Gradient Ascent Pulse Engineering Research Articles

Flexible preparation of the W state via the GRAPE algorithm in a nanoring-quantum-dot system

In this paper, we put forward a scheme for flexibly preparing W states in the presence of detunings and asymmetric couplings in a nanoring-quantum-dot (nanoring-QD) system. Based on the gradient ascent pulse engineering algorithm, the coupling strength modulation (CSM) and the energy level modulation (ELM) methods are proposed, respectively. In the CSM method, the waveforms of the coupling strengths are properly designed to compensate for detunings, thus the decline on the fidelity is effectively suppressed. For the asymmetric couplings, a high-fidelity W state can be generated by the ELM method, which alters the energy levels of QDs in a desired manner. Finally, we generalize the current scheme to realize the preparation of N-particle W states in nanoring-QD systems.

Hamiltonian Switching Control of Noisy Bipartite Qubit Systems

Abstract We develop a Hamiltonian switching ansatz for bipartite control that is inspired by the Quantum Approximate Optimization Algorithm (QAOA), to mitigate environmental noise on qubits. We demonstrate the control for a central spin coupled to bath spins via isotropic Heisenberg interactions, and then make physical applications to the protection of quantum gates performed on superconducting transmon qubits coupling to environmental two-level-systems (TLSs) through dipole-dipole interactions, as well as on such qubits coupled to both TLSs and a Lindblad bath. The control field is classical and acts only on the system qubits. We use reinforcement learning with policy gradient (PG) to optimize the Hamiltonian switching control protocols, using a fidelity objective for specific target quantum gates. 
 We use this approach to demonstrate effective suppression of both coherent and dissipative noise, with numerical studies achieving target gate implementations with fidelities over 0.9999 (four nines) in the majority of our test cases and showing improvement beyond this to values of 0.999999999 (nine nines) upon a subsequent optimization by Gradient Ascent Pulse Engineering (GRAPE). We analyze how the control depth, total evolution time, number of environmental TLS, and choice of optimization method affect the fidelity achieved by the optimal protocols and reveal some critical behaviors of bipartite control of quantum gates.

Optimal control in a magnetization-prepared rapid acquisition gradient-echo sequence.

This article proposes a numerical framework to determine the optimal magnetization preparation in a three-dimensional magnetization-prepared rapid gradient-echo (MP-RAGE) sequence to obtain the best achievable contrast between target tissues based on differences in their relaxation times. The benefit lies in the adaptation of the algorithm of optimal control, GRAdient Ascent Pulse Engineering (GRAPE), to the optimization of magnetization preparation in a cyclic sequence without full recovery between each cycle. This numerical approach optimizes magnetization preparation of an arbitrary number of radio frequency pulses to enhance contrast, taking into account the establishment of a steady state in the longitudinal component of the magnetization. The optimal control preparation offers an optimized mixed contrast in this traditional -weighted sequence. To show the versatility of the proposed method, numerical and in vitro results are described. Examples of contrasts acquired on brain regions of a healthy volunteer are presented for potential applications at 3 T.

GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controls

The GRadient Ascent Pulse Engineering (GRAPE) method is widely used for optimization in quantum control. GRAPE is gradient search method based on exact expressions for gradient of the control objective. It has been applied to various coherently controlled closed and open quantum systems. In this work, we adopt the GRAPE method for optimizing objective functionals in open quantum systems driven by both coherent and incoherent controls. In our case, a tailored or engineered environment acts on the controlled system as control via time-dependent decoherence rates or, equivalently, via spectral density of the environment. To develop the GRAPE approach for this problem, we compute gradient of various objectives for general N-level open quantum systems for the piecewise constant class of control. The case of a single qubit is considered in details and solved analytically. For this case, an explicit analytical expression for evolution and objective gradient is obtained via diagonalization of a matrix determining the system’s dynamics in the Bloch ball. The diagonalization is obtained by solving a cubic equation via Cardano’s method. The efficiency of the algorithm is demonstrated through numerical simulations for the state-to-state transition problem and its complexity is estimated. Robustness of the optimal controls is also studied.

