Session Y41: Quantum Control: Optimal Control

Optimal Control of Disordered Quantum Systems

We investigate several control strategies for the transport of an excitation along a spin chain. We demonstrate that high fidelity transport at the quantum speed limit can be achieved using protocols designed with differentiable programming. Building on this, show that this approach can be effectively adapted to control a disordered quantum system. We consider two settings: optimal control for a known, unwanted disorder pattern (i.e. a specific disorder realisation) and optimal control where only the disorder structure is known (i.e. optimising for high average fidelities). In the former, the disorder effects can be effectively countered for an appropriately chosen control protocol and extremely high fidelity control. However, in the latter setting the average fidelity can only be marginally improved.   

Quantum Noise Spectroscopy Informed Optimized Gates

In recent years, a number of quantum noise spectroscopy (QNS) protocols have been developed to characterize spatio-temporally correlated noise processes. Estimates of the noise power spectral density from QNS protocols are meant to inform optimized control protocols designed to mitigate noise while simultaneously implementing a particular quantum operation. While it is widely accepted that QNS should yield an added advantage to optimized control, there has yet to be an experimental demonstration of QNS-informed optimized control on a non-trivial gate. In this talk, we experimentally demonstrate the advantage of Gradient Ascent in Function Space (Filter GrAFS) optimized control, using injected noise as a probe. Injected noise is generated using the Schrodinger Wave Autoregressive Moving Average (SchWARMA) model, phase-modulating an ideal control signal to mimic the noise spectrum of pure-tone dephasing noise.  By subjecting optimized and non-optimized controls to different noise environments, we experimentally demonstrate the advantage of Filter GrAFS optimized control.   

Filter Function Tailored Noise-Optimized Single Qubit Gates

Noise suppression protocols represent a class of techniques that are meant to reduce gate error rates via specially designed control sequences. Here, we discuss a protocol that optimizes smooth control sequences to combat temporally-correlated noise, a class of noise that can be detrimental to quantum error correction. The control ansatz is specifically chosen to be a functional expansion of Slepians, a discrete time basis known to be optimally concentrated in time and frequency, and quite attractive when faced with experimental control hardware constraints. We leverage the filter function formalism to transform the control problem into a filter design problem, and show that the frequency response of a quantum system can be carefully tailored to avoid the most relevant dynamical contributions of the noise processes. Using gradient ascent, we obtain optimized filter functions and investigate the relationship between filter function design, control bandwidth, and noise characteristics. Even in the presence of multi-axis correlated noise, we show that as long as the noise spectral density is sufficiently weak over a range of frequencies, it is possible to generate control sequences that yield highpass or bandpass filter functions, and simultaneously produce high fidelity, arbitrary single qubit gate operations.   

Optimal control of open quantum systems

Optimal control theory (OCT) is a versatile tool that can be used to identify control strategies in the presence of decoherence, either avoiding or exploiting the environment. In this talk, I will discuss an example for each. First, for qubits encoded in the infinite-dimensional Hilbert space of bosonic modes, I will show how to employ OCT to systematically enhance performance in the implementation of strong and on-demand interactions between the qubits [1]. Second, I will discuss OCT in driven dissipative evolutions. For example, in trapped ion qubits, a protocol derived with OCT allows for the resource-efficient dissipative generation of an entangled state [2]. Finally, I will assess prospects for quantum control in reservoir engineering where the desired dissipation can be realized via quantum non-demolition measurements.

[1] D. Basilewitsch, Y. Zhang, S.M. Girvin, C.P. Koch, arXiv:2111.15573

[2] D.C. Cole, S.D. Erickson, G. Zarantonello, K.P. Horn, P.-Y. Hou, J.J.

Wu, D.H. Slichter, F. Reiter, C.P. Koch, D. Leibfried, arXiv:2108.03136   

Improving the performance of algorithms on NISQ-Devices through automated control design and implementation

Current NISQ era quantum computing devices suffer from various sources of noise, such as leakage, dephasing and cross-talks which makes running useful quantum algorithms especially challenging on such devices. Our team recently demonstrated that quantum control can be used to improve the performance of individual gates, through automated-closed loop optimization of microwave pulses. These control-design techniques consistently nullify coherent errors on at least half of the CNOT gates on a quantum device, with others limited by T1. Previously, it was not clear how these improvements would impact the ultimate performance of high level quantum algorithms.  We investigate how drop-in replacement of optimized control pulses can improve the performance of algorithms with potential NISQ advantage, namely, quantum fourier transform (QFT), Bernstein-Vazirani and QAOA in both simulation and experiment. We execute a fully autonomous parallel gate optimization across multiqubit systems, and with the resulting optimized gates see consistent improvements in algorithm performance.  In IBM systems with no further change to compilation strategy, drop-in gate replacement delivers at least a 30% improvement to the success probability of an accurate outcome. We observe that deterministic algorithms benefit most from reduction of coherent gate errors.   

Limitations of Average Hamiltonian Theory for Quantum Control

One path to Hamiltonian Engineering in solid-state spin systems is to use periodic pulse trains to make the system evolve under a desired effective Hamiltonian. These pulse trains are designed using Average Hamiltonian Theory, which uses the Magnus series expansion to obtain a time-independent description of a periodic time-dependent Hamiltonian at stroboscopic intervals. The most common application of these pulse trains has been to decouple internuclear magnetic dipolar interactions in solid state NMR. These sequences have been improved by increasing their length and complexity in order to cancel out higher order terms in this expansion. It has generally been believed that the Magnus expansion converges rapidly in the parameter regime in which these sequences are used, and that therefore the largest source of error in implementing these sequences comes from the lowest terms in the expansion that the pulse sequence does not account for. In an effort to explore the limits of control, we numerically explore if the maximum attainable fidelity of conventional AHT-designed dipolar decoupling sequences is due to their sensitivity to experimental error or due to poor convergence of the Magnus series.   

