Thermal Multi-scale Entanglement Renormalization Ansatz for Variational Gibbs State Preparation

Many simulation tasks require that one first prepare a system's Gibbs state. We present a family of quantum circuits for variational preparation of thermal Gibbs states on a quantum computer; we call them the thermal multi-scale entanglement renormalization ansatz (TMERA). TMERA circuits transform input qubits to wavepacket modes localized to varying length scales and approximate a systems Gibbs state as a mixed state of these modes. The TMERA is a based on the deep multi-scale entanglement renormalization ansatz (DMERA); a TMERA modifies a ground-state DMERA circuit by preparing each input qubit as a mixed state. The excitation probabilities for input qubits serve as variational parameters used to target particular temperature Gibbs states. Since a TMERA is a special case of the product spectrum ansatz for thermal states, it is simple to prepare, analyze, and optimize. We benchmark the TMERA on the transverse field Ising model in one dimension and find that for $D=6$ it produces global fidelities $\mathcal F > 0.4$ for 512-site systems across all temperatures.