Hermitian-preserving ansatz and variational open quantum eigensolver

TL;DR

This work tackles finding steady states of open quantum systems described by Lindblad master equations or non-Hermitian Hamiltonians. It introduces VOQE, which maps density matrices to pure states in a doubled Hilbert space via $|\rho\rangle = (1/C)\sum_{ij} \rho_{ij}|i,j\rangle$ and minimizes $C_L[|\rho\rangle]$ or $C_n[|\rho\rangle]$ to locate steady states. A Hermitian-preserving ansatz (HPA) with three block types is used to constrain the search space, and a post-selection measurement scheme enables efficient extraction of operator expectations. Numerical demonstrations on a driven XXZ LME and an Ising nHH spectrum show VOQE's ability to unify steady-state solutions for LMEs and nHHs and its potential applicability to NISQ devices.

Abstract

We propose a new variational quantum algorithm named Variational Open Quantum Eigensolver (VOQE) for solving steady states of open quantum systems described by either Lindblad master equations or non-Hermitian Hamiltonians. In VOQE, density matrices of mixed states are represented by pure states in doubled Hilbert space. We give a framework for building circuit ansatz which we call the Hermitian-preserving ansatz (HPA) to restrict the searching space. We also give a method to efficiently measure the operators' expectation values by post-selection measurements. We show the workflow of VOQE on solving steady states of the LMEs of the driven XXZ model and implement VOQE to solve the spectrum of the non-Hermitian Hamiltonians of the Ising spin chain in an imaginary field.

Figures (4)

• Figure 1: Variational Open Quantum Eigensolver (VOQE). (a): The sketch of VOQE. VOQE uses 2n-qubit (n qubits in row subsystem and n qubits in column subsystem) parameterized HPA to solve the steady state of n-qubit open quantum system equations including LME and nHH. Equations are first transformed into the vector form to obtain the cost function operator. Next, Hermitian states from the Conjugate Ansatzes are measured to evaluate the cost function value for classical optimization. After the steady state is obtained, a post-selection method is used to obtain operators' expectation values of the state. For nHHs, unlike LME, the trace-preserving term in Eq.(\ref{['nhh']}) leads to non-linear equations, which makes $Tr[\Gamma\rho]$ appear in $\mathcal{N}[\rho]$. Thus, there is an additional intermediate process as shown below. (b): Relations between the whole doubled Hilbert space, Hermitian state space, and Density matrix state space. Hermitian states have the Hermiticity restriction while density matrix states not only require the Hermiticity but also the positive semi-definiteness. (c): Basic HPA blocks. There are three basic blocks when one uses single or two-qubit gates to compose an HPA. Both types share the idea of pairing to satisfy the HPA conditions Eq. \ref{['lqg']}. Type 1 describes the unitary transformation process while Type 2 and Type 3 simulate the non-unitary process.
• Figure 2: Numerical experiments of VOQE. (a): The steady states of the LME of the driven open XXZ model. We set $\Delta=1$ and turn $\epsilon$ from 200 to 0.1. There will appear cosine spin profile $\langle\sigma_i^z\rangle=cos(\pi\frac{i-1}{n-1})$ as the $\epsilon$ increases to a large value. The problem size is 5-qubit and a 10-qubit HPA is used for training. (b): The complex spectrum of the nHH of the Ising spin chain in an imaginary field. The real part and the imaginary part are plotted respectively. The solid lines form the exact complex spectrum and the points are obtained from VOQE. We set $\lambda=0.5$ and turn $\kappa$ from -2 to 2, the spectrums of the Hamiltonians of the model are complex except for the PT-symmetry phases. For each $(\lambda,\kappa)$ setting, we run VOQE 30 times to make sure the majority of the spectrum is covered. The problem size is 3-qubit.
• Figure 3: A layer of the parameterized HPA for the numerical experiments on the driven open XXZ model. During the experiments, we fixed the layer depth to be 1 with additional single-qubit parameterized gates appended at the end of the ansatz. Here, depth means the number of HPA layers shown in this figure.
• Figure 4: Convergence of the cost functions of the driven open XXZ model with $\epsilon=1$ under random initialization as functions of iteration steps. For each cost function, we have re-scaled its range within $[0,1]$ to have a better presentation. The starting point (initial parameters) of each curve is chosen randomly. The optimizer is chosen to be the BFGS optimizer.