A Quantum Approximate Optimization Algorithm (1411.4028v1)

Abstract: We introduce a quantum algorithm that produces approximate solutions for combinatorial optimization problems. The algorithm depends on a positive integer p and the quality of the approximation improves as p is increased. The quantum circuit that implements the algorithm consists of unitary gates whose locality is at most the locality of the objective function whose optimum is sought. The depth of the circuit grows linearly with p times (at worst) the number of constraints. If p is fixed, that is, independent of the input size, the algorithm makes use of efficient classical preprocessing. If p grows with the input size a different strategy is proposed. We study the algorithm as applied to MaxCut on regular graphs and analyze its performance on 2-regular and 3-regular graphs for fixed p. For p = 1, on 3-regular graphs the quantum algorithm always finds a cut that is at least 0.6924 times the size of the optimal cut.

• The paper introduces QAOA as a quantum algorithm that employs alternating unitary operations to explore combinatorial spaces and enhance approximation quality.
• It demonstrates that tuning the parameter p improves circuit depth and performance, achieving near-optimal cuts in regular graphs.
• The study sets a benchmark for quantum-classical optimization, offering both theoretical insights and practical implications for complex problem solving.

Overview of "A Quantum Approximate Optimization Algorithm"

This paper presents a quantum algorithm designed to yield approximate solutions for combinatorial optimization problems, notably the MaxCut problem. Developed by Edward Farhi, Jeffrey Goldstone, and Sam Gutmann, the algorithm—termed the Quantum Approximate Optimization Algorithm (QAOA)—stands out for its dependency on a tunable integer parameter $p$, with the quality of the approximation improving as $p$ increases.

Core Contributions

1. Algorithm Structure: The QAOA applies a sequence of unitary gates to an initial quantum state, facilitating the exploration of combinatorial problem spaces. Specifically, it utilizes alternating layers of operators associated with the objective function and a mixing operator. The quantum circuit's depth is proportional to $p$ and scales linearly with the constraints in the problem.
2. Parameter $p$ and Performance: The algorithm's performance is dependent on the choice of $p$. For instance, with fixed $p$, classical preprocessing can efficiently determine optimal parameters for the quantum circuit. This is crucial in assessing the algorithm's effectiveness in identifying near-optimal solutions.
3. Application to MaxCut: The paper provides a detailed exploration of how the QAOA can be applied to MaxCut problems, particularly on regular graphs. Notable results include proving that for $p = 1$ in 3-regular graphs, the algorithm can achieve a solution that is at least 0.6924 times the size of the optimal cut.

Theoretical Implications

The algorithm links to the broader context of quantum computing's potential to solve optimization problems more efficiently than classical methods. The theoretical backbone is rooted in quantum mechanics and leverages Hilbert space properties to tackle computationally intensive tasks. This establishes a framework where quantum gates are applied iteratively, and their effects are measured to optimize combinatorial objectives.

Practical Implications

Practically, the QAOA suggests new pathways to address optimization challenges in fields ranging from operations research to machine learning. As quantum hardware continues to evolve, the feasibility of implementing these algorithms on real-world devices becomes increasingly viable. The ability to improve approximation ratios while maintaining a manageable quantum circuit depth places QAOA as a promising candidate for near-term quantum advantage cases.

Future Directions

Future research can extend this work by exploring:

• Scalability: As quantum computers grow more powerful, understanding how QAOA scales with larger problem instances will be critical.
• Algorithm Variants: Investigating alternate configurations and enhancements to QAOA, such as adaptive or problem-specific parameter tuning techniques.
• Comparison with Classical Approaches: Establishing clear benchmarks to evaluate how quantum approximate solutions fare against the best classical algorithms across different problems and parameters.

This paper lays the groundwork for ongoing explorations into quantum algorithms that have the potential to offer significant improvements over classical techniques in solving complex optimization challenges.