Solving quantum optimal control problems using projection-operator-based Newton steps

TL;DR

The paper addresses efficient quantum optimal control by extending the projection-operator-based Newton method (Q-PRONTO) with a regulator in the projection operator to stabilize trajectory tracking and allow larger Newton steps, improving convergence speed and potentially reaching superior local minima. It formulates the problem as a real-valued OCP and provides a detailed projection-based Newton descent framework, including first and second-order derivatives and an LQR-based update subproblem. The regulator design, including phase-aware and phase-agnostic projections, enables handling time-varying costs, undesirable transient populations, and multi-input scenarios, with demonstrations on beam-splitting, Lambda-system state transfer, and a Fluxonium X-gate. Numerical results show faster convergence, different optimal trajectories under regulation, and effective control of transient populations, highlighting practical benefits for quantum gate design and state transfers, with open-source release planned.

Abstract

The Quantum Projection Operator-Based NewtonMethod for Trajectory Optimization (Q-PRONTO) is a numerical method for solving quantum optimal control problems. This paper significantly improves prior versions of the quantum projection operator by introducing a regulator that stabilizes the solution estimate at every iteration. This modification is shown to not only improve the convergence rate of the algorithm, but also steer the solver towards better local minima compared to the unregulated case. Numerical examples showcase how Q-PRONTO can be used to solve multi-input quantum optimal control problems featuring time-varying costs and undesirable populations that ought to be avoided during the transient.

Figures (9)

• Figure 1: Value of the cost decrease $-Dg(\eta_k)\cdot\zeta_k$ at each iteration. The purple dashed line is the exit condition $\texttt{tol} = 10^{-4}$. The introduction of a regulator gain $K_r(t)\neq0$ causes the solver to converge faster ($20\%$ less iterations).
• Figure 2: Optimal control inputs obtained without regulator (Top: $K_r(t)=0$) and with regulator (Bottom: $K_r(t)\neq0$). The latter clearly features a lower frequency content, making it preferable from the implementation perspective.
• Figure 3: Population dynamics obtained without regulator (Top: $K_r(t)=0$) and with regulator (Bottom: $K_r(t)\neq0$). The former forces the direct transition $\ket0\!\to\!\ket3$, whereas the latter achieves the natural progression $\ket0\!\to\!\ket1\!\to\!\ket2\!\to\!\ket3$.
• Figure 4: Value of the cost function \ref{['eq:cost_split']} at each iteration of the two solvers. The solution obtained with the regulator has a lower cost than the one obtained without a regulator. The introduction of a regulator gain $K_r(t)\neq0$ causes the solver to converge to a better ($50\%$ cost reduction) local minimum.
• Figure 5: Optimal control input sequences obtained when the objectives are to minimize the control effort (Top) or prevent the system from populating $\ket 2$ (Bottom).

Theorems & Definitions (1)

• Remark 1