  A new computational method for finite-temperature properties of strongly
correlated electrons is proposed by extending the variational Monte Carlo
method originally developed for the ground state. The method is based on the
path integral in the imaginary-time formulation, starting from the
infinite-temperature state that is well approximated by a small number of
certain random initial states. Lower temperatures are progressively reached by
the imaginary-time evolution. The algorithm follows the framework of the
quantum transfer matrix and finite-temperature Lanczos methods, but we extends
them to treat much larger system sizes without the negative sign problem by
optimizing the truncated Hilbert space on the basis of the time-dependent
variational principle (TDVP). This optimization algorithm is equivalent to the
stochastic reconfiguration (SR) method that has been frequently used for the
ground state to optimally truncate the Hilbert space. The obtained
finite-temperature states allow an interpretation based on the thermal pure
quantum (TPQ) state instead of the conventional canonical-ensemble average. Our
method is tested for the one- and two-dimensional Hubbard models and its
accuracy and efficiency are demonstrated.
