Wiener KL Regularizer in Stochastic & Quantum Control

• Wiener KL Regularizer is a quadratic penalty that quantifies deviations in path-space drifts using the Cameron–Martin norm.
• It underpins variational and optimal control formulations by balancing control efficacy with the entropic cost of drift shifts.
• The regularizer’s implementation via Girsanov transformations and Euler–Lagrange conditions supports robust simulations in both classical and quantum frameworks.

The Wiener–KL regularizer is a quadratic penalty on path-space drifts that arises from minimizing the Kullback–Leibler (KL) divergence between Gaussian shift measures on classical Wiener space, with applications extending to stochastic control and open quantum dynamics. Its defining property is to penalize deviations in the drift of stochastic processes (relative to a Wiener reference) via the Cameron–Martin norm, and thus it appears as an Onsager–Machlup-type functional in variational and optimal control formulations. The regularizer is central to controlling path measures in both the mathematical theory of information projection and practical quantum stochastic control algorithms, mediating a tradeoff between control efficacy and entropic cost.

1. Classical Wiener–KL Regularizer: Foundations

Given classical Wiener space $\left(C_0[0,T],\|\cdot\|_\infty,\gamma\right)$, where $\gamma$ is the law of standard Brownian motion $B$, the Cameron–Martin space is

$$H = W_0^{1,2}[0,T] = \left\{h(t) = \int_0^t \dot h(s)\,ds : \dot h \in L^2[0,T]\right\}.$$

For $h \in H$, the translated measure $\mu_h = \gamma \circ (B \mapsto B + h)^{-1}$ is absolutely continuous with respect to $\gamma$, with Radon–Nikodym derivative

$$\frac{d\mu_h}{d\gamma} = \exp\left(\int_0^T \dot h(s)\,dB(s) - \frac{1}{2} \int_0^T \dot h(s)^2\,ds\right).$$

The KL divergence between shift measures is

$$D_{KL}(\mu_{h_1} \| \mu_{h_2}) = \frac{1}{2} \| h_1 - h_2 \|_H^2,$$

and in particular, $D_{KL}(\mu_h \| \gamma) = \frac{1}{2} \| h \|_H^2$. Thus, the Wiener–KL regularizer is naturally given by the quadratic integral $\gamma$0 (Selk et al., 2020).

2. Onsager–Machlup Functional and KL Projection

The Wiener–KL regularizer arises as a component of the Onsager–Machlup (OM) functional in the context of information projection on path space. For a target probability law $\gamma$1 on $\gamma$2,

$\gamma$3

with path-functional $\gamma$4, the problem of projecting $\gamma$5 onto shift measures $\gamma$6 in KL divergence is

$\gamma$7

This reduces (up to a constant) to the minimization of

$\gamma$8

where the second term is the Wiener–KL regularizer. Under suitable regularity of $\gamma$9, minimizers $B$0 exist and yield the most probable path (MAP estimator) with respect to the "twisted" law (Selk et al., 2020).

3. Variational Reformulation and Euler–Lagrange Conditions

The minimization of $B$1 may equivalently be framed as a calculus-of-variations problem for $B$2:

$B$3

with Lagrangian encoding the interaction of the shift with data-fit and penalty. For target measures driven by Girsanov density $B$4, the Lagrangian takes

$B$5

The necessary optimality condition is the Euler–Lagrange equation

$B$6

which, under convexity and coercivity, admits weak solutions attaining the minimum (Selk et al., 2020).

