Spectral estimation for Hamiltonians: a comparison between classical imaginary-time evolution and quantum real-time evolution

TL;DR

This paper compares classical Monte Carlo and quantum phase estimation methods for spectral estimation of Hamiltonians, highlighting their efficiencies, limitations, and potential for classical and quantum computational approaches.

Contribution

It introduces a classical Monte Carlo scheme for spectral estimation of stoquastic Hamiltonians and compares its efficiency with quantum methods, providing theoretical bounds and numerical validation.

Findings

Classical Monte Carlo can efficiently estimate a small number of eigenvalues for stoquastic Hamiltonians.

Quantum phase estimation can resolve a polynomial number of eigenvalues with quantum effort.

The paper quantifies the limitations of classical methods for spectral estimation of general Hamiltonians.

Abstract

We present a classical Monte Carlo (MC) scheme which efficiently estimates an imaginary-time, decaying signal for stoquastic (i.e. sign-problem-free) local Hamiltonians. The decay rates in this signal correspond to Hamiltonian eigenvalues (with associated eigenstates present in an input state) and can be classically extracted using a classical signal processing method like ESPRIT. We compare the efficiency of this MC scheme to its quantum counterpart in which one extracts eigenvalues of a general local Hamiltonian from a real-time, oscillatory signal obtained through quantum phase estimation circuits, again using the ESPRIT method. We prove that the ESPRIT method can resolve S = poly(n) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) quantum and classical effort through the quantum phase estimation circuits, assuming efficient preparation of the input state. We prove that our Monte Carlo scheme plus the ESPRIT method can resolve S = O(1) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) purely classical effort for stoquastic Hamiltonians, requiring some access structure to the input state. However, we also show that under these assumptions, i.e. S = O(1) eigenvalues, assuming a 1/poly(n) gap between them and some access structure to the input state, one can achieve this with poly(n) purely classical effort for general local Hamiltonians. These results thus quantify some opportunities and limitations of classical Monte Carlo methods for spectral estimation of Hamiltonians. We numerically compare the MC eigenvalue estimation scheme (for stoquastic Hamiltonians) and the QPE eigenvalue estimation scheme by implementing them for an archetypal stoquastic Hamiltonian system: the transverse field Ising chain.