Control of Stochastic Quantum Dynamics by Differentiable Programming

TL;DR

This paper introduces a differentiable programming framework for designing control strategies for stochastic quantum systems, demonstrated on qubit state preparation with neural networks achieving around 85% fidelity.

Contribution

It presents a novel automated control design method using neural networks and differentiable programming for stochastic quantum dynamics, including gradient backpropagation through SDEs.

Findings

Neural network controllers achieve ~85% fidelity in qubit state stabilization.

The framework effectively trains controllers using gradient backpropagation through stochastic differential equations.

Comparison shows neural network solutions outperform hand-crafted strategies.

Abstract

Control of the stochastic dynamics of a quantum system is indispensable in fields such as quantum information processing and metrology. However, there is no general ready-made approach to the design of efficient control strategies. Here, we propose a framework for the automated design of control schemes based on differentiable programming ($\partial \mathrm{P}$). We apply this approach to the state preparation and stabilization of a qubit subjected to homodyne detection. To this end, we formulate the control task as an optimization problem where the loss function quantifies the distance from the target state, and we employ neural networks (NNs) as controllers. The system's time evolution is governed by a stochastic differential equation (SDE). To implement efficient training, we backpropagate the gradient information from the loss function through the SDE solver using adjoint sensitivity methods. As a first example, we feed the quantum state to the controller and focus on different methods of obtaining gradients. As a second example, we directly feed the homodyne detection signal to the controller. The instantaneous value of the homodyne current contains only very limited information on the actual state of the system, masked by unavoidable photon-number fluctuations. Despite the resulting poor signal-to-noise ratio, we can train our controller to prepare and stabilize the qubit to a target state with a mean fidelity of around 85%. We also compare the solutions found by the NN to a hand-crafted control strategy.