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Physical Review Letters

Digital Optimal Robust Control

Meri Harutyunyan, Frédéric Holweck, Dominique Sugny, S. Guérin

Published Nov 14, 2023• 25 citations
Publisher

Abstract

The lack of ability to determine and implement accurately quantum optimal control is a strong limitation to the development of quantum technologies. We propose a digital procedure based on a series of pulses where their amplitudes and (static) phases are designed from an optimal continuous-time protocol for given type and degree of robustness, determined from a geometric analysis. This digitalization combines the ease of implementation of composite pulses with the potential to achieve global optimality, i.e., to operate at the ultimate speed limit, even for a moderate number of control parameters. We demonstrate the protocol on IBM's quantum computers for a single qubit, obtaining a robust transfer with a series of Gaussian or square pulses in a time T=382 ns for a moderate amplitude. We find that the digital solution is practically as fast as the continuous one for square subpulses with the same peak amplitudes.

DOI
10.1103/physrevlett.131.200801

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The lack of ability to determine and implement accurately quantum optimal control is a strong limitation to the development of quantum technologies. We propose a digital procedure based on a series of pulses where their amplitudes and (static) phases are designed from an optimal continuous-time protocol for given type and degree of robustness, determined from a geometric analysis. This digitalization combines the ease of implementation of composite pulses with the potential to achieve global optimality, i.e., to operate at the ultimate speed limit, even for a moderate number of control parameters. We demonstrate the protocol on IBM's quantum computers for a single qubit, obtaining a robust transfer with a series of Gaussian or square pulses in a time T=382 ns for a moderate amplitude. We find that the digital solution is practically as fast as the continuous one for square subpulses with the same peak amplitudes.
The ability of pulse-shaping devices to generate accurately quantum optimal control is a strong limitation to the development of quantum technologies. We propose and demonstrate a systematic procedure to design robust digital control processes adapted to such experimental constraints. We show to what extent this digital pulse can be obtained from its continuous-time counterpart. A remarkable efficiency can be achieved even for a limited number of pulse parameters. We experimentally implement the protocols on IBM quantum computers for a single qubit, obtaining an optimal robust transfer in a time T = 382 ns.
Fastness and robustness are both critical in the implementation of high-fidelity gates for quantum computation, but in practice a trade-off has to be made between them. In this paper, we investigate the robust time-optimal control problem that aims at the best balance. Based on the Taylor expansion of the system in terms of uncertainty parameters, we formulate the design problem as the optimal control of an augmented finite-dimensional system at its quantum speed limit (QSL), where the robustness is graded by the order of series truncation. The gradient-descent algorithm is then introduced to sequentially seek QSLs corresponding to different orders of robustness. Numerical simulations are carried out for single-qubit systems with frequency and field amplitude uncertainties, and the obtained time-optimal control pulses can effectively suppress gate errors to the prescribed robustness order. These results provide a practical guide for selecting pulse lengths in the pulse-level compilation of quantum circuits.
Achieving high-fidelity control of quantum systems is essential for realization of a practical quantum computer. Composite pulse sequences which suppress different types of errors can be nested to suppress a wide variety of errors but the result is often not optimal, especially in the presence of constraints such as bandwidth limitations. Robust smooth pulse shaping provides flexibility but obtaining such analytical pulse shapes is a nontrivial problem and choosing the appropriate parameters typically requires a numerical search in a high-dimensional space. In this work, we extend a previous analytical treatment of robust smooth pulses to allow the determination of pulse parameters without numerical search. We also show that the problem can be reduced to a set of coupled ordinary differential equations which allows for a more streamlined numerical treatment.
Abstract Precise control of quantum systems is one of the most important milestones for achieving practical quantum technologies, such as computation, sensing, and communication. Several factors deteriorate the control precision and thus their suppression is strongly demanded. One of the dominant factors is systematic errors, which are caused by discord between an expected parameter in control and its actual value. Error-robust control sequences, known as composite pulses, have been invented in the field of nuclear magnetic resonance (NMR). These sequences mainly focus on the suppression of errors in one-qubit control. The one-qubit control, which is the most fundamental in a wide range of quantum technologies, often suffers from detuning error. As there are many possible control sequences robust against the detuning error, it will practically be important to find ‘optimal’ robust controls with respect to several cost functions such as time required for operation, and pulse-area during the operation, which corresponds to the energy necessary for control. In this paper, we utilize the Pontryagin’s maximum principle (PMP), a tool for solving optimization problems under inequality constraints, to solve the time and pulse-area optimization problems. We analytically obtain pulse-area optimal controls robust against the detuning error. Moreover, we found that short-CORPSE, which is the shortest known composite pulse so far, is a probable candidate of the time optimal solution according to the PMP. We evaluate the performance of the pulse-area optimal robust control and the short-CORPSE, comparing with that of the direct operation.
Optimal control theory is applied to analyze the time-optimal solution with a single scalar control knob in a two-level quantum system without quantum decoherence. Emphasis is placed on the dependence on the maximum control strength umax{u}_{\text{max}}umax​. General constraints on the optimal protocol are derived and used to rigorously parametrize the time-optimal solution. Two concrete problems are investigated. For generic state preparation problems, both multiple bang-bang and bang-singular-bang are legitimate and should be considered. Generally, the optimal is bang-bang for small umax{u}_{\text{max}}umax​, and there exists a state-dependent critical amplitude above which singular control emerges. For the X-gate operation of a qubit, the optimal protocol is exclusively multiple bang-bang. The minimum gate time is about 80% of that based on the resonant Rabi $\ensuremath{\pi}$-pulse over a wide range of control strength; in the ${u}_{\text{max}}\ensuremath{\rightarrow}0$ limit, this ratio is derived to be $\ensuremath{\pi}/4$. To develop practically feasible protocols, we present methods to smooth the abrupt changes in the bang-bang control while preserving perfect gate fidelity. The presence of bang-bang segments in the time-optimal protocol indicates that the high-frequency components and a full calculation (instead of the commonly adopted rotating wave approximation) are essential for the ultimate quantum speed limit.
High-fidelity quantum gates are crucial for achieving fault-tolerant quantum computing; however, decoherence significantly reduces gate fidelities during long operation times. Although optimal control techniques can theoretically minimize these operation times, they often neglect realistic uncertainties in system parameters. In this work, we demonstrate that by using robust optimal control strategies, the cross-resonance gate in superconducting systems can be operated within 64 ns, achieving fidelities of [Formula: see text] while maintaining robustness against up to 10% uncertainty in a single parameter. Alternatively, by extending the control time to 71 ns, we achieve fidelities of [Formula: see text] with robustness against up to 3% uncertainty. Our results identify the minimal control times attainable with experimentally feasible pulses and system parameters, as well as the maximum allowable static parameter error for high-fidelity operations. Furthermore, we demonstrate simultaneous robustness against both static and time-dependent errors by generating 100 ns control pulses ([Formula: see text]) that maintain robustness against 10% static parameter error and time-dependent parameter fluctuations two orders of magnitude stronger than typical experimental noise. These findings demonstrate a viable open-loop strategy for implementing fast, high-fidelity quantum gates in the presence of realistic system uncertainties that would otherwise degrade conventional control pulses.