4. Interpretations and Explicit Examples

The Wiener–KL regularizer encodes a penalty for "open-loop" deterministic drifts in the sense that it quantifies the entropic cost (relative entropy to Wiener measure) of the drift. In representative cases:

• Linear cost functional: For $B$7, the OM minimizer is $B$8, yielding $B$9.
• Quadratic cost functional: For $$H = W_0^{1,2}[0,T] = \left\{h(t) = \int_0^t \dot h(s)\,ds : \dot h \in L^2[0,T]\right\}.$$0, the minimizer collapses to $$H = W_0^{1,2}[0,T] = \left\{h(t) = \int_0^t \dot h(s)\,ds : \dot h \in L^2[0,T]\right\}.$$1 (i.e., $$H = W_0^{1,2}[0,T] = \left\{h(t) = \int_0^t \dot h(s)\,ds : \dot h \in L^2[0,T]\right\}.$$2) (Selk et al., 2020).

In each case, the Wiener–KL regularizer's role is to discourage large deterministic shifts from the reference (Brownian) dynamics, concentrating the feasible drift within the Cameron–Martin space.

5. Wiener–KL Regularizer in Open Quantum Control

The Wiener–KL construction has been generalized to open quantum systems under the diffusive (homodyne) unravelling, in which the measurement record $$H = W_0^{1,2}[0,T] = \left\{h(t) = \int_0^t \dot h(s)\,ds : \dot h \in L^2[0,T]\right\}.$$3 for each channel satisfies

$$H = W_0^{1,2}[0,T] = \left\{h(t) = \int_0^t \dot h(s)\,ds : \dot h \in L^2[0,T]\right\}.$$4

Comparing the path measure $$H = W_0^{1,2}[0,T] = \left\{h(t) = \int_0^t \dot h(s)\,ds : \dot h \in L^2[0,T]\right\}.$$5 (induced by control parameter $$H = W_0^{1,2}[0,T] = \left\{h(t) = \int_0^t \dot h(s)\,ds : \dot h \in L^2[0,T]\right\}.$$6) to the Wiener reference $$H = W_0^{1,2}[0,T] = \left\{h(t) = \int_0^t \dot h(s)\,ds : \dot h \in L^2[0,T]\right\}.$$7, Girsanov's theorem yields

$$H = W_0^{1,2}[0,T] = \left\{h(t) = \int_0^t \dot h(s)\,ds : \dot h \in L^2[0,T]\right\}.$$8

directly paralleling the classical formulation (Moody et al., 18 Jun 2026).

6. Integration into Quantum Optimal Control Functionals

In stochastic quantum control, the Wiener–KL regularizer is added to the primary objective, generally the average state-transfer infidelity,

$$H = W_0^{1,2}[0,T] = \left\{h(t) = \int_0^t \dot h(s)\,ds : \dot h \in L^2[0,T]\right\}.$$9

yielding the composite cost

$h \in H$0

with $h \in H$1 a tunable parameter. This cost is minimized over the control parametrization, typically via gradient-based methods on sample trajectories of the stochastic Schrödinger equation (SSE). The gradient of $h \in H$2 with respect to control parameters $h \in H$3 admits the closed form

$h \in H$4

with $h \in H$5 determined by the pathwise sensitivity of the measurement record's drift. In practice, the expectation is estimated by averaging over batches of simulated trajectories with automatic differentiation through the SSE integrator (Moody et al., 18 Jun 2026).

7. Computational Aspects and Domain of Applicability

Implementation of $h \in H$6 utilizes discretized-time SSE solvers (e.g., Euler–Maruyama), parallel simulation on GPU hardware, and automatic differentiation to compute gradients with respect to control parameters. Boundary conditions include normalization of quantum trajectories, initial state specification, and parametrizations of the control fields as piecewise-constant or Fourier modes. The construction is specific to the diffusive (Itô) unravelling and does not extend directly to jump-type (photon-counting) processes (Moody et al., 18 Jun 2026).

The Wiener–KL regularizer differs fundamentally from standard penalties on control amplitude or fluence, acting directly on the observable consequences of control (the effects on decoherence channels) instead of the control field magnitude. Empirically, its inclusion improves state-to-state transfer fidelity, robustness to noise-model mismatch, and suppression of forbidden state occupancy in open quantum systems (Moody et al., 18 Jun 2026).