Enabling Quantum Speedup of Markov Chains using a Multi-level Approach

Quantum speedup for mixing a Markov chain can be achieved based on the construction of slowly-varying r Markov chains where the initial chain can be easily prepared and the spectral gaps have uniform lower bound. The overall complexity is proportional to r. We present a multi-level approach to construct such a sequence of r Markov chains by varying a resolution parameter h. We show that the density function of a low-resolution Markov chain can be used to warm start the Markov chain with high resolution. We prove that in terms of the chain length the new algorithm has O(1) complexity rather than O(r).