QuOCS: The quantum optimal control suite

Quantum optimal control includes a family of pulse-shaping algorithms that aim to unlock the full potential of a variety of quantum technologies. The Quantum Optimal Control Suite (QuOCS) unites experimental focus and model-based approaches in a unified framework. Easy usage and installation presented here and the availability of various combinable optimization strategies is designed to improve the performance of many quantum technology platforms, such as color defects in diamond, superconducting qubits, atom- or ion-based quantum computers. It can also be applied to the study of more general phenomena in physics. In this paper, we describe the software and the toolbox of gradient-free and gradient-based algorithms. We then show how the user can connect it to their experiment. In addition, we provide illustrative examples where our optimization suite solves typical quantum optimal control problems, in both open- and closed-loop settings. Integration into existing experimental control software is already provided for the experiment control software Qudi (Binder et al., 2017 [41]), and further extensions are investigated and highly encouraged.QuOCS is available from GitHub, under Apache License 2.0, and can be found on the PyPI repository. Program summaryProgram Title: QuOCS - Quantum Optimal Control SuiteCPC Library link to program files:https://doi.org/10.17632/wjjch757fk.1Developer's repository link:https://github.com/Quantum-OCS/QuOCSLicensing provisions: Apache-2.0Programming language: PythonExternal routines: NumPy [1], SciPy [1], JAX [2]Nature of problem: Quantum systems are typically controlled by time-dependent electromagnetic fields to perform a certain set of quantum operations. Those operations may in turn provide building blocks for various quantum information processing tasks such as quantum computation, communication, simulation, sensing, and metrology. Numerous control strategies exist to design and improve such operations [3]. While some strategies are constructed to target a rather specific problem with high efficiency, others are more general to solve a wide range of applications [4,5]. To access the different algorithms, one has to download different optimization suites with different input and output parameters, making them hard to compare and combine. To benefit from the variety of algorithms, we have devised a customizable and intuitive optimization suite that simultaneously provides access to some of the most popular quantum optimal control algorithms.Solution method: We combine, in a unified framework, some of the frequently used optimal control algorithms which are the dressed Chopped Random Basis method (dCRAB) [6], and Gradient Ascent Pulse Engineering (GRAPE) [7], with an extension to make use of Automatic Differentiation (AD) [8]. The software is able to connect to both models of quantum dynamics in simulations and real-time quantum experiments to perform open- and closed-loop optimization, respectively. With minimal knowledge of optimal control theory, the user can manage to run optimizations of quantum processes using a variety of additional features such as stopping criteria and drift compensation. Logging and data management of the optimization progress and results are also handled by the suite. Its modular structure allows for extensions that accommodate customized algorithms and can be implemented by the user straightforwardly.Additional comments including unusual features: The connection to the experiments is performed by an extension that enables a direct integration to a laboratory control software Qudi [9].

Binary Control Pulse Optimization for Quantum Systems

Quantum control aims to manipulate quantum systems toward specific quantum states or desired operations. Designing highly accurate and effective control steps is vitally important to various quantum applications, including energy minimization and circuit compilation. In this paper we focus on discrete binary quantum control problems and apply different optimization algorithms and techniques to improve computational efficiency and solution quality. Specifically, we develop a generic model and extend it in several ways. We introduce a squared L2-penalty function to handle additional side constraints, to model requirements such as allowing at most one control to be active. We introduce a total variation (TV) regularizer to reduce the number of switches in the control. We modify the popular gradient ascent pulse engineering (GRAPE) algorithm, develop a new alternating direction method of multipliers (ADMM) algorithm to solve the continuous relaxation of the penalized model, and then apply rounding techniques to obtain binary control solutions. We propose a modified trust-region method to further improve the solutions. Our algorithms can obtain high-quality control results, as demonstrated by numerical studies on diverse quantum control examples.

Improved quantum computing with higher-order Trotter decomposition

In designing quantum control, it is generally required to simulate the controlled system evolution with a classical computer. However, computing the time evolution operator can be quite resource-consuming since the total Hamiltonian is often hard to diagonalize. In this paper, we mitigate this issue by substituting the time evolution segments with their Trotter decompositions, which reduces the propagator into a combination of single-qubit operations and fixed-time system evolutions. The resulting procedure can provide substantial speed gain with acceptable costs in the propagator error. As a demonstration, we apply the proposed strategy to improve the efficiency of the gradient ascent pulse engineering algorithm for searching optimal control fields. Furthermore, we show that the higher-order Trotter decompositions can provide efficient Ans\"atze for the variational quantum algorithm, leading to improved performance in solving the ground-state problem. The strategy presented here is also applicable for many other quantum optimization and simulation tasks.

