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. Layden et al. [Nature 619, 282 (2023)] proposed a promising application by introducing a Quantum-enhanced Markov Chain Monte Carlo (QeMCMC) approach to reduce the thermalization time required when sampling from hard probability distributions. In QeMCMC 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 considerably 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 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; in this case only 6 simulated qubits suffice to gain advantage compared to standard classical approaches when investigating the magnetization of a 36 spin system. Our method can be easily combined with other classical and quantum techniques and is adaptable to various quantum hardware specifications -- in particular those with limited connectivity.