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Extending the variational quantum eigensolver to finite temperatures
We present a variational quantum thermalizer (VQT), called quantum-VQT (qVQT), which extends the variational quantum eigensolver to finite temperatures. The qVQT makes use of an intermediate measurement between two variational circuits to encode a density matrix on a quantum device. A classical optimization provides the thermal state and, simultaneously, all associated excited states of a quantum mechanical system. We demonstrate the capabilities of the qVQT for two different spin systems. First, we analyze the performance of qVQT as a function of the circuit depth and the temperature for a one-dimensional Heisenberg chain. Second, we use the excited states to map the complete, temperature dependent phase diagram of a two-dimensional J1-J2 Heisenberg model. Numerical experiments on both quantum simulators and real quantum hardware demonstrate the efficiency of our approach, which can be readily applied to study various quantum many-body systems at finite temperatures on currently available noisy intermediate-scale quantum devices.