  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.
