (left) with spacing $\delta^{(1)}$
• Figure 2: The block encoding circuit of $(A-\mu I)^{\dagger}(A-\mu I)/\alpha_\mu^2+\nu I$ with normalization factor $1+\nu$. $R(\theta) = \cos\theta-\sin\theta\sin\theta\cos\theta$ and $\theta = \arccos(\sqrt{1/(1+\nu)})$. The dashed line indicates that the unitary $U_\mu$ does not act on these qubits.
• Figure 3: Illustration of Case 2. $\lambda_{\min}$ lies outside all the circles centered at $\mu^{(l)}_j$ with radius $\delta^{(l)}$ (only partial arcs of these circles are shown in the diagram). Particularly, $\lambda_{\min}$ is outside the circle centered at the origin with radius $R_1^{(l)}+\delta^{(l)}$.

Theorems & Definitions (20)

• Lemma 1
• Lemma 2
• Theorem 1: Quantum eigenvalue estimation
• proof : Sketch of the proof
• Theorem 2: Quantum eigenvector preparation
• proof
• Theorem 3: Real eigenvalues
• proof
• Theorem 4: Extreme eigenvalue
• proof : Sketch of the proof