Relaxational Quantum Eigensolver: State Characterization and Thermometry

Many quantum algorithms, such as QAOA or the Variational Quantum Eigensolver (VQE), focus on minimizing a classical or quantum problem Hamiltonian through adiabatic preparation-like ansatze. However, these algorithms typically must race against proliferating gate error, limiting their usefulness for problems needing high circuit depths. Drawing on ideas from bath engineering, open quantum systems, and variational algorithms, we develop an algorithm exhibiting continuous, approximate error correction, which we call the Relaxational Quantum Eigensolver (RQE). In RQE, we weakly couple a second register of auxiliary "shadow" qubits to the primary system in Trotterized evolution, thus engineering an approximate zero-temperature bath by periodically resetting the auxiliary qubits during the algorithm's runtime. Balancing the infinite temperature bath of random gate error, RQE returns states with an average energy equal to a constant fraction of the ground state. In this work we focus on better understanding the steady state, its "temperature" T as a function of error rate, and methods for estimating both T and deviations from thermal behavior. This basic proof of concept demonstrates stabilization of finite temperature states of many-body Hamiltonians against random error.