Quantum Speedup for Nonreversible Markov Chains

TL;DR

The paper addresses accelerating sampling from stationary distributions of nonreversible Markov chains using quantum algorithms. It develops two complementary workflows that leverage reversibilizations and quantum polynomial transforms (GQET/GQSVT) to construct reflections through target stationary states, enabling quantum Monte Carlo speedups. A key insight is that when the reversibilization time is smaller than the mixing time, the quantum methods achieve a speedup on the order of √(τ_rev τ), potentially surpassing the standard quadratic speedup for reversible chains. The approach introduces the geometric reversibilization and reversibility-on-π-average as practical conditions to realize efficient reflections, with implications for applications in physics, chemistry, biology, and finance where nonreversible dynamics are prevalent.

Abstract

Quantum algorithms can potentially solve a handful of problems more efficiently than their classical counterparts. In that context, it has been discussed that Markov chains problems could be solved significantly faster using quantum computing. Indeed, previous work suggests that quantum computers could accelerate sampling from the stationary distribution of reversible Markov chains. However, in practice, certain physical processes of interest are nonreversible in the probabilistic sense and reversible Markov chains can sometimes be replaced by more efficient nonreversible chains targeting the same stationary distribution. This study constructs Markov chain reversibilizations and develops quantum algorithmic techniques to accelerate nonreversible processes. Such an up-to-exponential quantum speedup goes beyond the predicted quadratic quantum acceleration for reversible chains and is likely to have a decisive impact on many applications ranging from statistics and machine learning to computational modeling in physics, chemistry, biology and finance.

Figures (3)

• Figure 1: Algorithmic workflow for the curved and flat discriminant approaches. Both algorithms start by taking classical steps of the Markov chain until a condition is met. Either through singular value or eigenvalue transform, a polynomial is applied to the corresponding discriminant in order to obtain a projected unitary encoding of the sought operator $\ket\pi\bra\pi$.
• Figure 2: Figure $\mathbf a$ shows the relationship between the spectra of the additive and geometric reversibilizations of a Markov kernel $P^j$, as a function of the number of Markov chain steps $j$. The additive reversibilization $(P^j+(P^\star)^j)/2$ is the process that goes forward or backward in time with equal probabilities. The geometric reversibilization $Q_j$ is the process whose spectral gap $\gamma(Q_j)$ governs the complexity of the quantum algorithm. Figure $\mathbf b$ shows that the condition of reversibility on $\pi$-average, $1-\braket{\pi|D_j|\pi}\ll \gamma(Q_j)$, is verified long before the mixing time $\tau(1/4)$. The quantum algorithm does provide a speedup if this condition is satisfied.
• Figure 3: Nonreversible walk on a graph with bottleneck. The condition of reversibility on $\pi$-average $1-\braket{\pi|D_j|\pi}\ll \gamma(Q_j)$ is verified for $j\in \mathbb N$ much smaller than the mixing time. As a consequence, $\gamma(Q_j)$ is of the same order as $\gamma_\infty\left(P^j\right)$, the pseudo-spectral gap of $P^j$.

Theorems & Definitions (16)

• Proposition 1
• Proposition 2
• Proposition 3
• Theorem 1
• Theorem 2
• Definition 1
• Proposition 4
• Proposition 5
• Proposition 6
• Corollary 1