Benchmarking a heuristic Floquet adiabatic algorithm for the Max-Cut problem

TL;DR

This paper introduces a Floquet adiabatic algorithm with fixed Trotter steps for Max-Cut, significantly reducing gate counts and demonstrating effective classical simulation, with implications for quantum computational resources.

Contribution

It proposes a novel Floquet adiabatic approach with fixed Trotter steps, enabling efficient classical simulation and potential quantum advantage for Max-Cut.

Findings

Reduces gate count by orders of magnitude compared to continuous adiabatic evolution.

Successfully solves Max-Cut on 3-regular graphs with low bond dimensions.

Provides resource estimates for quantum computers to outperform classical solvers.

Abstract

According to the adiabatic theorem of quantum mechanics, a system initially in the ground state of a Hamiltonian remains in the ground state if one slowly changes the Hamiltonian. This can be used in principle to solve hard problems on quantum computers. Generically, however, implementation of this Hamiltonian dynamics on digital quantum computers requires scaling Trotter step size with system size and simulation time, which incurs a large gate count. In this work, we argue that for classical optimization problems, the adiabatic evolution can be performed with a fixed, finite Trotter step. This "Floquet adiabatic evolution" reduces by several orders of magnitude the gate count compared to the usual, continuous-time adiabatic evolution. We give numerical evidence using matrix-product-state simulations that it can optimally solve the Max-Cut problem on $3$-regular graphs in a large number of instances, with surprisingly low runtime, even with bond dimensions as low as $D=2$. Extrapolating our numerical results, we estimate the resources needed for a quantum computer to compete with classical exact or approximate solvers for this specific problem.