Quantum Speedups for Multiproposal MCMC

Multiproposal Markov chain Monte Carlo (MCMC) algorithms choose from multiple proposals to generate their next chain step in order to sample from challenging target distributions more efficiently. However, on classical machines, these algorithms require $𝒪\left(P\right)$ target evaluations for each Markov chain step when choosing from $P$ proposals. Recent work demonstrates the possibility of quadratic quantum speedups for one such multiproposal MCMC algorithm. After generating $P$ proposals, this quantum parallel MCMC (QPMCMC) algorithm requires only $𝒪\left(\sqrt{P}\right)$ target evaluations at each step, outperforming its classical counterpart. However, generating $P$ proposals using classical computers still requires $𝒪\left(P\right)$ time complexity, resulting in the overall complexity of QPMCMC remaining $𝒪\left(P\right)$ . Here, we present a new, faster quantum multiproposal MCMC strategy, QPMCMC2. With a specially designed Tjelmeland distribution that generates proposals close to the input state, QPMCMC2 requires only $𝒪\left(1\right)$ target evaluations and $𝒪\left(\text{log}P\right)$ qubits when computing over a large number of proposals $P$ . Unlike its slower predecessor, the QPMCMC2 Markov kernel (1) maintains detailed balance exactly and (2) is fully explicit for a large class of graphical models. We demonstrate this flexibility by applying QPMCMC2 to novel Ising-type models built on bacterial evolutionary networks and obtain significant speedups for Bayesian ancestral trait reconstruction for 248 observed salmonella bacteria.