Quantum-enhanced Markov Chain Monte Carlo for Combinatorial Optimization

TL;DR

This work presents a near-term quantum optimization framework that combines quantum-enhanced MCMC (QeMCMC) with warm-starting and parallel tempering to tackle Maximum Independent Set problems. By using a quantum circuit to generate proposals, a warm-started QAOA-like initialization, and replica exchanges across temperatures, the approach aims to overcome rugged energy landscapes in MIS. Empirically, the method recovers MIS optima on instances up to 117 variables using IBM hardware and shows early scaling advantages over classical MCMC in selected cases, with hardware results sometimes outperforming noisier simulations. The study demonstrates a practical pathway to quantum-assisted optimization on current devices and highlights the value of benchmarking libraries (QOBLIB) for assessing progress toward quantum advantage in optimization.

Abstract

Quantum computing offers an alternative paradigm for addressing combinatorial optimization problems compared to classical computing. Despite recent hardware improvements, the execution of empirical quantum optimization experiments at scales known to be hard for state-of-the-art classical solvers is not yet in reach. In this work, we offer a different way to approach combinatorial optimization with near-term quantum computing. Motivated by the promising results observed in using quantum-enhanced Markov chain Monte Carlo (QeMCMC) for approximating complicated probability distributions, we combine ideas of sampling from the device with QeMCMC together with warm-starting and parallel tempering, in the context of combinatorial optimization. We demonstrate empirically that our algorithm recovers the global optima for instances of the Maximum Independent Set problem (MIS) up to 117 decision variables using 117 qubits on IBM quantum hardware. We show early evidence of a scaling advantage of our algorithm compared to similar classical methods for the chosen instances of MIS. MIS is practically relevant across domains like financial services and molecular biology, and, in some cases, already difficult to solve to optimality classically with only a few hundred decision variables.

Figures (8)

• Figure 1: The algorithm workflow. We see a hot, medium and cold replica working together such that the coldest replica eventually converges to the target objective. Objective values of proposed solutions to the MIS problem are plotted against iterations. Each iteration requires drawing samples from the QPU and then classically processing these counts, necessitating a close exchange of information between quantum and classical hardware.
• Figure 2: Parallel tempering experiment run on ibm_boston for the 117-node MIS problem. The left axis indicates the progress of objective values per iteration. We see the global optimum is found after $151$ iterations, with $10,000$ shots per iteration and a random sample taken from a smaller portion of these. The Hamming distance between accepted solutions is also plotted for each iteration for the winning replica in orange, corresponding to the right axis.
• Figure 3: Comparison of experimental setups for 117 node problem instance experiments. Simulations of the quantum-enhanced algorithm are compared to classical MCMC methods used to find the target objective. In the top plot, we present an MPS simulation of the quantum-enhanced algorithm. Across ten repeats, the target objective is found after a median of $\sim90,000$ iterations for the coldest replica, with $1-$shot per iteration. The middle plot shows a comparison to a classical MCMC simulation, which also uses $1-$shot per iteration. Across ten repeats, the target objective is found in $4,000 - 8,000$ iterations for the coldest replica, with a median of $\sim6,000$ shots, or iterations. Finally, in the bottom plot, we display another classical MCMC simulation, where across ten repeats, the target objective is found in $4,000 - 8,000$ iterations for the coldest replica, with a median of $\sim5,000$ iterations, in this case using $10,000$ shots per iteration and a random sample taken from the best ten proposals.
• Figure 4: A summary of anticipated performance scaling in terms of shots and iterations of the classical vs. quantum-enhanced algorithm. Median iterations to converge are reported for the 17, 52 and 117 node graphs after ten repeated tests of the classical and quantum-enhanced algorithms. A line of best fit provides us with a naïve indication of scaling behavior as to how we anticipate each method to perform with increasing problem instance size. On the left, we present anticipated performance scaling in terms of shots for the classical vs. simulated quantum-enhanced algorithm, using the Qiskit Aer MPS simulator. On the right, we see anticipated performance scaling in terms of iterations of the classical vs. quantum-enhanced algorithm, both simulated and run on real hardware.
• Figure 5: Theoretical distribution for a five node MIS problem instance, for a range of (dimensionless) temperatures $T$.