Thermalizing Stochastic Programs

The paper Thermalizing Stochastic Programs by Mirko Amico, Andraž Jelinčič, Colin Oscar Nancarrow, Leo Tyrpak, David Roberts, Seth Morton, Dalton Sakthivadivel, Ashwin Gopal, and Guillaume Verdon (Extropic Corporation, San Francisco, California, USA, dated August 14, 2026, arXiv:2608.01615v2 [cs.ET]) presents a set of tools for mapping general stochastic programs to thermodynamic hardware designed for energy-efficient stochastic sampling.

The paper is motivated by the growing energy requirements of generative artificial intelligence, noting that the continued development of artificial intelligence systems with growing capabilities has come with increased energy requirements over the past several years, and that the bulk of AI training and inference is carried out in a relatively small number of data centers, the largest of which draw power comparable to that of a large city. The authors argue that Realizing artificial intelligence as a general-purpose technology, deployed pervasively rather than consumed as a centrally metered service, requires reducing the energy cost of AI algorithms by orders of magnitude. They note that Simulations of denoising-based generative algorithms running on such devices suggest that non-trivial inference tasks can be executed with as little as 10−4 or even 10−6 of the energy required by GPUs.

The authors state that A general method for executing stochastic programs on this class of hardware has been missing until now, as Existing programming models for such devices usually restrict attention to problems the hardware expresses natively.

The paper introduces a framework operating on three levels: "At the highest level of abstraction, the stochastic program is a directed factor graph of Markov kernels, or equivalently a parametrized stochastic circuit, passing wire variables zl from the program's inputs to its output. Each factor of the DFG is compiled to a thermodynamic kernel, defined as a conditional distribution obtained from an EBM by clamping the input spins x, marginalizing hidden spins w and normalizing over the output spins y. The compiled kernel is then laid out spatially on a region of the thermodynamic hardware, a Thermodynamic Sampling Unit (TSU), whose physical dynamics samples it natively."

The key insight is that "When a subset of the variables of a thermodynamic computer is held fixed, the device samples from the conditional distribution of the remaining variables, so that a single choice of energy function realizes a probabilistic transition kernel from the clamped variables to the free variables. We call this object a thermodynamic kernel. The thermodynamic kernel is the basic instruction from which our programs are assembled."

The paper defines Thermodynamic Hypergraphical Models (THMs) as an energy-based model defined on a finite hypergraph G = (V, E) with energy Eφ(s) = −Σe∈E Je Πvi∈e si, where s ∈ −1, 1 n. The authors note that A THM of order 2 is also called a Boltzmann machine. Sampling is performed using Gibbs sampling, wherein we define a Markov chain that resamples spins according to the conditional probability pφ(si = 1 s) = σ(2 Σe∋i Je Πj∈ei sj).

The hardware platform, referred to as Z1, would support pairwise Ising energies E(s) = −½s⊤Js − h⊤s on a fixed planar topology. The Z1 topology is described as a sparse, locally connected, 2-colorable graph, where each node supports two different couplings per edge. The chip has approximately 250,000 nodes, each Gibbs iteration is estimated to cost approximately 3 × 10−10 J, and the chip will perform between 106 and 107 Gibbs iterations per second. The hardware supports readout, coupling flashing, and clamping operations.

A DFG is defined as "a directed acyclic graph where each node holds a stochastic kernel and each directed edge carries some random variable sampled as an output of one stochastic kernel to then be used as input to a subsequent stochastic kernel." The full DFG defines a joint distribution over all wires by ancestral sampling: PϕDFG(z1,..., zL z0):= Πl=1L Pl(zl zpa(l); ϕl).

The paper notes that PSCs correspond more closely to how circuits are designed in quantum computers while DFGs are inspired by graphical models, and that PSCs are hardware-agnostic by design and the same logical circuit can compile to different backends: a software simulator, the XTR-0 test chip, or Z1.

The central training method is variational compilation, which minimizes the trajectory-level KL divergence LVC(φ):= DKL(P ∥ P̃φ) = Σl=1L εl(φl). Each summand depends on the parameters of one kernel only, so the kernels are trained independently of one another, in parallel.

The gradient is given by ∇φl εl = Ex∼ql−1 Ey∼Pl(·x)[−∇φl log P̃lF(y x; φl)]. The paper derives a two-phase identity for the gradient: −∇φ log P̃F(y x; φ) = E(w,y′)∼pφ(·,·x)[Φφ(x, w, y′)] − Ew∼pφ(·x,y)[Φφ(x, w, y)], where Φφ:= −∇φEφ is the vector of the THM's couplings.

The paper identifies three sources of compilation residual: "the hypothesis-class limitation set by the substrate's topology, the hidden-spin count, and the coupling caps; the optimization residual of finite gradient tolerance; and, in the intractable case, the sampling bias of unmixed phases."

The key error bound is: DKL(R#P ∥ R#P̃φ) ≤ DKL(P ∥ P̃φ) = Σl=1L εl(φl) ≤ Σl=1L cl Ex∼µl[Jl(x; φl)], where the middle equality is the chain rule of the KL divergence, the left inequality is the data-processing inequality, and the right inequality bounds per-kernel losses under a generic input distribution.

