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

Figures (6)

• Figure 1: Quantum algorithm to find eigenvalues and eigenstates for non-Hermitian operators
• Figure 2: Three-qubit unitary rotation entangled ansatz with nearest-neighbor CNOT gates. The unitary rotation is applied to each qubit to allow complete Hilbert space exploration and CNOT gates to entangle those qubits together. This classic hardware-efficient entangling circuit allows maximum expressibility with shallower circuit depth.
• Figure 3: Plot showing convergence of the evaluated cost function from the matrix $M_1$ to zero for different initial guesses of parameters fixed to a single value converging at the computational limit of the system ($10^{-14} - 10^{-16}$, shaded grey).
• Figure 4: Plot showing real part of the variance as cost function, for the matrix $M_1$ whose ansatz depends on two parameters $\theta_1$ and $\theta_2$. The eigenvalues corresponding to those eigenstates are obtained at the points $P_1$ to $P_5$ at which the cost function is zero.
• Figure 5: Plot showing correlation between the eigenvalue for matrix $A$ (dashed blue line) and real part of the variance as cost function (solid red line), whose ansatz depends only on a single parameter $\theta$. The maxima and minima of the eigenvalue expectations lie exactly on the global minima, which is zero for those particular values of the parameter $\theta$