Quantum random power method for ground state computation

TL;DR

This paper introduces a hybrid quantum-classical random power method that efficiently approximates ground states of Hamiltonians, leveraging quantum polynomial filtering and classical iterative algorithms, with proven convergence and noise robustness.

Contribution

It presents a novel hybrid approach combining quantum filtering with classical iteration for ground state computation, offering advantages over purely classical methods.

Findings

Method converges to ground state approximation with high probability.

Quantum circuit depth is independent of initial overlap and scales favorably with spectral gap.

Numerical validation shows robustness to quantum noise and sparsity conditions.

Abstract

We present a quantum-classical hybrid random power method that approximates a ground state of a Hamiltonian. The quantum part of our method computes a fixed number of elements of a Hamiltonian-matrix polynomial via quantum polynomial filtering techniques with either Hamiltonian simulation or block encoding. The use of the techniques provides a computational advantage that may not be achieved classically in terms of the degree of the polynomial. The classical part of our method is a randomized iterative algorithm that takes as input the matrix elements computed from the quantum part and outputs an approximation of ground state of the Hamiltonian. We prove that with probability one, our method converges to an approximation of a ground state of the Hamiltonian, requiring a constant scaling of the per-iteration classical complexity. The required quantum circuit depth is independent of the initial overlap and has no or a square-root dependence on the spectral gap. The iteration complexity scales linearly as the dimension of the Hilbert space when the quantum polynomial filtering corresponds to a sparse matrix. We numerically validate this sparsity condition for well-known model Hamiltonians. We also present a lower bound of the fidelity, which depends on the magnitude of noise occurring from quantum computation regardless of its charateristics, if it is smaller than a critical value. Several numerical experiments demonstrate that our method provides a good approximation of ground state in the presence of systematic and/or sampling noise.