Hermitian-preserving ansatz and variational open quantum eigensolver

We propose a 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 a circuit ansatz which we call the Hermitian-preserving ansatz to restrict the searching space. We also give a method to efficiently measure the operators' expectation values by postselection measurements. We show the workflow of VOQE on solving steady states of the Lindblad master equations 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.