Exponential quantum advantages for practical non-Hermitian eigenproblems

TL;DR

This work addresses the challenge of solving non‑Hermitian eigenproblems on quantum hardware, where eigenvalues can be complex or defective. It introduces a fuzzy quantum eigenvalue detector (FQED) combined with a divide‑and‑conquer strategy and quantum singular value transformation (QSVT) to isolate eigenvalues near a reference line in the complex plane, achieving a provable exponential quantum speedup over classical methods. The framework extends to line-gap and point-gap problems, Liouvillian gaps in open quantum systems, PT‑symmetry breaking witnesses, and relaxation times of Markov processes, with explicit block-encoding and circuit‑level constructions. Overall, it presents a broadly applicable quantum algorithmic paradigm for non‑Hermitian spectral problems with potential impact across quantum physics and complex systems.

Abstract

Non-Hermitian physics has emerged as a rich field of study, with applications ranging from $PT$-symmetry breaking and skin effects to non-Hermitian topological phase transitions. Yet most studies remain restricted to small-scale or classically tractable systems. While quantum computing has shown strong performance in Hermitian eigenproblems, its extension to the non-Hermitian regime remains largely unexplored. Here, we develop a quantum algorithm to address general non-Hermitian eigenvalue problems, specifically targeting eigenvalues near a given line in the complex plane -- thereby generalizing previous results on ground state energy and spectral gap estimation for Hermitian matrices. Our method combines a fuzzy quantum eigenvalue detector with a divide-and-conquer strategy to efficiently isolate relevant eigenvalues. This yields a provable exponential quantum speedup for non-Hermitian eigenproblems. Furthermore, we discuss the broad applications in detecting spontaneous $PT$-symmetry breaking, estimating Liouvillian gaps, and analyzing classical Markov processes. These results highlight the potential of quantum algorithms in tackling challenging problems across quantum physics and beyond.

Figures (8)

• Figure 1: Illustration of energy gaps. (a) The energy gap for Hermitian matrices with real reference point $\boldsymbol{P}$. (b) Point gap for non-Hermitian matrices with complex reference point $\boldsymbol{P}$. (c) The line gap for non-Hermitian matrices with reference line $\boldsymbol{L}$.
• Figure 2: (a) sketch of FQED$(\mu,\varepsilon_{\text{th}})$. Inner disk $\mathcal{D}(\mu,\varepsilon_{\text{th}})$ and outer disk $\mathcal{D}(\mu,2K\varepsilon_{\text{th}})$ are marked with yellow and green colors. (b) Each iteration for solving Problem \ref{['prob:lg']}. Inner disks cover the line represented by $g_{\min}$, while the outer disks do not overlap with the real axis. If at least one of the FQEDs returns True, we update $g_{\max}$, otherwise we update $g_{\min}$. (c) The guess region of the minimum eigenvalue shrinks iteratively until it is sufficiently small. In the last step, we output $\lambda_{\min}$ as the center of the last FQED returning True.
• Figure 3: Applications of eigenvalue problems in non-Hermitian many-body physics.
• Figure S1: Sketch of Algorithm. \ref{['alg:21']} for solving Problem. \ref{['def:1f']}. Yellow and green disks represents the inner and outer disks of FQEDs (see Fig. \ref{['fig:alg0']}(a) in main text). The initial and updated guess region is enclosed by grey circles. Once an FQED returns True, the guess region is updated. (c) The guess region is updated iteratively until its area is sufficiently small.
• Figure S2: Sketch of the process of solving Problem. \ref{['prob:pgf']}. The initial guess region is a ring enclosed by two grey circles. At each iteration, we query a set of FQEDs which are encapsulated as $\mathscr{S}_{\text{ring}}$ in Algorithm. \ref{['alg:erss0']}. If one of the FQEDs returns True, the ERSS returns True, otherwise the ERSS returns False. The updated guess region is a ring enclosed by a grey circle and red or blue circles. This process is repeated iteratively until the guess region is sufficiently small.

Theorems & Definitions (14)

• Lemma 1
• Lemma 2
• Lemma 3: Adapted from Theorem 19 in Gilyen.19
• Lemma 1
• proof
• Definition 1: fuzzy quantum eigenvalue detector for general matrices
• Lemma 2
• Lemma 3
• Lemma 4
• Theorem 1