  In this paper we develop a quantum algorithm to realize finite temperature
simulation on a quantum computer. As quantum computers use real-time evolution
we did not use the imaginary time methods popular on classical algorithms.
Instead, we implemented a real-time thermo field dynamics formalism, which has
the added benefit of being able to compute quantities that are both time- and
temperature-dependent. To implement thermo field dynamics we apply a unitary
transformation to discrete quantum mechanical operators to make new
Hamiltonians with encoded temperature dependence. The method works normally for
fermions, which have a finite representation, but needs some modification to
work with bosons. These Hamiltonians are then processed into a Pauli matrix
representation in order to be used as input for IBM's Qiskit package. We then
use IBM's quantum simulator to calculate an approximation to the Hamiltonaian's
ground state energy via the variational quantum eigensolver (VQE) algorithm.
This approximation is then compared to a classically calculated value for the
exact energy. The thermo field dynamics quantum algorithm has general
applications to material science, high-energy physics and nuclear physics,
particularly in those situations involving real-time evolution at high
temperature.
