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

Fernando G. S. L. Brandão, Michael Broughton+3 more|arXiv (Cornell University)|Dec 10, 2018

Quantum Computing Algorithms and Architecture61 citations

TL;DR

The paper shows that for fixed QAOA depth p, the objective value concentrates over typical problem instances drawn from a distribution, implying the landscape is largely instance-independent and enabling potential parameter reuse across instances.

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.

Motivation & Objective

• Motivate studying QAOA performance on random problem instances.
• Show that the QAOA objective value concentrates across typical instances at fixed depth.
• Demonstrate instance-independence of the objective landscape for practical problem classes.

Proposed method

• Express the QAOA state as alternating U(B, beta) and U(C, gamma) applied to the uniform state |s>.
• Decompose the cost expectation F_p(gamma, beta) as a sum over local edge contributions for MaxCut.
• Analyze the p=1, 3-regular MaxCut case to prove concentration for large n.
• Extend arguments to higher depth circuits using the Law of Large Numbers and numerical evidence.
• Provide numerical simulations to illustrate concentration and parameter transfer across sizes.
• Discuss implications for reducing outer-loop optimization and reuse of parameters.

Experimental results

Research questions

• RQ1 Does the QAOA objective value concentrate over typical instances for fixed p and large n?
• RQ2 Is the concentration phenomenon instance-independent for the problem classes considered?
• RQ3 How does the depth p affect concentration, and can this be observed numerically beyond p=1?
• RQ4 Can concentration enable parameter reuse across different instance sizes or distributions?
• RQ5 What implications do these results have for practical optimization strategies on near-term quantum devices?

Key findings

• For 3-regular MaxCut graphs at depth p=1, the QAOA objective concentrates as n grows.
• Numerical results indicate that concentration persists for higher depths, and the authors argue this via the Law of Large Numbers.
• The landscape becomes nearly identical across typical instances, not just at optimal parameters.
• Simulations suggest parameters found at smaller bit-precision (e.g., 10 bits) perform well at larger sizes (e.g., 24 bits).
• This concentration implies potential strategies to reduce or eliminate outer-loop optimization and to obtain good solutions with fewer quantum computer calls.