Operator-Level Quantum Acceleration of Non-Logconcave Sampling

TL;DR

The paper tackles the challenge of sampling from Gibbs distributions $\sigma \propto e^{-\beta V}$ when $V$ is non-convex, where classical Langevin methods suffer metastability. It develops an operator-level quantum framework that encodes the target Gibbs measure into the ground state of the Witten Laplacian and performs Gibbs sampling by preparing the encoded state $|\sqrt{\sigma}\rangle$ through quantum singular value thresholding on a block-encoded operator, avoiding time discretization errors. The main contributions include a provable quantum speedup for general non-log-convex sampling with complexity scaling as $\tilde{O}(\sqrt{C_{\rm PI}})$, a quartic improvement over MALA in key regimes, and the first quantum acceleration of replica-exchange Langevin diffusion via a generalized Witten Laplacian. The work also shows warm-start preparation via Lindblad dynamics, analyzes discretization and block-encoding costs, and provides numerical demonstrations illustrating quantum versus classical performance, highlighting potential practical impact for high-dimensional, rugged landscapes.

Abstract

Sampling from probability distributions of the form $σ\propto e^{-βV}$, where $V$ is a continuous potential, is a fundamental task across physics, chemistry, biology, computer science, and statistics. However, when $V$ is non-convex, the resulting distribution becomes non-logconcave, and classical methods such as Langevin dynamics often exhibit poor performance. We introduce the first quantum algorithm that provably accelerates a broad class of continuous-time sampling dynamics. For Langevin dynamics, our method encodes the target Gibbs measure into the amplitudes of a quantum state, identified as the kernel of a block matrix derived from a factorization of the Witten Laplacian operator. This connection enables Gibbs sampling via singular value thresholding and yields up to a quartic quantum speedup over best-known classical Langevin-based methods in the non-logconcave setting. Building on this framework, we further develop the first quantum algorithm that accelerates replica exchange Langevin diffusion, a widely used method for sampling from complex, rugged energy landscapes.

Figures (8)

• Figure 1: A schematic diagram of operator-level quantum acceleration of classical Gibbs sampling.
• Figure 2: For Langevin dynamics, the Poincaré constant $C_{\rm PI}$ scales as $\mathcal{O}(\gamma^{-1})$ for the convex potential, but as $\exp(\Omega(\beta\Gamma))$ for the non-convex potential. The mixing time of the continuous-time Langevin dynamics scales linearly in $C_{\rm PI}$.
• Figure 3: Left y-axis: Value of the filter function $p(s)$ that is approximately equal to $1$ on $[0,\mathrm{Gap}/4]$, and approximately equal to $0$ on $[3\mathrm{Gap}/4, 1]$. Right y-axis: Projection $|c_i|$ to each right singular vector before and after QSVT. Here $\mathrm{Gap}=\mathrm{Gap}\left(\mathbb{L}\right)/\alpha$, and $\alpha$ is the block encoding subnormalization factor for $\mathbb{L}$.
• Figure 4: Quantum acceleration of Langevin dynamics. Each column corresponds to a different inverse temperature $\beta$. The overlap $\left\lvert\bra{\sqrt{\sigma}} \ket{\phi}\right\rvert$ measures the similarity between the sampled and true distribution, where $\ket{\sqrt{\sigma}}$ is the encoded Gibbs state and $\ket{\phi}$ represents either (1) the square root of the output probability density by MALA, or (2) the output pure state by our method. Note that the overlap definition can be generalized to accommodate mixed states; see \ref{['eqn:mixed-state-overlap']}. Top row: Sample distributions obtained using MALA with the number of iterations increasing by a factor of 9. Middle row: Distributions obtained by our quantum algorithm with the degree of polynomials increasing by a factor of 3. Bottom row: True distribution (Gibbs state).
• Figure 5: Quantum acceleration of replica exchange. Left: a non-convex 1D potential. Right; Inverse spectral gaps for LD, RELD, and their quantum counterparts as functions of $\beta$. The spectral gap of LD decays exponentially in $\beta$, while that of RELD shows a much milder decay. The quantum algorithms achieve square-root improvements in the gaps in both cases.

Theorems & Definitions (34)

• Theorem 1: Informal
• Theorem 2: Informal
• Lemma 3
• proof
• Definition 4: Block-encoding of a rectangular matrix
• Theorem 5: QSVT with even polynomials
• Lemma 6: gilyen2019quantum
• Proposition 7: Singular value thresholding
• proof
• Theorem 8