Optimal control for state preparation in two-qubit open quantum systems driven by coherent and incoherent controls via GRAPE approach

In this work, we consider a model of two qubits driven by coherent and incoherent time-dependent controls. The dynamics of the system is governed by a Gorini–Kossakowski–Sudarshan–Lindblad master equation, where coherent control enters into the Hamiltonian and incoherent control enters into both the Hamiltonian (via Lamb shift) and the dissipative superoperator. We consider two physically different classes of interaction with coherent control and study the optimal control problem of state preparation formulated as minimization of the Hilbert–Schmidt distance’s square between the final density matrix and a given target density matrix at some fixed target time. Taking into account that incoherent control by its physical meaning is a non-negative function of time, we derive an analytical expression for the gradient of the objective and develop optimization approaches based on adaptation for this problem of GRadient Ascent Pulse Engineering (GRAPE). We study evolution of the von Neumann entropy, purity, and one-qubit reduced density matrices under optimized controls and observe a significantly different behavior of GRAPE optimization for the two classes of interaction with coherent control in the Hamiltonian.

Optimization of a Controlled- Z Gate with Data-Driven Gradient-Ascent Pulse Engineering in a Superconducting-Qubit System

The experimental optimization of a two-qubit controlled-Z (CZ) gate is realized following two different data-driven gradient ascent pulse engineering (GRAPE) protocols in the aim of optimizing the gate operator and the output quantum state, respectively. For both GRAPE protocols, the key computation of gradients utilizes mixed information of the input Z-control pulse and the experimental measurement. With an imperfect initial pulse in a flattop waveform, our experimental implementation shows that the CZ gate is quickly improved and the gate fidelities subject to the two optimized pulses are around 99%. Our experimental study confirms the applicability of the data-driven GRAPE protocols in the problem of the gate optimization.

Generalizable control for multiparameter quantum metrology

Quantum control can be employed in quantum metrology to improve the precision limit for the estimation of unknown parameters. The optimal control, however, typically depends on the actual values of the parameters and thus needs to be designed adaptively with the updated estimations of those parameters. Traditional methods, such as gradient ascent pulse engineering (GRAPE), need to be rerun for each new set of parameters encountered, making the optimization costly, especially when many parameters are involved. Here we study the generalizability of optimal control, namely, optimal controls that can be systematically updated across a range of parameters with minimal cost. In cases where control channels can completely reverse the shift in the Hamiltonian due to a change in parameters, we provide an analytical method which efficiently generates optimal controls for any parameter starting from an initial optimal control found by either GRAPE or reinforcement learning. When the control channels are restricted, the analytical scheme is invalid, but reinforcement learning still retains a level of generalizability, albeit in a narrower range. In cases where the shift in the Hamiltonian is impossible to decompose to available control channels, no generalizability is found for either the reinforcement learning or the analytical scheme. We argue that the generalization of reinforcement learning is through a mechanism similar to the analytical scheme. Our results provide insights into when and how the optimal control in multiparameter quantum metrology can be generalized, thereby facilitating efficient implementation of optimal quantum estimation of multiple parameters, particularly for an ensemble of systems with ranges of parameters.

Quantum control landscape for ultrafast generation of single-qubit phase shift quantum gates

Mathematical analysis of quantum control landscapes, which aims to prove either absence or existence of traps for quantum control objective functionals, is an important topic in quantum control. In this work, we provide a rigorous analysis of quantum control landscapes for ultrafast generation of single-qubit quantum gates and show, combining analytical methods based on a sophisticated analysis of spectrum of the Hessian, and numerical optimization methods such as gradient ascent pulse engineering, differential evolution, and dual annealing, that control landscape for ultrafast generation of phase shift gates is free of traps.

