High-Order Hermite Optimization: Fast and Exact Gradient Computation in Open-Loop Quantum Optimal Control using a Discrete Adjoint Approach

TL;DR

This paper introduces High-Order Hermite Optimization (HOHO), a open-loop discrete adjoint method that yields exact gradients when forward dynamics are solved with high-order Hermite Runge-Kutta timestepping for quantum optimal control. By deriving a discrete adjoint framework tailored to Hermite methods and exploiting a matrix-free, gradient-accumulation strategy, HOHO achieves substantial speedups and memory savings over conventional approaches, enabling efficient gradient-based gate design for complex, multi-qudit systems. The authors validate the method on Rabi oscillations and on a three-subsystem gate design involving qudits coupled to a resonator bus, reporting speedups up to 775× and memory reductions by several orders of magnitude, particularly at tight fidelity targets. The work demonstrates the practical impact of high-order discrete adjoints for stiff, oscillatory quantum dynamics and outlines promising directions for future improvements and applications in larger-scale quantum devices.

Abstract

This work introduces the High-Order Hermite Optimization (HOHO) method, an open-loop discrete adjoint method for quantum optimal control. Our method is the first of its kind to efficiently compute exact (discrete) gradients when using continuous, parameterized control pulses while solving the forward equations (e.g. Schrodinger's equation or the Linblad master equation) with an arbitrarily high-order Hermite Runge-Kutta method. The HOHO method is implemented in QuantumGateDesign$.$jl (https://github.com/leespen1/QuantumGateDesign.jl), an open-source software package for the Julia programming language, which we use to perform numerical experiments comparing the method to Juqbox$.$jl (https://github.com/LLNL/Juqbox.jl). For realistic model problems we observe speedups up to 775x.

Figures (6)

• Figure 1: Control pulse implementing the Hadamard gate for the Hamiltonian $H(t) = 0.5\omega_0 \sigma_z + c(t)\sigma_x$.
• Figure 2: Generalized infidelity of a CNOT gate, as computed by a low-accuracy and high-accuracy numerical method. The generalized infidelity is optimized using a small number of timesteps. For each set of control parameters found at each iteration of the optimization, we also simulate the system using these control parameters and a larger number of timesteps in order to obtain a high-accuracy solution, which is used to compute a more accurate generalized infidelity and the error in the low-accuracy solution at the final time.
• Figure 3: Comparison of the number of timesteps used, the time taken to compute the gradient, and the relative error in the numerical approximation of the state, for the three subsystem example, starting in the initial states of the gate design problem: $|000\rangle, |001\rangle, |010\rangle$, and $|011\rangle$. The HOHO method is used with orders 2, 4, 6, 8, 10, and 12. We also compare with the second-order method Störmer-Verlet, as implemented in the Julia package Juqbox.jl.
• Figure 4: Optimization of a CNOT gate using two qudits coupled to a resonator bus as the mode, using B-spline curves multiplied carrier waves as the control pulse ansatz, using the methods and numbers of timesteps prescribed by Table \ref{['tab:cnot3_stepsize']} in order to reach target relative final state errors on the orders of $10^{-1}$, $10^{-3}$, $10^{-5}$, and $10^{-7}$. We do not include the second-order method for target errors of $10^{-5}$ and $10^{-7}$ due to the memory needed to store the state history for the large number of timesteps required. The second-order method could be used with checkpointing or the "memory-lean" approach, but it is clear that at these levels of accuracy the higher-order methods are significantly faster.
• Figure 5: Time evolution of the state populations for the CNOT gate implemented by the best (lowest generalized infidelity) control pulses found by the optimizations performed with a target relative final state error of $10^{-7}$ for the two qudits plus resonator model (see Figure \ref{['fig:cnot3_optimization']}). The results shown here were obtained using the sixth-order Hermite method with 5,409 timesteps ($\Delta t \approx 0.102$ nanoseconds), which is the order and number of timesteps used in the optimization which found this control pulse.

Theorems & Definitions (9)

• Remark 1
• Theorem 1
• proof
• Remark 2
• Theorem 2
• proof
• Remark 3
• Remark 4
• Remark 5