Counterdiabatic Hamiltonian Monte Carlo

Reuben Cohn-Gordon; Uroš Seljak; Dries Sels

stat.MLFeb 24, 2026

Counterdiabatic Hamiltonian Monte Carlo

Authors: Reuben Cohn-Gordon, Uroš Seljak, Dries Sels

Organizations: University of California, Berkeley · New York University

Abstract

Hamiltonian Monte Carlo (HMC) is a state of the art method for sampling from distributions with differentiable densities, but can converge slowly when applied to challenging multimodal problems. Running HMC with a time varying Hamiltonian, in order to interpolate from an initial tractable distribution to the target of interest, can address this problem. In conjunction with a weighting scheme to eliminate bias, this can be viewed as a special case of Sequential Monte Carlo (SMC) sampling \cite{doucet2001introduction}. However, this approach can be inefficient, since it requires slow change between the initial and final distribution. Inspired by \cite{sels2017minimizing}, where a learned counterdiabatic term added to the Hamiltonian allows for efficient quantum state preparation, we propose Counterdiabatic Hamiltonian Monte Carlo (CHMC), which can be viewed as an SMC sampler with a more efficient kernel. We establish its relationship to recent proposals for accelerating gradient-based sampling with learned drift terms, and demonstrate on simple benchmark problems.

Figures & tables

Figure 1: Illustration of lag induced by a time-varying Hamiltonian (top row), and the correction introduced by a learned counterdiabatic term (bottom row). The lefthand column shows a Gaussian with time-varying mean, and the righthand column an interpolation between a Gaussian and a mixture of two Gaussians. In the simple case, parametrizing $A$ as a sum of polynomials with learned coefficients suffices, and in the more complex case, $A$ is parameterized by a neural network.

q_{\text{half}}\leftarrow q+\tfrac{\varepsilon}{2}\Big(p+\dot{\lambda}_{k}\,\nabla_{p}A_{\phi}(q,p)\Big)

Algorithm 2 Counterdiabatic HMC (CHMC) with population and weighting

Figure 2: Here, the path of distributions is a geometric path between a standard normal and a mixture of two

System	True $E[x^{2}]$	Naive HMC	CHMC
Gaussian Moving Mean	2.0	1.12	2.1
Gaussian Annealing	0.1	86.5	0.65
Double Well	7.34	2.06	4.22

Table 4

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Generating samples from a continuous probability density is a central algorithmic problem across statistics, engineering, and the sciences. For high-dimensional settings, Hamiltonian Monte Carlo (HMC) is the default algorithm across mainstream software packages. However, despite the extensive line of work on HMC and its widespread empirical success, it remains unclear how many iterations of HMC are required as a function of the dimension

d

. On one hand, a variety of results show that Metropolized HMC converges in

O(d^{1/4})

iterations from a warm start close to stationarity. On the other hand, Metropolized HMC is significantly slower without a warm start, e.g., requiring

Ω(d^{1/2})

iterations even for simple target distributions such as isotropic Gaussians. Finding a warm start is therefore the computational bottleneck for HMC. We resolve this issue for the well-studied setting of sampling from a probability distribution satisfying strong log-concavity (or isoperimetry) and third-order derivative bounds. We prove that non-Metropolized HMC generates a warm start in

\tilde{O}(d^{1/4})

iterations, after which we can exploit the warm start using Metropolized HMC. Our final complexity of

\tilde{O}(d^{1/4})

is the fastest algorithm for high-accuracy sampling under these assumptions, improving over the prior best of

\tilde{O}(d^{1/2})

. This closes the long line of work on the dimensional complexity of MHMC for such settings, and also provides a simple warm-start prescription for practical implementations.

Abstract

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