For Fixed Control Parameters the Quantum Approximate Optimization Algorithm's Objective Function Value Concentrates for Typical Instances

TL;DR

This paper proves that for fixed parameters, the QAOA objective function concentrates around a typical value for many instances, simplifying optimization and potentially reducing quantum resource requirements.

Contribution

It demonstrates that the QAOA objective function concentrates for fixed parameters across typical instances, with proofs for low-depth circuits and numerical validation for higher depths.

Findings

Objective function concentrates for fixed parameters across instances.

Concentration holds for low-depth MaxCut on large 3-regular graphs.

Numerical evidence suggests concentration persists at higher depths.

Abstract

The Quantum Approximate Optimization Algorithm, QAOA, uses a shallow depth quantum circuit to produce a parameter dependent state. For a given combinatorial optimization problem instance, the quantum expectation of the associated cost function is the parameter dependent objective function of the QAOA. We demonstrate that if the parameters are fixed and the instance comes from a reasonable distribution then the objective function value is concentrated in the sense that typical instances have (nearly) the same value of the objective function. This applies not just for optimal parameters as the whole landscape is instance independent. We can prove this is true for low depth quantum circuits for instances of MaxCut on large 3-regular graphs. Our results generalize beyond this example. We support the arguments with numerical examples that show remarkable concentration. For higher depth circuits the numerics also show concentration and we argue for this using the Law of Large Numbers. We also observe by simulation that if we find parameters which result in good performance at say 10 bits these same parameters result in good performance at say 24 bits. These findings suggest ways to run the QAOA that reduce or eliminate the use of the outer loop optimization and may allow us to find good solutions with fewer calls to the quantum computer.