Path integral control in the complex plane

Abstract

Path integral control is a stochastic optimal control framework that computes optimal policies as weighted averages over state trajectories. It was recently shown that, combined with stochastic quantum trajectories, the path integral control approach constitutes a promising alternative for quantum optimal control. In this work, we propose an extension of path integral control to the complex plane using circular complex Wiener processes. The change of measure is defined on the driving noise space under an exact control-noise matching condition, which also accommodates rank-deficient state diffusions. Applied to stochastic Schrödinger equations, the resulting formalism gives a compact treatment of paired non-Hermitian jump channels with unequal positive dissipation rates and yields cross-entropy-optimal feedback and open-loop control formulas.

The paper asks how path-integral stochastic control can be formulated directly for complex quantum trajectories, particularly with unequal excitation and relaxation rates. It uses circular complex Wiener noise and imposes exact control–noise matching, $g_1=g_2G$. A Wiener-space Girsanov construction avoids requiring an invertible state-space diffusion, allowing rank-deficient diffusions and matching at dark states. The resulting identity is $C[u]=D_{\mathrm{KL}}(P^u\Vert P^*)-\log Z$, where $P^*$ is the Gibbs-tilted uncontrolled measure. A separate variational construction minimizes the reverse KL, yielding cross-entropy-optimal feedback, open-loop and piecewise-constant pulse formulas. These solve the original control problem only when the projections coincide; exact attainability requires the Gibbs noise shift to satisfy $h^*=Gu^*$ within the admissible control class. The derivations use Girsanov likelihoods, change of measure and variations of a quadratic action. For linear stochastic Schrödinger equations, operator identities provide state-independent matching matrices for conjugate jump pairs with unequal strictly positive rates. For deterministic controls and costs linear in the unnormalized density matrix, the expected objective equals a Lindblad-control objective. An illustrative two-qubit calculation uses eight pulse intervals and $10^3$ trajectories per iteration; after 40 iterations, Lindblad-evaluated Bell-state fidelity increases from $0.490$ to $0.971$. Every complex diffusion considered has an equivalent doubled-real representation. Cross-entropy projections remain unraveling- and lift-dependent, with no general performance guarantee for the original objective. Exactly vanishing rates are singular, and unmatched single-jump controls lie outside the framework.