Solving Free Fermion Problems on a Quantum Computer

The simulation of time-dynamics and thermal states of free fermions on N = 2^n modes are known to require poly(2^n) computational classical resources. We present several such free fermion problems that can be solved by a quantum algorithm with exponentially-improved, poly(n) cost. The key technique is the block-encoding of the correlation matrix into a unitary. We demonstrate how such a unitary can be efficiently realized as a quantum circuit, in the context of dynamics and thermal states of tight-binding Hamiltonians. We prove that the problem of free fermion time-dynamics is BQP-complete, thus ensuring a general exponential speedup of our approach.