Classical Combinatorial Optimization Scaling for Random Ising Models on 2D Heavy-Hex Graphs

TL;DR

This paper investigates the classical computational hardness of Ising models on heavy-hex graphs, revealing that classical algorithms can efficiently solve sparse instances, thus emphasizing the need for more complex problems to demonstrate quantum advantage.

Contribution

It provides empirical analysis of classical scaling for solving specific Ising models relevant to near-term quantum hardware, highlighting the importance of problem complexity in quantum advantage.

Findings

Gurobi solves large sparse Ising models in linear or weakly quadratic time.

Simulated annealing exhibits exponential scaling on certain Ising models.

Classical algorithms can efficiently solve these models, questioning their use as benchmarks for quantum advantage.

Abstract

Motivated by near term quantum computing hardware limitations, combinatorial optimization problems that can be addressed by current quantum algorithms and noisy hardware with little or no overhead are used to probe capabilities of quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA). In this study, a specific class of near term quantum computing hardware defined combinatorial optimization problems, Ising models on heavy-hex graphs both with and without geometrically local cubic terms, are examined for their classical computational hardness via empirical computation time scaling quantification. Specifically the Time-to-Solution metric using the classical heuristic simulated annealing is measured for finding optimal variable assignments (ground states), as well as the time required for the optimization software Gurobi to find an optimal variable assignment. Because of the sparsity of these Ising models, the classical algorithms are able to find optimal solutions efficiently even for large instances (i.e. $100,000$ variables). The Ising models both with and without geometrically local cubic terms exhibit average-case linear-time or weakly quadratic scaling when solved exactly using Gurobi, and the Ising models with no cubic terms show evidence of exponential-time Time-to-Solution scaling when sampled using simulated annealing. These findings point to the necessity of developing and testing more complex, namely more densely connected, optimization problems in order for quantum computing to ever have a practical advantage over classical computing. Our results are another illustration that different classical algorithms can indeed have exponentially different running times, thus making the identification of the best practical classical technique important in any quantum computing vs. classical computing comparison.