Krylov-subspace approach for the efficient control of quantum many-body dynamics

The Gradient Ascent Pulse Engineering (GRAPE) is a celebrated control algorithm with excellent converging rates, owing to a piece-wise-constant ansatz for the control function that allows for cheap objective gradients. However, the computational effort involved in the exact simulation of quantum dynamics quickly becomes a bottleneck limiting the control of large systems. In this paper, we propose a modified version of GRAPE that uses Krylov approximations to deal efficiently with high-dimensional state spaces. Even though the number of parameters required by an arbitrary control task scales linearly with the dimension of the system, we find a constant elementary computational effort (the effort per parameter). Since the elementary effort of GRAPE is super-quadratic, this speed up allows us to reach dimensions far beyond. The performance of the K-GRAPE algorithm is benchmarked in the paradigmatic XXZ spin-chain model.

Assessing three closed-loop learning algorithms by searching for high-quality quantum control pulses

Designing a high-quality control is crucial for reliable quantum computation. Among the existing approaches, closed-loop leaning control is an effective choice. Its efficiency depends on the learning algorithm employed, thus deserving algorithmic comparisons for its practical applications. Here, we assess three representative learning algorithms, including GRadient Ascent Pulse Engineering (GRAPE), improved Nelder-Mead (NMplus) and Differential Evolution (DE), by searching for high-quality control pulses to prepare the Bell state. We first implement each algorithm experimentally in a nuclear magnetic resonance system and then conduct a numerical study considering the impact of some possible significant experimental uncertainties. The experiments report the successful preparation of the high-fidelity target state with different convergence speeds by the three algorithms, and these results coincide with the numerical simulations when potential uncertainties are negligible. However, under certain significant uncertainties, these algorithms possess distinct performance with respect to their resulting precision and efficiency. This study provides insight to aid in the practical application of different closed-loop learning algorithms in realistic physical scenarios.

Universal nonselective excitation and refocusing pulses with improved robustness to off-resonance for Magnetic Resonance Imaging at 7 Tesla with parallel transmission.

In MRI at ultra-high field, the -point and spiral nonselective (SPINS) pulse design techniques can be advantageously combined with the parallel transmission (pTX) and universal pulse techniques to create uniform excitation in a calibration-free manner. However, in these approaches, pulse duration is typically increased as compared to standard hard pulses, and excitation quality in regions exhibiting large resonance frequency offsets often suffer. This limitation is inherent to structure of -point or SPINS pulse, and likely can be mitigated using parameterization-free pulse design approaches. The Gradient Ascent Pulse Engineering (GRAPE) algorithm was used to design parameterization-free RF and magnetic field gradient (MFG) waveforms for creating excitation, up to scalable refocusing and inversion, nonselectively across the brain. Simulations were performed to provide flip angle normalized root-mean-squares error (FA-NRMSE) estimations for the and the -point, SPINS, and GRAPE pulses. GRAPE pulses were tested experimentally with anatomical head scans at 7T. As compared to -points and SPINS, GRAPE provided substantial improvement of excitation, refocusing, and inversion quality at off-resonance while at least preserving the same global FA-NRMSE performance. As compared to -points, GRAPE allowed for a substantial reduction of the pulse duration for the excitation and the refocusing. Parameterization-free universal nonselective pTX-pulses were successfully computed using GRAPE. Performance gains as compared to -points were validated numerically and experimentally for three imaging protocols. In its current implementation, the computational burden of GRAPE limits its use to applications where pulse computations are not subject to time constraints.

Krotov method for optimal control of closed quantum systems

The mathematics of optimal control of quantum systems is of great interest in connection with fundamental problems of physics as well as with existing and prospective applications to quantum technologies. One important problem is the development of methods for constructing controls for quantum systems. One of the commonly used methods is the Krotov method, which was initially proposed outside of quantum control theory in articles by Krotov and Feldman (1978, 1983). This method was used to develop a novel approach to finding optimal controls for quantum systems in [64] (Tannor, Kazakov, and Orlov, 1992), [65] (Somlói, Kazakov, and Tannor, 1993), and in many other works by various scientists. Our survey discusses mathematical aspects of this method for optimal control of closed quantum systems. It outlines various modifications with different forms of the improvement function (for example, linear or linear-quadratic), different constraints on the control spectrum and on the admissible states of the quantum system, different regularisers, and so on. The survey describes applications of the Krotov method to controlling molecular dynamics and Bose–Einstein condensates, and to quantum gate generation. This method is compared with the GRAPE (GRadient Ascent Pulse Engineering) method, the CRAB (Chopped Random-Basis) method, and the Zhu–Rabitz and Maday–Turinici methods. Bibliography: 158 titles.