Supercharging Quantum Optimal Control with Efficient Automatic Differentiation

Numerical optimal control theory has proven an essential tool for achieving a core requirement of both quantum information and quantum metrology: the creation of particular non-classical states, respectively of quantum processes that produce such states. Finding the most general solutions to the relevant control problems would require the direct optimization of, e.g., measures for entanglement, spin squeezing, or the quantum Fisher information. To date, such measures have generally not been considered suitable for optimal control due to the lack of an analytic derivative. We show how the use of automatic differentiation (AD) allows to directly optimize these non-classical measures, and virtually any other functional.  We further show how AD can be \emph{combined} with existing numerical techniques of quantum control to allow the formulation of a ``semi-automatic differentiation'' approach that eliminates the often exponential overhead associated with previous attempts to use AD in quantum control. Thus, our methods scale to large Hilbert space dimensions and open quantum systems. We illustrate the use of the technique for the optimization of entangling quantum gates and the creation of spin-squeezed states.

The proposed methods are realized in a new open-source software framework for numerical quantum control, QuantumControl.jl, written in the Julia language. Designed for extensibility and performance, the framework consists of a collection of packages that implement a wide variety of control algorithms, including Krotov's method, several variations of gradient-ascent methods (GRAPE, GROUP, GOAT), and gradient-free methods such a CRAB. These are enhanced with methods of automatic differentiation for unparalleled flexibility.   

Push-pull optimization of quantum controls (PPOQC)

Optimization of quantum controls to achieve a target process is centered around an objective function comparing the realized process with the target. We propose an objective function that incorporates not only the target operator but also a set of its orthogonal operators whose combined influence leads to an efficient exploration of the parameter space, faster convergence, and extraction of superior solutions. The push-pull optimization, as we call it, can be adopted in various quantum control scenarios. We describe adopting it for gradient-based and variational-principle-based approaches. Numerical analysis of quantum registers with up to seven qubits reveals significant benefits of push-pull optimization. We describe applying the push-pull optimization to prepare a long-lived singlet order in a two-qubit system using NMR techniques.   

Optimizing quantum control pulses with complex constraints and few variables through Tensorflow

Applying optimal control algorithms on realistic quantum systems confronts two key challenges: to efficiently adopt physical constraints in the optimization and to minimize the variables for the convenience of experimental tune-ups. In order to resolve these issues, we propose a novel algorithm by incorporating multiple constraints into the gradient optimization over piece-wise pulse constant values, which are transformed to contained numbers of the finite Fourier basis for bandwidth control. Such complex constraints and variable transformation involved in the optimization introduce extreme difficulty in calculating gradients. We resolve this issue efficiently utilizing auto-differentiation on Tensorflow. We test our algorithm by finding smooth control pulses to implement single-qubit and two-qubit gates for superconducting transmon qubits with always-on interaction, which remains a challenge of quantum control in various qubit systems. Our algorithm provides a promising optimal quantum control approach that is friendly to complex and optional physical constraints.   

Increasing memory and runtime performance of GRAPE for control in large quantum systems

Gradient Ascent Pulse Engineering (GRAPE) is a popular technique in quantum optimal control. Recent implementations of the GRAPE algorithm are based on automatic differentiation (AD). AD helps avoid the need for coding of analytical gradients but incurs a large memory cost. Specifically, AD stores intermediate states and propagators at all time steps, thus posing a severe bottleneck for quantum systems with large Hilbert space dimension. To address this issue, we implement hard-coded analytical gradients in a scheme that avoids propagator storage and significantly reduces the storage of states. We further succeed in enhancing runtime performance with an improved algorithm for state propagation and propagator derivatives. We benchmark the performance and memory cost of our code against AD-based implementations for large-dimensional state transfer and gate optimization problems. Results confirm the expected improvements that will allow us to tackle optimal control of much larger quantum systems in the future.   

Robust quantum computing on qubit arrays with fixed coupling

We show how robust and scalable quantum computing can be realized on an arbitrarily large two dimensional arrays of qubits with fixed longitudinal couplings despite significant uncertainty in all the qubit-qubit and drive-qubit coupling strengths. This opens the possibility for bypassing the fabrication complexity associated with tunable couplers required in conventional quantum computing hardware. Our approach is based on driving a chosen subarray of qubits such that the total multi-qubit Hamiltonian can be decomposed into a sum of commuting few-qubit blocks and on efficient optimization of the unitary evolution within each block. Robust optimal control is then employed to implement a universal set of quantum gates with fidelities around 99.99% despite 1% uncertainty in the Hamiltonian's parameters. This robust feature is crucial for scaling up as uncertainty in the properties of qubits is substantial in large devices.   

Optimizing cross resonance gates using recurrent neural networks

Quantum control is a powerful technique for implementing quantum circuits and designing high-fidelity quantum gates. Besides analytical and gradient-based control protocols, recently machine learning has evolved as a tool to generate control pulses for implementing high fidelity quantum gates. In this work, we demonstrate a new deep learning model based on encoder-decoder architecture using Long Short Term Memory (LSTM) units combined with convolution layers to implement high fidelity fast quantum gates. The model architectures are trained to generate optimized pulse sequences for single and two-qubit gates with infidelities in the order of 10-5 and 10-4 respectively. Furthermore, we incorporate real-world hardware limitations by incorporating pulse constraints like amplitude and bandwidth limitations. We apply these techniques to coupled transmon qubits and study the optimal sequences with attention to leakage out of computational subspace and the effect of decoherence.