Analytic gradients for low-rank quantum optimal control

Overview

Quantum optimal control is essential for high-fidelity operations in superconducting circuits, but simulating open quantum systems is computationally expensive. As system sizes grow, the density matrix's degrees of freedom scale quadratically, making standard gradient-based optimization methods like GRAPE or automatic differentiation (AD) prohibitively slow and memory-intensive. This paper introduces Low-Rank Optimal Control (LROC), a method that exploits the fact that quantum computing protocols are designed to maintain high purity, allowing the density matrix to be represented by a small number of pure states.

LROC approximates the density matrix as a convex mixture of a few pure states, truncating the representation to a rank $M$. By deriving the adjoint equation for this factorized form, the authors enable the computation of analytic gradients for any differentiable objective function. This approach avoids the memory overhead associated with storing the full computational graph required by reverse-mode automatic differentiation. The algorithm integrates a piecewise-constant pulse parametrization, similar to GRAPE, to compute gradients with a complexity that scales as $O(NM^2)$ in time and $O(NM)$ in memory, where $N$ is the Hilbert space dimension.

Performance and Applications

The authors demonstrate LROC on four critical superconducting-circuit tasks: preparing a five-qubit GHZ state, implementing a cross-resonance CNOT gate, performing qubit readout, and executing an error correction primitive. In each case, the method reaches fidelities consistent with intrinsic dissipation limits. The results show that LROC can effectively navigate complex dynamics, including leakage into non-computational levels and strong drives, while maintaining a significantly lighter computational footprint than standard full-density-matrix approaches.