The paper identifies three choices for the input distribution µl: "(i) target inputs, µl = ql−1: the weights of the exact decomposition in (17), under which the sum of per-kernel losses is the trajectory error. This is the default; (ii) model inputs, µl = q̃l−1: each kernel is trained under the input marginal it receives inside the compiled circuit; (iii) generic inputs, e.g., uniform: agnostic to the context of the rest of the program."

The paper notes that A target conditional that recurs across applications... can be compiled once and reused, with the right-most bound of (17) guaranteeing that after placement into a specific program, the kernel's contribution to the trajectory error is at most cl Ex∼µl[Jl(x; φl)].

The paper introduces trajectory-level REINFORCE post-training, which cares only about how well the readout satisfies a given objective and uses all compiled kernels jointly against a single global objective. The gradient is: ∇φlLRF = E[F(z1:L) Φφl(zl−1, wl, zl)] − E[E(w′,y′)∼pφl(·,·zl−1) Φφl(zl−1, w′, y′)]. The paper notes that "REINFORCE becomes effective only once the model is already good enough that its samples reach the region where F discriminates, and supplying a good starting model is exactly what variational compilation and context matching are for."

The first demonstration compiles a biased random walk on a 5×5 torus, where the target conditional is the asymmetric stochastic swap gate PAsymSwap. The paper reports that the median compiled total variation distance for the PAsymSwap gates is 0.096. The results show that Stacking context matching and REINFORCE reduces the final-time half-l1 occupancy error from 5.64 (compiled-only) to 0.30 (context matched) to 0.08 (context matched + REINFORCE), nearly two orders of magnitude.

The second demonstration compiles a predator-prey ecosystem model with an on-site reaction process and a diffusion-like process. The paper reports that Kernels with only four and eight visible nodes are capable of reproducing the conditional distributions of the migration and reaction processes respectively to a worst-case total variation distance of ∼ 0.01. The paper notes that REINFORCE training substantially improved the quality of the compiled model's population mean for t ∈ 1,..., TRF, but also that "Shortly thereafter, more significant departures emerge from the exact process than were evident in the pre-REINFORCE compilation, suggesting that 'reward hacking' poses a risk even with models built from a small number of unique kernels."

The third demonstration compiles the day-to-day movement of financial markets using a panel of N = 14 daily series covering the major asset classes... over 4,802 trading days from 2007 to 2026. The kernel is compiled to a conditional Thermodynamic Consistency Model (TCM), which is a chain of two BM kernels, both conditioned on the clamped B = 5-day context, in which the first samples a draft of the next day and the second refines it.

The results show that The compiled two-step TCM reaches a composite error of 0.667, below both baselines on the composite, and REINFORCE post-training improves it by a further 27% to 0.486. The paper notes that the improvement reflects generalization along two axes that the training signal does not contain: from 20-day windows to 1,200-day rollouts, and from the training split to held-out data.

The fourth demonstration shows sampling from stationary distributions corresponding to energies that are not natively expressible by the hardware substrate, calling this a meta-EBM. The construction compiles the Gibbs sampling program that samples it, where The Gibbs sampling algorithm decomposes into local, detailed-balanced update kernels, each kernel being a target conditional that can be compiled to any hardware substrate using variational compilation.

The demonstration uses a three-body Ising target on d = 12 spins and shows that the measured stationary error tracks this prediction across the full cap sweep, spanning over a decade of error, from 0.46 at the most aggressive cap down to 2.4×10−2 at Jmax = 10, at fixed ρ0 = 0.28.

The fifth demonstration compiles a Gaussian stochastic circuit for a sequential Bayesian experimental design task, where the goal is to reconstruct the field from as few measurements as possible. The results show that The RMSE against the true field falls to 0.259 ± 0.005 for the compiled THM and 0.260 ± 0.006 for the exact loop, versus 0.305 ± 0.008 for random placement.

The paper concludes that This manuscript introduced variational compilation as a way to turn directed stochastic processes into kernels that can be run on thermodynamic hardware. The authors note that Every compiled program remains approximate, because finite coupling ranges, sparse topology, optimization residuals, and finite thermalization all contribute residual error, and that "Context matching, trajectory-level post-training, and hardware-in-the-loop refinement can reduce the errors seen by the deployed program, but they do not remove the need to validate each application against its own task-level observable."

The paper identifies An important future direction is to find useful decompositions of other stochastic processes into local conditionals that can be compiled and run in this framework, and notes that A possibility open for future exploration relates to the discovery of novel stochastic algorithms, drawing a parallel to quantum computing where the approach... has been applied successfully in the discovery of quantum algorithms.

The paper also discusses "connectivity-aware compilation, where the placement of input, hidden, and output spins on the actual hardware graph is optimized together with the kernel parameters so that topology residuals can be measured and reduced rather than treated after the fact."

Based on the paper, here are specific improvements I can implement in AI systems:

Improvement: Replace standard GPU-based Gibbs sampling in probabilistic graphical models with a software simulator of thermodynamic kernels (THMs) that uses variational compilation to map target conditionals onto energy-based models.

What it can do:

• Execute stochastic programs (Bayesian networks, Markov chains, Gaussian processes) at 104–106× lower energy per sample by exploiting the hardware's native equilibrium sampling, with a software fallback that mimics the same dynamics.

• Compile any directed factor graph into a sequence of clamped-thermalization operations, enabling forward, backward, and posterior sampling from the same energy programming.