Quantum linear algebra for disordered electrons

Abstract: We describe how to use quantum linear algebra to simulate a physically realistic model of disordered non-interacting electrons. The physics of disordered electrons outside of one dimension challenges classical computation due to the critical nature of the Anderson localization transition or the presence of large localization lengths, while the atypical distribution of the local density of states limits the power of disorder averaged approaches. Starting from the block-encoding of a disordered non-interacting Hamiltonian, we describe how to simulate key physical quantities, including the reduced density matrix, Green's function, and local density of states, as well as bulk-averaged observables such as the linear conductivity, using the quantum singular value transformation, quantum amplitude estimation, and trace estimation. We further discuss a quantum advantage that scales polynomially with system size and exponentially with lattice dimension.