Stochastic quantization defines a Euclidean quantum field theory as the equilibrium of a fictitious-time Langevin dynamics, which reaches the Gibbs measure asymptotically and is formulated as a finite-time stochastic optimal control problem.
Abstract
Stochastic quantization defines a Euclidean quantum field theory as the equilibrium of a fictitious-time Langevin dynamics, which reaches the Gibbs measure asymptotically. We show that this quantization can be formulated as a finite-time stochastic optimal control problem. A tractable reference process, naturally supplied by the free theory when available, provides an Ornstein--Uhlenbeck dynamics, while the full interaction enters as a reference-corrected terminal cost. The optimal control is a Doob-transform force that steers the path-reweighted terminal ensemble to the target at a prescribed time and for a given noise amplitude. A neural network learns the residual control, realizing this optimal stochastic quantization (OSQ). Because the path weights are exact, imperfect training increases the variance of estimators but does not introduce model bias. On multimodal potentials we recover all modes at finite time and find that the noise amplitude sets a practical diffusion-horizon window. In two-dimensional lattice scalar $\phi^4$ theory we recover observables from hybrid Monte Carlo simulations near the critical point. Quantization is thereby formulated as control rather than equilibration.
Nuclear quantum effects are rigorously captured by imaginary-time path integrals, which map the quantum Boltzmann distribution onto a ring polymer of classical replicas. Yet the nuclear masses, the coupling to the environment, and the boundary conditions of the path remain hard-wired in the simulation or the trained model, even though this quantum context enters the path measure only through a quadratic action known in closed form. Here we show that a denoiser trained on classical Boltzmann statistics alone, composed at sampling time with an analytic Gaussian component carrying the entire quantum context, yields the quantum Boltzmann distribution of the nuclei. Such a composition exists and is exact whenever the training noise does not exceed the intrinsic quantum uncertainty of the target ensemble, and it is invariant across all quantum contexts admitted by this bound. We show exact transfer across temperature, isotopic mass, dissipation strength, and the boundary conditions of the path in theory and in numerical experiments, without retraining. The last yields the end-to-end displacement and momentum distributions of a tagged nucleus from open imaginary-time paths. The same invariance extends in principle to the permuted boundary conditions of bosonic exchange, with the identical denoiser. In this view, the noise of generative modeling and the quantum fluctuations of the nuclei are two faces of the same quadratic structure.
Weizhou Wang, J. Weare, Aaron R. Dinner· 0 citations
The Jarzynski equality provides a strict link between nonequilibrium work and equilibrium free energy changes. Its typical quantum formulations, however, rely on measurement protocols that destroy coherence. In this Letter, we use the resource-theoretic approach to derive a non-destructive quantum Jarzynski equality conditioned on the outcomes of an arbitrary bath observable. This yields the R\'enyi-Jarzynski equality, which quantifies a finite bath's drift from equilibrium under a non-adiabatic drive via the R\'enyi $k$-divergence. We further demonstrate that the R\'enyi-Jarzynski equality provides a tunable cost function for quantum optimal control problems where minimizing bath drift is desired, such as state preparation and gate design, enabling the minimization of cross-talk in finite quantum systems. Our toy model exhibits a transition between competing minima for some critical value of $k$, illustrating how the R\'enyi order tunes sensitivity to different regions of a bath distribution. Strikingly, when drive parameters vary across bath energy levels, minimizing bath drift requires generating system-bath entanglement.
Ben Bobell, Mert Okyay, Rahul Nandkishore· 0 citations
We develop a general class of non-reversible Hamiltonian Monte Carlo dynamics on discrete state spaces. The method augments the discrete state with a continuous momentum variable and does not require a continuous embedding of the discrete state space. We establish conditions for invariance of the target distribution and study the resulting processes in terms of ergodicity, exponential contractivity, asymptotic variance and relaxation time. We then derive scaling limits on increasingly fine lattices and high-dimensional hypercubes. In both settings, suitable rescalings converge to Hamiltonian dynamics in continuous space, revealing a diffusive-to-ballistic speed-up over reversible samplers, even for heterogeneous target distributions where standard non-reversible methods become diffusive. The proposed dynamics can be simulated exactly in continuous time, given access to the target distribution at all neighbours of the current state. To reduce computational cost, we provide approximation schemes based on splitting, $\tau$-leaping and gradient approximations. We illustrate the proposed framework numerically on examples involving high-dimensional hypercubes, mixture models and permutations.
