Quantum-enhanced Markov Chain Monte Carlo for systems larger than your Quantum Computer

Quantum computers theoretically promise computational advantage in many tasks, but it is much less clear how such advantage can be maintained when using existing and near-term hardware that has limitations in the number and quality of its qubits. One promising application was proposed in Layden et al [Nature 619, 282-287 (2023)] where a method to reduce the thermalisation time required when sampling from hard probability distribution was introduced as a Quantum-enhanced Markov Chain Monte Carlo (QeMCMC) approach. In [Nature 619, 282-287 (2023)] the size of the required quantum computer scales linearly with the problem, putting limitations on the sizes of systems that one can consider. In this work we introduce a framework to coarse grain the algorithm in such a way that the quantum computation can be performed using, multiple times, smaller quantum computers and we term the method the Coarse Grained Quantum-enhanced Markov Chain Monte Carlo (CGQeMCMC). Example strategies within this framework are put to the test, with the quantum speedup of [Nature 619, 282-287 (2023)] persisting while using only $\sqrt{n}$ simulated qubits where $n$ is the number of qubits required in the original QeMCMC -- a quadratic reduction in resources. The coarse graining framework has the potential to be practically applicable in the near term as it requires very few qubits to approach classically intractable problem instances, here only 6 simulated qubits suffice to gain advantage compared to standard classical approaches when investigating the magnetisation of a 36 spin system. Our method is also adjustable to quantum hardware specifications, and it appears that it can be easily combined with other techniques both classical and quantum.