Random pulse layers boost quantum control with fewer parameters

Marco Dall’Ara and colleagues at the Fraunhofer Institute for Applied Solid State Physics IAF have developed new methods to streamline quantum control by constructing pulse sequences from random-amplitude pulses organized into layers, each governed by a single optimization parameter. The researchers demonstrate that increasing the number of these pulses leads to random unitaries converging exponentially fast, efficiently exploring the possibilities for controlling quantum states.

This approach lowers the total number of parameters needed for optimization, and in gradient-free optimization, the RALLY methods are orders of magnitude more accurate with fewer evaluations. These findings offer a path toward advancing quantum machine learning and variational quantum algorithms.

Random Layers Enable Exponential Unitary Space Exploration

This convergence signifies an efficient exploration of the unitary space, a longstanding challenge in quantum optimal control. Two distinct methods, termed $RALL{Y}_T$ and $RALL{Y}_A$, were developed to control these layers. $RALL{Y}_T$ focuses on optimizing the time durations of each layer while maintaining randomly selected pulse amplitudes, potentially using discrete values. Alternatively, $RALL{Y}_A$ optimizes a single scaling factor applied to the random pulse amplitudes within each layer, keeping pulse durations constant.

“Layers – grouping pulses into layers, with one optimization parameter per layer, reduces the number of optimization parameters and simplifies the optimization,” the researchers state, highlighting the core benefit of this layered structure. Numerical validation of both $RALL{Y}_T$ and $RALL{Y}_A$ demonstrated their effectiveness in problems involving unitary synthesis, ground-state preparation and state transfer across diverse quantum systems.

For all tested problems, these methods approached an information-theoretic lower bound on the number of optimization parameters, consistently outperforming other commonly used algorithms. In gradient-free optimization scenarios, the RALLY methods achieved substantially higher accuracy with fewer evaluations of the figure of merit. The work demonstrates that this pulse-sequence structure converges exponentially fast with increasing layers to the Haar-random ensemble, providing exponential expressivity.

RALLY_T and RALLY_A Methods for Quantum Optimal Control

The efficiency with which quantum systems explore possible states increases exponentially as the number of pulse layers grows, according to a new approach to quantum optimal control. This design contrasts with conventional methods that typically demand a larger number of adjustable parameters to achieve comparable control over quantum dynamics. This flexibility allows researchers to tailor the optimization process to specific quantum systems and control requirements. This means the methods achieve high accuracy with a minimal set of adjustments, a critical advantage for complex quantum computations.

The researchers also found that the resulting random unitaries converge rapidly toward the uniform Haar-random ensemble, indicating an efficient exploration of the entire space of possible quantum states. “The ensemble of reachable unitaries converges exponentially fast to the uniform Haar-random ensemble with the total number of pulses,” the paper states, highlighting the speed and thoroughness of this exploration. Beyond immediate applications in quantum control, the principles behind RALLY methods have implications for broader areas of quantum information processing.

The random-layer structure can be directly incorporated into parameterized circuits used in variational quantum algorithms and quantum machine learning, potentially accelerating progress in these fields. The researchers suggest that these techniques could even inspire the development of more efficient classical optimal control methods, demonstrating a cross-disciplinary impact. Further research will explore the application of these random-layer concepts to pulse sequences based on different mathematical representations, such as the Fourier basis.

Layered Pulse Sequences Reduce Optimization Parameters

This counterintuitive result stems from the way RALLY methods construct pulse sequences, allowing the number of pulses to increase without a corresponding increase in parameters needed for adjustment. Analytical proof demonstrates the ensemble of unitaries generated by these methods approaches a uniform distribution, which is important for efficient optimization of arbitrary target unitaries, with convergence accelerating exponentially with layer size. The convergence behavior holds true even when sampling only the layer-level optimization parameters, a feature that enhances optimization efficiency. For all problems considered, the RALLY methods outperform other commonly used algorithms.

Haar-Random Ensemble Convergence with Increasing Layer Size

Layered pulse sequences offer a pathway to efficient quantum control by minimizing the number of parameters needing optimization, a benefit stemming from the convergence behavior observed when increasing layer size. Numerical evidence indicates that the ensemble of unitaries achievable with these methods exhibits the same convergence characteristics regardless of whether optimization samples all parameters or only the N_L layer-level parameters. This feature allows expressivity to increase with layer size N_P without increasing the number of optimization parameters N_L, streamlining the optimization process.

