Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices (1812.01041v2)

Abstract: The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical variational algorithm designed to tackle combinatorial optimization problems. Despite its promise for near-term quantum applications, not much is currently understood about QAOA's performance beyond its lowest-depth variant. An essential but missing ingredient for understanding and deploying QAOA is a constructive approach to carry out the outer-loop classical optimization. We provide an in-depth study of the performance of QAOA on MaxCut problems by developing an efficient parameter-optimization procedure and revealing its ability to exploit non-adiabatic operations. Building on observed patterns in optimal parameters, we propose heuristic strategies for initializing optimizations to find quasi-optimal $p$-level QAOA parameters in $O(\text{poly}(p))$ time, whereas the standard strategy of random initialization requires $2^{O(p)}$ optimization runs to achieve similar performance. We then benchmark QAOA and compare it with quantum annealing, especially on difficult instances where adiabatic quantum annealing fails due to small spectral gaps. The comparison reveals that QAOA can learn via optimization to utilize non-adiabatic mechanisms to circumvent the challenges associated with vanishing spectral gaps. Finally, we provide a realistic resource analysis on the experimental implementation of QAOA. When quantum fluctuations in measurements are accounted for, we illustrate that optimization will be important only for problem sizes beyond numerical simulations, but accessible on near-term devices. We propose a feasible implementation of large MaxCut problems with a few hundred vertices in a system of 2D neutral atoms, reaching the regime to challenge the best classical algorithms.

• The paper shows that QAOA, when applied to MaxCut problems, identifies optimal parameter patterns enabling polynomial-time heuristic strategies for efficient optimization.
• The paper demonstrates that QAOA yields exponential improvements in average performance for unweighted graphs and stretched-exponential gains for weighted graphs compared to traditional methods.
• The paper reveals that QAOA overcomes limitations of quantum annealing by leveraging non-adiabatic transitions, offering a practical route for near-term quantum advantage.

Quantum Approximate Optimization Algorithm: Performance and Practical Considerations

The paper examines the Quantum Approximate Optimization Algorithm (QAOA), emphasizing its potential for combinatorial optimization on near-term quantum devices. Despite its limited depth, QAOA's performance surpasses traditional quantum annealing, especially in complex scenarios where small spectral gaps hinder adiabatic processes.

Key Findings and Methodological Insights

1. Parameter Optimization and Patterns: The research identifies specific patterns in the optimal parameters of QAOA when applied to MaxCut problems. Optimal parameters demonstrate smooth variations, which have been exploited to develop heuristic strategies for efficient optimization. These strategies, termed INTERP and FOURIER, allow parameter optimization in polynomial time, contrasting with the exponential scaling required for random initialization.
2. Performance Beyond Low Depth: Numerical simulations reveal that QAOA's average performance improves exponentially for unweighted graphs and stretched-exponentially for weighted graphs. The results underscore QAOA's applicability even at intermediate depths (i.e., 1<p<∞), bridging the gap between theoretical promise and practical implementation.
3. Comparison with Quantum Annealing: On instances where quantum annealing fails due to small spectral gaps, QAOA efficiently finds solutions by leveraging non-adiabatic transitions. This ability to learn and optimize beyond the adiabatic limit challenges prevalent assumptions about the necessity of slow evolution for success in quantum algorithms.
4. Experimental Considerations: In real-world implementations, finite measurement samples introduce projection noise, affecting the precision and measurement costs. The paper considers these computational costs, providing an operational framework for large-scale implementations, notably with platforms like neutral Rydberg atoms.
5. Resource Analysis for Implementation: The implementation of QAOA with physical systems, such as Rydberg atoms, is projected to be feasible for problems with hundreds of vertices. The architecture leverages protocols to manage interaction patterns and coherence time limitations, proposing a pathway toward realizing a quantum advantage.

Implications and Future Directions

The results indicate that QAOA could be superior to classical algorithms for solving specific instances of combinatorial optimization problems, particularly those characterized by complex interactions and narrow gaps. The practical implications extend to quantum computing architectures aiming to solve real-world problems within feasible coherence times.

The methodology emphasizes the need for further research into heuristic optimization strategies and highlights the role of non-adiabatic transitions in achieving quantum speedup. Moving forward, experimental demonstrations with larger quantum systems will be pivotal in validating the theoretical outcomes and benchmarking against classical state-of-the-art solutions.

In conclusion, QAOA presents a promising algorithmic advance for near-term quantum computation, with the potential to address complex optimization tasks efficiently. By leveraging parameter patterns and innovative heuristic strategies, QAOA bridges theoretical potential and practical implementation, paving the way for significant developments in quantum algorithms.