For Fixed Control Parameters the Quantum Approximate Optimization Algorithm's Objective Function Value Concentrates for Typical Instances (1812.04170v1)

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.

• The paper shows that fixed QAOA control parameters lead to nearly constant objective function values across typical instances on 3-regular graphs.
• It employs both theoretical analysis and numerical simulations to validate the concentration phenomenon using graph theory and the Law of Large Numbers.
• Moreover, the findings indicate that parameters optimized on smaller instances can be effectively scaled to larger cases, lowering computational costs.

Analysis of Objective Function Concentration in QAOA

The paper "For Fixed Control Parameters the Quantum Approximate Optimization Algorithm's (QAOA) Objective Function Value Concentrates for Typical Instances" presents a theoretical and empirical investigation into the behavior of QAOA's objective function across typical instances of combinatorial optimization problems. The work primarily explores the concentration phenomena observed when the algorithm is employed on families of instances of the MaxCut problem on graphs, particularly focusing on 3-regular graphs.

The paper demonstrates that when applying the QAOA with fixed control parameters, the resultant objective function values exhibit concentration, meaning they remain nearly constant for most problem instances derived from a reasonable distribution. This concentration is observed not only for the parameters that optimize the QAOA performance but generally across the parameter landscape, implying an independence from specific instance details to a significant extent. Such findings promise practical advantages in reducing computational costs, as the need to optimize parameters individually for each instance can be bypassed.

The authors focus on the MaxCut problem, particularly on large 3-regular graphs using low-depth quantum circuits. They show that, under the assumption of typical instances being drawn from suitable distributions, instance-independence holds for the parameter-dependent objective function. For verifying these theoretical insights, the paper includes numerical simulations that show strong concentration even for higher-depth quantum circuits. Through these simulations, the authors encapsulate the remarkable predictive utility of these concentrated objective function values in determining the quantum expectation — a crucial insight for improving computational efficiency.

Another significant observation discussed in the paper is that parameters optimized at lower bit instances tend to hold their utility when applied to higher bit instances. This suggests a promising strategy for practitioners aiming to scale QAOA without exhaustive re-optimization: parameters derived from smaller test cases can be extended to larger instances, maintaining performance fidelity. This not only simplifies and hastens the parameter search process but also underscores how low-cost computational experiments could yield insights applicable to larger, more complex quantum systems.

From a theoretical standpoint, the findings advance understanding by leveraging graph theory results relating to concentration of graph properties, supported by the Law of Large Numbers, providing a robust framework for assessing the correlation structure across graph instances. Practically, these insights could guide future work to refine outer-loop optimization processes in QAOA implementations, a step towards making quantum computers more pragmatic for combinatorial search problems by reducing instance-specific optimization efforts.

Speculatively, this work could significantly influence the future of heuristic quantum algorithms, potentially leading to methods combining analytical insights and learning-based approaches to enhance instance generation, shedding further light on the nature of quantum optimization landscapes. A synergistic approach capitalizing on known fixed-parameter behaviors could guide the design and testing of algorithms, aligning physical quantum device constraints with algorithmic adaptability. The broad applicability beyond MaxCut to other constrained graph problems further points to exciting avenues for future research.

Overall, the paper's analytical approaches and empirical confirmations offer a promising trajectory for effectively utilizing variational quantum algorithms in an optimized and resource-efficient way, emphasizing advancements in algorithmic strategies that are rationalized from both theoretical and practical perspectives.