Methods for non-variational heuristic quantum optimisation

TL;DR

The paper addresses the need for non-variational quantum heuristics for combinatorial optimisation and proposes two hybrid algorithms, Quantum-enhanced Simulated Annealing (QeSA) and Quantum-enhanced Parallel Tempering (QePT), built on quantum real-time evolution applied to classical MCMC. It provides a framework, definitions, and proof-of-principle numerics on Sherrington-Kirkpatrick instances, suggesting potential quantum advantages in exploration and thermalisation while emphasizing robustness to noise and parallelizability on near-term devices. The work distinguishes its approach from fault-tolerant quantum speedups by employing a quantum subroutine for proposals within predominantly classical processes, and it discusses practical HPC-QC considerations such as heterogeneous chain enhancement and hardware overheads. Overall, the results indicate a promising direction for scalable quantum-augmented optimisation, with clear future work needed to quantify speedups and validate performance on real quantum hardware.

Abstract

Optimisation plays a central role in a wide range of scientific and industrial applications, and quantum computing has been widely proposed as a means to achieve computational advantages in this domain. To date, research into the design of noise-resilient quantum algorithms has been dominated by variational approaches, while alternatives remain relatively unexplored. In this work, we introduce a novel class of quantum optimisation heuristics that forgo this variational framework in favour of a hybrid quantum-classical approach built upon Markov Chain Monte Carlo (MCMC) techniques. We introduce Quantum-enhanced Simulated Annealing (QeSA) and Quantum-enhanced Parallel Tempering (QePT), before validating these heuristics on hard Sherrington-Kirkpatrick instances and demonstrate their superior scaling over classical benchmarks. These algorithms are expected to exhibit inherent robustness to noise and support parallel execution across both quantum and classical resources with only classical communication required. As such, they offer a scalable and potentially competitive route toward solving large-scale optimisation problems with near-term quantum devices.

Figures (10)

• Figure 1: Visual representation of MCMC, SA and PT. MCMC attempts to thermalize a single Markov chain of a given temperature, $T_{\text{low}}$. SA starts at a high temperature and anneals to $T_{\text{low}}$, while PT runs multiple chains in parallel at different temperatures, swapping configurations between chains occasionally.
• Figure 2: The probability of finding global minima, against the number of steps taken. Classical SA (green) quickly finds the global minima occasionally, however if it is not in the local region of the initial state, then it gets stuck in local minima and can take an extremely long time to escape. QeSA (blue), however, does not suffer from such locality, and longer annealing runs are rewarded with a very high probability of finding the global minima. Each data point is the average of 100 different models, where 100 anneals are performed for each step value.
• Figure 3: Effort (Eq. \ref{['eqn:comb_opt']}) required to find the global minima against the number of Markov chain steps for $n=10$ for SA (green) and QeSA (blue). Although the classical approach works well when repeating many short anneals, the quantum approach can quickly reduce the effort by performing fewer, longer anneals. A subset of low effort values are selected and a simple quadratic function (black) over these are used to approximate the optimal effort in each case. The optimal number of steps, represented by vertical dotted lines, are $\ell_{SA} \approx 43$ and $\ell_{QeSA} \approx 151$
• Figure 4: Optimal effort for SA (green, circle) and QeSA (blue, triangle) as the number of spins is increased. For each $n$, the optimal $\ell$ is found (as in Fig. \ref{['fig:10_effort']}), and $100$ new randomly initialised Ising models are then used to generate this figure, with each anneal now being repeated $1000$ times.
• Figure 5: Comparison between PT (left) and QePT (right) which displays our hypothesis that not all Markov chains must be Quantum-enhanced to produce a quantum speed-up.