Increasing the layer size N_P enhances these convergence properties without requiring more optimization parameters N_L, offering a strategy to avoid over-parametrization while maintaining a larger total pulse count. Fluctuations in convergence behavior become more pronounced when reducing the number of spins in an Ising spin chain, a detail explored further in the paper’s appendices.

This typical behavior underpins the efficacy of the RALLY methods, which rely on constructing parametrized pulse sequences from random-amplitude pulses. The researchers demonstrate that the first four moments of the ensemble of unitaries reachable by the $RALL{Y}_T$ propagator converge exponentially to the corresponding moments of a Haar-random distribution, as visualized in the supporting figures.

RALLY_T Accommodates Experimental Constraints and Shorter Durations

$RALL{Y}_T$ readily incorporates practical limitations encountered in quantum hardware, a feature distinguishing it from many optimization techniques. Finite bandwidth constraints, often problematic for pulse sequences relying on abrupt changes, are easily addressed within the $RALL{Y}_T$ framework. $RALL{Y}_T$ offers flexibility in managing pulse duration. An additional penalty term can be introduced to minimize the total pulse-sequence duration without sacrificing optimization efficiency. The researchers demonstrate that $RALL{Y}_T$ maintains performance even when pulse amplitudes are restricted to a discrete set of values, further enhancing its adaptability to experimental setups.

For state transfer problems involving an Ising spin chain, $RALL{Y}_T$ and GRAPE exhibited comparable results when applied to a six-spin system. “Constraints of the experimental setting, such as a finite bandwidth, are readily accommodated,” the researchers note, highlighting the method’s practical advantages. The approach also allows for the incorporation into existing algorithms like CRAB and dCRAB, functioning as an enhanced set of basis functions.

Numerical Validation: Unitary Synthesis and State Preparation

RALLY_T demonstrated computational efficiency in simulations applicable to larger systems than many existing quantum-optimal-control algorithms, a benefit stemming from its design for streamlined calculations. Researchers specifically targeted three control tasks to benchmark the methods: synthesizing a three-qubit gate, preparing the ground state on a Rydberg-atom platform, and facilitating state transfer within an Ising spin chain.

The Rydberg-atom platform’s Hamiltonian incorporates Van der Waals interactions between atoms and external driving forces, while the Ising spin chain provides a different test environment for evaluating transfer capabilities. The ability of $RALL{Y}_T$ and $RALL{Y}_A$ to function in both gradient-based and gradient-free optimization offers versatility in pulse optimization strategies, accommodating both closed-loop and open-loop control schemes.

RALLY Methods Outperform Algorithms in Gradient-Free Optimization

This improved accuracy is particularly notable as it was achieved with a reduced number of optimization parameters, simplifying the control process. $RALL{Y}_T$, with minor modifications, also generates pulse sequences with shorter total durations, potentially reducing experimental overhead. By randomly selecting pulse amplitudes from just two discrete values and using a gradient-based optimization routine, $RALL{Y}_T$ accurately optimized systems containing two additional qubits compared to the GRAPE algorithm within the same computational timeframe.

The method also readily accommodates realistic limitations imposed by experimental setups, such as finite bandwidth constraints. Each layer within the RALLY framework consists of a series of constant-amplitude pulses, with the number of pulses per layer defining the layer size. Optimization parameters are either layer durations (in RALLY_T) or scaling factors for control amplitudes (in RALLY_A), denoted as and respectively.

RALLY_T Scales to Larger Systems Compared to GRAPE

This scaling capability stems from the method’s computational efficiency, allowing for simulations of larger quantum systems than previously possible with comparable quantum-optimal-control algorithms. Numerical simulations confirm that $RALL{Y}_T$ maintains accuracy while expanding the scale of addressable systems, a critical step toward practical quantum control. The ability to minimize the number of parameters is important for reducing the complexity of quantum computations and accelerating the development of scalable quantum technologies.

The design of $RALL{Y}_T$ also allows for realistic constraints, such as limited bandwidth, to be readily incorporated into the optimization process. By adding an explicit penalty on total pulse duration, $RALL{Y}_T$ efficiently identifies short pulses suitable for implementation in large systems. “Within the maximal runtime of 2× ${10}^4$ s, $RALL{Y}_T$ is able to handle systems with two spins more than GRAPE,” the researchers report, highlighting the practical implications of this approach. The simulations, averaging results from ten independent calculations, consistently show $RALL{Y}_T$’s superior performance in scaling to more complex quantum systems.

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