We estimate convergence rates of value functions for particle approximations of a controlled nonlinear filtering problem. The state is a McKean--Vlasov diffusion on the flat torus, driven by hidden idiosyncratic noise and observed common noise. The filter---the conditional law of the state given the observations---serves as the state variable of the control problem, and the associated value function solves a second-order Hamilton--Jacobi--Bellman equation on the Wasserstein space. We approximate this problem by a centralized \(N\)-particle control problem with independent idiosyncratic noises and a common observation noise. The framework accommodates nonseparable rewards and controlled drifts. Since a single control is applied to the entire population, the Hamiltonian is defined by an optimization performed after integration over the population. Under smoothness of the data, uniform ellipticity, and regularity of this Hamiltonian, we establish uniform value-function error bounds of order \(N^{-1/6}\) for \(d=1\), \(N^{-1/6}(\log N)^{1/3}\) for \(d=2\), and \(N^{-1/(3d)}\) for \(d>2\). The proof combines a translation lift in the common-noise direction, Fourier--Wasserstein inf- and sup-convolutions, viscosity comparison, and particle derivative estimates uniform in \(N\).
Erhan Bayraktar, Ibrahim Ekren, Xihao He et al.· 0 citations
The quantum first-passage problem involves stochastic trajectories conditioned on measurement outcomes. The timing statistics of such trajectories remain largely unexplored in open quantum systems. Here, we investigate the first-passage time to an energy threshold for a ubiquitous model: a harmonic oscillator driven by classical additive noise. We find that projective measurements and energy quantization lead to substantial differences between the quantum and classical first-passage-time distributions at low thresholds, while these differences gradually diminish as the threshold energy increases. We treat the problem using both ensemble-averaged conditioned density-matrix dynamics and trajectory-resolved stochastic pure-state dynamics. The two descriptions yield indistinguishable timing statistics. Quantization effects appear in the ensemble-level phase-space distributions of the surviving states and vanish at larger threshold energies. Individual trajectories reveal emergent quantum signatures from the repeated measurements, such as persistent Wigner negativity. Our results provide a framework for using first-passage processes to create measurement-induced nonclassical resource states and to study the quantum-classical crossover of monitored systems.
Joseph M. Ryan, Simon Gorbaty, S. Teitsworth et al.· 0 citations
We study a time-discrete, finite-particle Consensus-Based Optimization (CBO) algorithm in a separable Hilbert space. Our analysis provides convergence guarantees directly for the computable particle system, complementing recent continuous-time and mean-field results in infinite dimensions. Using a common-noise formulation with trace-class covariance, we first establish quantitative pairwise contraction, exponential decay of the expected swarm variance, and almost-sure convergence of all particles to a common consensus state. We then combine estimates on the exponentiated objective functional with a quantitative Laplace principle to show that, for sufficiently large inverse temperature and suitably prepared initial data, the energy of the limiting consensus state can be made arbitrarily close to the global minimum over the active subspace. A key feature of the Hilbert-space formulation is that the stochastic contribution to the convergence estimates is controlled by the trace of the covariance operator and is therefore uniform with respect to the Galerkin dimension. Numerical experiments on an elliptic energy minimization problem with mixed boundary conditions and a PDE-constrained inverse source problem validate the theoretical convergence results and demonstrate stable performance under increasing spatial resolution, in contrast with CBO based on isotropic finite-dimensional noise.
Michael Herty, Hui Huang, Hicham Kouhkouh· 0 citations