Real Variance-Based Variational Quantum Eigensolver for Non-Hermitian Matrices

TL;DR

This paper introduces a novel variational quantum algorithm tailored for non-Hermitian matrices, enabling accurate eigenvalue computation in open quantum systems using only Hermitian measurements.

Contribution

It proposes a systematic RVVQE method with a guaranteed convergence cost function, suitable for non-Hermitian operators, and demonstrates its scalability through numerical results.

Findings

The algorithm guarantees convergence to true eigenstates.

It uses only Hermitian measurements, simplifying implementation.

Numerical results show scalability to larger matrices.

Abstract

Non-Hermitian operators naturally arise in the description of open quantum systems, which exhibit features such as resonances and decay processes, where the associated eigenvalues are complex. Standard quantum algorithms, including the Variational Quantum Eigensolver (VQE), are designed for Hermitian operators and are ineffective in recovering correct eigenvalues for non-Hermitian matrices. We present a systematic formulation based on a Real Variance-based Variational Quantum Eigensolver (RVVQE) for non-Hermitian operators. A correct cost function that guarantees convergence to the true eigenstates is identified. Our implementation utilizes Hermitian measurements only, rendering the algorithm easily deliverable. The performance and scalability of the proposed algorithm on a hierarchy of dense non-Hermitian matrices of increasing dimension are demonstrated with numerical results and computational metrics.