Sparse random Hamiltonians are quantumly easy

TL;DR

This paper demonstrates that for most sparse random Hamiltonians, quantum phase estimation can efficiently prepare low-energy states from the maximally mixed state, revealing their quantum computational simplicity.

Contribution

It shows that the maximally mixed state is a good trial state for phase estimation on most sparse random Hamiltonians, providing theoretical guarantees and complexity insights.

Findings

Phase estimation efficiently finds low-energy states for most sparse random Hamiltonians.

The maximally mixed state serves as an effective initial trial state.

Low-energy states have nonnegligible quantum circuit complexity.

Abstract

A candidate application for quantum computers is to simulate the low-temperature properties of quantum systems. For this task, there is a well-studied quantum algorithm that performs quantum phase estimation on an initial trial state that has a nonnegligible overlap with a low-energy state. However, it is notoriously hard to give theoretical guarantees that such a trial state can be prepared efficiently. Moreover, the heuristic proposals that are currently available, such as with adiabatic state preparation, appear insufficient in practical cases. This paper shows that, for most random sparse Hamiltonians, the maximally mixed state is a sufficiently good trial state, and phase estimation efficiently prepares states with energy arbitrarily close to the ground energy. Furthermore, any low-energy state must have nonnegligible quantum circuit complexity, suggesting that low-energy states are classically nontrivial and phase estimation is the optimal method for preparing such states (up to polynomial factors). These statements hold for two models of random Hamiltonians: (i) a sum of random signed Pauli strings and (ii) a random signed $d$-sparse Hamiltonian. The main technical argument is based on some new results in nonasymptotic random matrix theory. In particular, a refined concentration bound for the spectral density is required to obtain complexity guarantees for these random Hamiltonians.