[Paper Review] Sub-universal variational circuits for combinatorial optimization problems

Gal Weitz, Lirandë Pira+2 more|arXiv (Cornell University)|Aug 29, 2023

Quantum Computing Algorithms and ArchitectureComputer Science3 citations

TL;DR

This paper introduces PAOA (Probabilistic Approximate Optimization Algorithm), a classical variational circuit using two-bit stochastic matrices to solve combinatorial optimization problems like Max-Cut. It demonstrates that PAOA outperforms QAOA on various graph types, especially larger ones, achieving higher approximation ratios with minimal hyperparameter tuning, suggesting that sub-universal classical circuits may serve as strong benchmarks for quantum advantage claims.

ABSTRACT

Quantum variational circuits have gained significant attention due to their applications in the quantum approximate optimization algorithm and quantum machine learning research. This work introduces a novel class of classical probabilistic circuits designed for generating approximate solutions to combinatorial optimization problems constructed using two-bit stochastic matrices. Through a numerical study, we investigate the performance of our proposed variational circuits in solving the Max-Cut problem on various graphs of increasing sizes. Our classical algorithm demonstrates improved performance for several graph types to the quantum approximate optimization algorithm. Our findings suggest that evaluating the performance of quantum variational circuits against variational circuits with sub-universal gate sets is a valuable benchmark for identifying areas where quantum variational circuits can excel.

Motivation & Objective

• To develop a classical analog to quantum variational circuits that can fairly benchmark quantum algorithms like QAOA.
• To investigate whether sub-universal classical circuits can match or exceed the performance of quantum variational circuits on combinatorial optimization problems.
• To provide a practical, hardware-implementable classical alternative to quantum approaches like QAOA for near-term applications.
• To evaluate the performance of variational circuits with restricted gate sets as a proxy for identifying where quantum advantage might emerge.
• To explore the potential of p-bits and probabilistic circuits as cost-effective, high-performance alternatives to quantum solutions.

Proposed method

• Proposes PAOA, a classical variational circuit based on parameterized two-bit stochastic matrices, inspired by QAOA but operating entirely within classical probability space.
• Uses p-bits—classical probabilistic bits that represent mixtures of 0 and 1—enabling physical implementation with current technology.
• Employs gradient-based optimization to train the circuit parameters, similar to QAOA, ensuring a fair comparison.
• Introduces variants like Reduced PAOA and Min PAOA with controlled parameter counts to study the impact of model capacity on optimization performance.
• Tests the method on Max-Cut problems across diverse graph types, including Erdős–Rényi and Barabási–Albert networks, with increasing size.
• Compares results to QAOA, brute force, random guessing, and the Goemans-Williamson algorithm using approximation ratio as the primary metric.

Figure 1: Perspective on the relationship between three model classes considered in this work. Parameterized quantum circuits (PQCs) are the paradigm that represents the largest class of parameterized quantum algorithms. QAOA (Farhi et al. [ 4 ] ) is a popular example. PAOA represents a class of pro

Experimental results

Research questions

• RQ1 Can a classical variational circuit using sub-universal gate sets outperform quantum variational circuits like QAOA on combinatorial optimization problems?
• RQ2 Does a classical probabilistic circuit with p-bits offer a viable, hardware-implementable alternative to near-term quantum algorithms?
• RQ3 What is the impact of model depth and parameter count on the performance of classical variational circuits for Max-Cut?
• RQ4 Can PAOA serve as a fairer benchmark than random guessing or SDP relaxations for evaluating quantum advantage in optimization?
• RQ5 Is there a performance gap between quantum and classical variational circuits that emerges only at scale, despite similar training protocols?

Key findings

• Reduced PAOA achieved an approximation ratio of 0.89 on a 28-node Erdős–Rényi graph, outperforming QAOA (0.71–0.81) and random guessing (0.82).
• On a 20-node Erdős–Rényi graph, Reduced PAOA achieved an average approximation ratio of 0.88, surpassing QAOA (0.84–0.85) and PAOA (0.80).
• PAOA demonstrated consistent performance across different graph types, including Barabási–Albert networks, with reliable improvement over random guessing on graphs up to 100 nodes.
• The Min PAOA variant with one layer achieved a 0.83 approximation ratio on the 20-node Erdős–Rényi graph, indicating strong performance with minimal complexity.
• QAOA performance degraded with increasing depth (e.g., 0.71 at 6 layers), while PAOA variants maintained or improved performance, suggesting better optimization landscape.
• PAOA achieved high-quality results 'out of the box' with no hyperparameter tuning, whereas QAOA required optimization to match performance, indicating PAOA's robustness.

Figure 2: Illustrative example of the Max-Cut algorithms for a 3-node graph $G$ showing both QAOA and PAOA circuits. (Left) $H$ is the Hadamard gate and $p$ is the circuit depth (see Section III for the in-depth explanation). (Right) $R$ denotes a random initial state, and $P$ represents the probabi