[Paper Review] Iterative Quantum Algorithms for Maximum Independent Set: A Tale of Low-Depth Quantum Algorithms

Lucas T. Brady, Stuart Hadfield|arXiv (Cornell University)|Sep 22, 2023

Quantum Computing Algorithms and ArchitectureComputer Science3 citations

TL;DR

This paper introduces Iterative Quantum Algorithms (IQA) for the Maximum Independent Set (MIS) problem, generalizing the Recursive QAOA framework. It shows that at depth $ p=1 $, IQA performs identically to classical greedy algorithms, but for $ p>1 $, quantum-enhanced iterative steps yield improved performance, demonstrating that quantum advantage in near-term devices may stem from hybrid classical-quantum design rather than pure quantum speedup.

ABSTRACT

Quantum algorithms have been widely studied in the context of combinatorial optimization problems. While this endeavor can often analytically and practically achieve quadratic speedups, theoretical and numeric studies remain limited, especially compared to the study of classical algorithms. We propose and study a new class of hybrid approaches to quantum optimization, termed Iterative Quantum Algorithms, which in particular generalizes the Recursive Quantum Approximate Optimization Algorithm. This paradigm can incorporate hard problem constraints, which we demonstrate by considering the Maximum Independent Set (MIS) problem. We show that, for QAOA with depth $p=1$, this algorithm performs exactly the same operations and selections as the classical greedy algorithm for MIS. We then turn to deeper $p>1$ circuits and other ways to modify the quantum algorithm that can no longer be easily mimicked by classical algorithms, and empirically confirm improved performance. Our work demonstrates the practical importance of incorporating proven classical techniques into more effective hybrid quantum-classical algorithms.

Motivation & Objective

• To develop a general framework for iterative quantum optimization (IQA) applicable to combinatorial problems like MIS.
• To investigate whether iterative quantum algorithms can outperform classical counterparts by leveraging quantum enhancements in hybrid settings.
• To analyze the performance of QAOA-based IQA at low circuit depth ($ p=1 $) and compare it to classical greedy algorithms.
• To explore how increasing quantum circuit depth ($ p>1 $) enables access to richer structural information, improving solution quality beyond classical heuristics.
• To assess the potential for quantum advantage in near-term quantum devices by embedding proven classical techniques into quantum hybrid workflows.

Proposed method

• Proposes a novel iterative quantum algorithm for MIS based on QAOA, where each iteration reduces graph size using quantum-assisted vertex selection.
• Uses QAOA with depth $ p=1 $ to compute expectation values for vertex selection, showing equivalence to classical greedy selection rules.
• Employs a recursive reduction strategy: select a vertex based on quantum output, remove it and its neighbors, and repeat on the remaining graph.
• Extends the framework to higher-depth QAOA ($ p>1 $), enabling access to larger local neighborhoods and improved solution quality.
• Introduces variants like MINQ and MAXQ that use quantum decisions to guide bit-fixing in iterative steps, generalizing classical greedy heuristics.
• Employs empirical evaluation on Erdős–Rényi graphs to compare approximation ratios across classical, non-iterative QAOA, and iterative quantum algorithms.

Experimental results

Research questions

• RQ1 Does the iterative quantum algorithm with $ p=1 $ QAOA achieve the same results as classical greedy algorithms for MIS?
• RQ2 Can deeper QAOA circuits ($ p>1 $) provide improved performance over classical greedy algorithms in the iterative quantum framework?
• RQ3 To what extent do quantum-enhanced iterative algorithms benefit from classical algorithmic insights, and how does this affect claims of quantum advantage?
• RQ4 How does the performance of iterative quantum algorithms scale with graph size and edge density compared to non-iterative QAOA?
• RQ5 Can the iterative quantum approach be generalized to other NP-hard problems via problem transformations, such as Maximum Clique or Minimum Vertex Cover?

Key findings

• For $ p=1 $, the iterative quantum algorithm performs identically to the classical greedy algorithm, making the same vertex selections at each iteration.
• At $ p=1 $, the quantum algorithm accesses no additional information beyond what is available to classical greedy algorithms, and thus offers no quantum advantage in this regime.
• For $ p>1 $, the quantum algorithm accesses information about larger local neighborhoods, enabling improved solution quality over classical greedy algorithms.
• Empirical results show that iterative quantum algorithms with $ p=2 $ achieve significantly higher approximation ratios than non-iterative QAOA, which drops from 0.69 at $ n=5 $ to 0.495 at $ n=16 $.
• The performance gap between iterative quantum algorithms and classical greedy algorithms increases with circuit depth, suggesting that quantum enhancement arises from richer information access.
• The framework is compatible with various quantum algorithms, including QAOA variants, hardware-efficient ansätze, and quantum annealing, enabling broad applicability to optimization problems.