Martingale Strategy for Modeling Quantum Adiabatic Evolution

TL;DR

This paper introduces a 'martingale' strategy based on a $1/Z$ expansion to efficiently model adiabatic quantum evolution, reducing computational complexity while capturing key system behaviors.

Contribution

It presents a novel $1/Z$ expansion approach combined with a martingale strategy to model quantum adiabatic evolution with reduced computational costs.

Findings

The method can identify when the energy gap closes.

It reduces computational complexity from exponential to polynomial.

Illustrated with two-spin and multi-qubit systems.

Abstract

We propose a strategy for modeling the behavior of an adiabatic quantum computer described by an Ising Hamiltonian with $N$ sites and the coordination number $Z$. The method is based on the $1/Z$ expansion for the density matrix of the system. In each order, the ground state energy is found neglecting the higher-order correlations between the sites, as long as the set of equations remains non-singular. The conditions of the appearance of a singularity, equivalent to the disappearance of energy gap in the given approximation, can be directly obtained from the equations. Then the next order in the expansion must be used, at the price of an $N$-fold increase in computational resources. This "martingale" strategy allows reducing the computational costs to a power of $N$ rather than $2^N$, with a finite probability of success. The strategy is illustrated by the case of a two-spin system and extended to a large number of qubits. Comparing the predictions to the experimental results obtained by using an adiabatic quantum computer would help quantify the importance of multi-site correlations, and the influence of decoherence, on its operation.