Quantum Speedups for Multiproposal MCMC

TL;DR

The paper tackles the bottleneck in multiproposal MCMC where each iteration naively requires $O(P)$ target evaluations. It introduces QPMCMC2, a quantum algorithm that achieves $O(1)$ target evaluations per iteration with $O( obreak ext{log} P)$ qubits, while preserving exact detailed balance for a broad class of graphical models; amplitude amplification can further boost reliability. The framework maps proposal generation, target evaluation, and selection into quantum subroutines, yielding a per-iteration cost that no longer scales with $P$ and improving both convergence speed and effective sample size. Demonstrations on Ising-type models on phylogenetic networks, including a bacterial Neighbor-Net and 248 Salmonella isolates, show significant speedups and higher ESS per oracle, highlighting practical potential for Bayesian ancestral-trait reconstruction in complex, reticulate evolutionary settings.

Abstract

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 $\mathcal{O}(P)$ 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 $\mathcal{O}(\sqrt{P})$ target evaluations at each step, outperforming its classical counterpart. However, generating $P$ proposals using classical computers still requires $\mathcal{O}(P)$ time complexity, resulting in the overall complexity of QPMCMC remaining $\mathcal{O}(P)$. 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 $\mathcal{O}(1)$ target evaluations and $\mathcal{O}(\log P)$ 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.

Figures (9)

• Figure 1: This phylogenetic tree $\mathcal{G}$ has $M_o=3$ leaf nodes, $M_o-1=2=M_a$ internal nodes and $2M_o-1=5=M_{tot}$ total nodes. Leaf nodes represent observed taxa, and internal nodes are unobserved ancestors. We observe a binary trait variable $\sigma_m$ for each of the leaf nodes and model all (both observed and unobserved) traits $\sigma_m$ using an Ising model with interactions $j_{mm'}$ which condition on weights $w_{mm'}>0$.
• Figure 2: Reticulate evolution. This stylized bacterial phylogenetic network includes a reticulation (dashed line) that characterizes the exchange of genetic material between microbes. Whereas the network deviates from the bifurcating tree hypothesis of Figure \ref{['fig:phylo']}, the problem of ancestral trait reconstruction is still meaningful.
• Figure 3: A Neighbor-net phylogenetic network describes the shared evolutionary history of 248 salmonella bacteria isolates. The extremal nodes correspond to the 248 observed isolates, and the $M_a=3{,}065$ interior nodes correspond to unobserved ancestors. Interior squares are potential reticulation events. Colors (red, resistance; blue, no resistance) are observed and posterior mode resistances to the antibiotic ampicillin for observed microbes and unobserved ancestors, respectively.
• Figure 4: Comparison between trace plots generated by the QPMCMC2, QPMCMCholbrook2023quantum and \ref{['alg:multiProp']} (PMCMC) for $P=3$ and $P=50$. For implementation of one MCMC iteration, QPMCMC2 requires 1 oracle calls of $\pi_{\bar{\boldsymbol{\theta}}}^*(\cdot)$, while QPMCMC requires $\mathcal{O}(\sqrt{P})$ calls and multiproposal MCMC requires $P+1$ calls. In these cases, only QPMCMC2 improves the converge rate when using a larger $P$.
• Figure 5: Trace plots generated by the QPMCMC2 algorithm for different numbers of proposals $P$ and Metropolis-Hastings algorithm, tested on the square lattice graph. For both the single-trait and multi-trait problem, increasing $P$ accelerates convergence to higher posterior probability states. Here, we observe that QPMCMC2 significantly outperforms the Metropolis-Hastings algorithm: for $P=300$, QPMCMC2 achieves a 3.8-fold improvement in convergence rate compared to the Metropolis-Hastings algorithm.

Theorems & Definitions (6)

• Theorem 3.1
• Definition 3.2
• Corollary 3.2.1
• proof
• Theorem 4.1
• proof