Random layers for quantum optimal control with exponential expressivity

Abstract

A long-standing challenge in quantum optimal control is finding an optimal pulse structure that leads to an efficient exploration of the unitary space with a minimal number of optimization parameters. We solve this challenge by constructing parametrized pulse sequences from random constant-amplitude pulses grouped in layers with one optimization parameter per layer. We show that, when increasing the number of pulses, the resulting random unitaries converge exponentially fast to the uniform Haar-random ensemble. Grouping the pulses into layers allows to lower the total number of optimization parameters. We focus on two random-layer (RALLY) methods: In RALLY$_\text{T}$, time durations of the layers are optimized while the pulse amplitudes are randomly chosen beforehand, possibly even from a few discrete values. RALLY$_\text{A}$ optimizes a joint scaling factor of the random pulse amplitudes in each layer. We numerically validate the two methods by applying them to problems of unitary synthesis, ground-state preparation and state transfer in different quantum systems. For all problems considered, both methods approach an information-theoretic lower bound on the number of optimization parameters and outperform other commonly used algorithms. In gradient-free optimization, the RALLY methods are orders of magnitude more accurate with fewer figure-of-merit evaluations. The RALLY methods are also applicable for enhanced quantum machine learning and variational quantum algorithms.

Figures (13)

• Figure 1: (a) Sketch of a pulse-optimization procedure in quantum optimal control. The dynamics of a quantum system with internal Hamiltonian ${\cal H}_{0}$ is controlled by applying a pulse sequence $u(t)$ that couples to the system via the control Hamiltonian ${\cal H}_{c}$. The optimal pulse is found by minimizing a figure of merit $J[u]$ in a feedback loop. (b) In RALLY, the control pulse sequence $u(t)$ is grouped into $N_\text{L}$ layers (different colors) with $N_\text{P}$ pulses per layer. The pulse amplitudes are chosen randomly. In RALLY$_\text{T}$, the layer durations $\tau_\ell$ are optimized. In RALLY$_\text{A}$, the scaling factors $\xi_\ell$ of the control amplitudes are optimized. (c) Illustration of how the expressivity increases for larger $N_\text{P}$. When changing $\tau$, the dynamics of a single layer traces out a trajectory on the Bloch sphere. With increasing $N_\text{P}$, this trajectory becomes more complex and covers an increasing area of the Bloch sphere. Single-spin trajectories are shown for the Hamiltonian $\mathcal{H}(t) = 5\sigma_z + u(t)\sigma_x$, starting from the initial state $\lvert 0 \rangle$, with evolution time $\tau = 100$ and $u(t)$ composed of fixed control amplitudes whose values are randomly sampled from the interval $[-2,2]$.
• Figure 2: Exponential convergence of the first four moments of the ensemble of unitaries reachable by the RALLY$_\text{T}$ propagator $U_\text{R,T}$ to the corresponding moments of a Haar random distribution. The difference $\delta_t$\ref{['eq:delta_t']} between the corresponding moments is shown for $t=1,2,3,4$ as a function of $N_{\mathrm{L}}N_{\mathrm{P}}$. Orange points show $\delta_t$ with layer durations drawn uniformly from $[0,10]$ and control amplitudes drawn uniformly from $[-1,1]$. The dashed red line is a linear fit on the logarithmic scale. More importantly, results for $\delta_t$ follow the same decay curve also when only layer durations are sampled and the control amplitudes are fixed (shown by the 10 green lines). We attribute the fluctuations of the green lines to the finite system sizes, see Appendix \ref{['app_typ']}. The plateaus at larger values of $N_{\mathrm{L}}N_{\mathrm{P}}$ are due to the finite sample size and they are consistent with the expected values from the Haar distribution (black dotted lines); see Appendix \ref{['App:Harr_conv']} for details. For each value of $N_{\mathrm{L}}N_{\mathrm{P}}$, where $N_{\mathrm{L}}\in\{17, 50, 83\}$ and $N_{\mathrm{P}}\in\{1,2,3,4\}$ we use $10^7$ samples and average over 10 repetitions for one set of randomly chosen values of $h_{ix}$ and $h_{iz}$, cf. Eq. \ref{['eq:random_spin_H']}.
• Figure 3: Illustration of the interpolation between successive control amplitudes to accommodate the hardware's finite bandwidth in RALLY$_\text{T}$. A limited bandwidth $\Delta\Omega$ imposes a finite rise time $\tau_\text{rise}$, implemented in the pulse via an interpolating function that depends only on the amplitudes and not on the pulse durations. Because RALLY$_\text{T}$ samples amplitudes before optimization and optimizes time durations, the propagators for these interpolations, denoted $U_{\text{rise}}$, can be precomputed and included in the figure of merit, and eventually in its gradient, as constants.
• Figure 4: Performance of RALLY methods for the unitary synthesis of a 3-qubit gate, cf. Eqs. \ref{['eq:target_unitary']}, in a globally-driven Rydberg-atom platform. As a function of the number of optimization parameters ($N_\text{L}$ for RALLY), the figure shows the median unitary infidelity $J_\text{u}$\ref{['eqn_unitary']} (top), success probability of achieving $J_\text{u}\le 10^{-3}$ over 10 independent runs (middle) and the median number of figure-of-merit evaluations required to reach $J_\text{u}\le 10^{-3}$ (bottom). The RALLY methods with $N_\text{P}=5$ rapidly converge to the target precision of $10^{-9}$ and efficiently saturate the information-theoretic lower bound (vertical dashed line) on the number of optimization parameters, whereas dCRAB with Fourier and piecewise constant (pwc) bases converges more slowly. The RALLY methods also need fewer figure-of-merit (FoM) evaluations than dCRAB. Missing bars in the bottom part correspond to cases where no successful optimizations were achieved. All methods use the same optimizer (adaptive Nelder–Mead) and identical stopping criteria.
• Figure 5: Success probability of achieving $J_\text{u}\le 10^{-3}$ for different total pulse-sequence durations and layer sizes $N_\text{P}$ using RALLY$_\text{T}$. The results indicate that the effectiveness of the method improves as the layer size increases while keeping the total pulse-sequence duration fixed. For each pair of total duration and $N_\text{P}$, we run at least 100 optimizations with different random initializations of $\tau_\ell$ and group the results by their final total pulse-sequence durations. Due to the abundance of local minima in the optimization landscape, the layer durations typically remain close to their initial values, so the heatmap effectively samples different total durations at fixed $N_\text{P}$.

Theorems & Definitions (4)

• Theorem D.1: Finite uniform-weight $t$-design bound
• proof : Proof: (See Ref Scott_2008)
• Theorem D.2: Continuous measure version
• proof : Proof: (Same of Ref Scott_2008, replace sums by integrals)