Solving Free Fermion Problems on a Quantum Computer

Abstract

Simulating noninteracting fermion systems is a common task in computational many-body physics. In absence of translational symmetries, modeling free fermions on $N$ modes usually requires poly$(N)$ computational resources. While often moderate, these costs can be prohibitive in practice when large systems are considered. We present several free-fermion problems that can be solved by a quantum algorithm with substantially reduced computational costs. The memory costs are exponentially improved, poly log$(N)$. The runtime improvement, compared to the best known classical algorithms, is either exponential or significantly polynomial, depending on the geometry of the problem. The simulation of free-fermion dynamics belongs to the BQP-hard complexity class. This implies (under standard assumptions) that our algorithm yields an exponential speedup for any classical algorithm at least for some geometries. The key technique in our algorithm is the block-encoding of objects such as correlation matrices and Green's functions into a unitary. We demonstrate how such unitaries can be efficiently realized as quantum circuits, in the context of dynamics and thermal states of tight-binding Hamiltonians. The special cases of disordered and inhomogeneous lattices, as well as large non-lattice graphs, are presented in detail. Finally, we show that our simulation algorithm generalizes to other promising targets, including free boson systems.