Optimal control methods for quantum gate preparation: a comparative study

In this study, optimal control methods for quantum gate preparation are investigated. Quantum computation demands very high fidelity and requires controls to be easily tunable to achieve different computational tasks. Here, NOT and Controlled-NOT gates are prepared using four optimal control approaches on single- and two-qubit spin systems. Techniques we employed and compared are Krotov method, gradient ascent pulse engineering (GRAPE), chopped random basis optimization (CRAB) and gradient optimization of analytic controls (GOAT). For the preparation of NOT gate both unitary and Lindbladian dynamics are considered. From the numerical simulations, it is observed that GOAT achieves better results as compared to Krotov, GRAPE and CRAB, in terms of minimum infidelity, algorithmic simplicity and analyticity.

Optimal control of mirror pulses for cold-atom interferometry

Atom matterwave interferometry requires mirror and beamsplitter pulses that are robust to inhomogeneities in field intensity, magnetic environment, atom velocity and Zeeman sub-state. Pulse shapes determined using quantum control methods offer significantly improved interferometer performance by allowing broader atom distributions, larger interferometer areas and higher contrast. We have applied gradient ascent pulse engineering (GRAPE) to optimise the design of phase-modulated mirror pulses for a Mach-Zehnder light-pulse atom interferometer, with the aim of increasing fringe contrast when averaged over atoms with an experimentally relevant range of velocities, beam intensities, and Zeeman states. Pulses were found to be highly robust to variations in detuning and coupling strength, and offer a clear improvement in robustness over the best established composite pulses. The peak mirror fidelity in a cloud of $\sim 80\ \mu$K ${}^{85}$Rb atoms is predicted to be improved by a factor of 2 compared with standard rectangular $\pi$ pulses.

Quantum optimal control in a chopped basis: Applications in control of Bose-Einstein condensates

We discuss quantum optimal control of Bose-Einstein condensates trapped in magnetic microtraps. The objective is to transfer a condensate from the ground state to the first-excited state. This type of control problem is typically solved using derivative-based methods in a high-dimensional control space such as gradient-ascent pulse engineering (GRAPE) and Krotov's method or derivative-free methods in a reduced control space such as Nelder-Mead with a chopped random basis (CRAB). We discuss how these methods can be combined in gradient optimization using parametrization (GROUP) including the finite bandwidth of the control electronics. We compare these methods and find that GROUP converges much faster than Nelder-Mead with CRAB and achieves better results than GRAPE and Krotov's method on the control problem presented here.

Numerical optimal control of spin systems at zero magnetic field

Zero-field nuclear magnetic resonance (NMR) has been recently developed as a complementary tool for nondestructive structural investigations of matter. However, precise and efficient control in such systems is still at the beginning. Based on the controllability of spin systems, we theoretically study the issue of quantum control in zero-field NMR by using a numerical optimization method based on the gradient ascent pulse engineering (GRAPE) algorithm. A set of quantum gates, including single-qubit and two-qubit gates, are achieved as examples. In particular, we show that homonuclear spins can be individually manipulated in zero-field NMR by the GRAPE method, which extends the ability to manipulate spin systems at zero magnetic field. Quantum control realized by the numerical GRAPE method features the properties of high fidelity, robustness, and hardware-friendly characteristics. High-fidelity quantum control in zero-field NMR systems might provide promising applications in material science, chemical analysis, quantum information processing, and fundamental physics.

Practical pulse engineering: Gradient ascent without matrix exponentiation

Since 2005, there has been a huge growth in the use of engineered control pulses to perform desired quantum operations in systems such as nuclear magnetic resonance quantum information processors. These approaches, which build on the original gradient ascent pulse engineering algorithm, remain computationally intensive because of the need to calculate matrix exponentials for each time step in the control pulse. In this study, we discuss how the propagators for each time step can be approximated using the Trotter–Suzuki formula, and a further speedup achieved by avoiding unnecessary operations. The resulting procedure can provide substantial speed gain with negligible costs in the propagator error, providing a more practical approach to pulse engineering.