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arXiv:0805.2902 · Journal of Modern Optics

Implementation of quantum gates via optimal control

Sonia G. Schirmer

Published Feb 4, 2009• 33 citations
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Abstract

Starting with the basic control system model often employed in nuclear magnetic resonance pulse design, we derive more realistic control system models taking into account effects such as off-resonant excitation for systems with fixed inter-qubit coupling controlled by globally applied electromagnetic fields, as well as for systems controlled by a combination of a global fields and local control electrodes. For both models optimal control is used to find controls that implement a set of two- and three-qubit gates with fidelity of at least 99.99%.

DOI
10.1080/09500340802344933

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Starting with the basic control system model often employed in nuclear magnetic resonance pulse design, we derive more realistic control system models taking into account effects such as off-resonant excitation for systems with fixed inter-qubit coupling controlled by globally applied electromagnetic fields, as well as for systems controlled by a combination of a global fields and local control electrodes. For both models optimal control is used to find controls that implement a set of two- and three-qubit gates with fidelity of at least 99.99%.
With the development of quantum information science, the active manipulation of quantum systems is becoming an important research frontier. To build realistic quantum information processors, one of the challenges is to implement arbitrary desired operations with high precision on quantum systems. A large number of quantum control methods and relevant numerical techniques have been put forward in recent years, such as quantum optimal control and quantum feedback control. Nuclear magnetic resonance (NMR) spin systems offer an excellent testbed to develop benchmark tools and techniques for controlling quantum systems. In this review paper, we briefly introduce some of the basic control ideas developed for NMR systems in recent years. We first explain, for the liquid spin systems, the physics of various couplings and the causes of relaxation effects. These mechanisms govern the system dynamics, and thus are crucial for constructing rigorous and efficient control models. We also identify three types of available control means: 1) raido-frequency fields as coherent controls; 2) phase cycling, gradient fields and relaxation effects as non-unitary controls; 3) radiation damping effect as feedback control mechanism. Then, we elucidate some important control tasks, which may arise from the conventional NMR spectroscopy (e.g., pulse design and polarization transfer) or from quantum information science (e.g., algorithmic cooling and pseudo-pure state preparation). In the last part, we review some of the most important control methods that are applicable to NMR control tasks. For systems with a relatively small number of spins, it is possible to use analytic optimal control theory to realize the target unitary operations. However, for larger systems, numerical methods are necessary. The gradient ascent pulse engineering algorithm and pulse compiler techniques are the most successful techniques for implementing complicated quantum networks currently. There are some interesting topics of utilizing radiation damping and relaxation effects to achieve more powerful controls. Finally, we give an outline of the possible future work.
Quantum optimal control for gate optimization aims to provide accurate, robust, and fast pulse sequences to achieve gate fidelities on quantum systems below the error correction threshold. Many methods have been developed and successfully applied in simulation and on quantum hardware. In this paper, we establish a connection between the iterative linear quadratic regulator and quantum optimal control by adapting it to gate optimization of quantum systems. We include constraints on the controls and their derivatives to enable smoother pulses. We achieve high-fidelity simulation results forX\mathbf{X}Xand cross-resonance gates on one- and two-qubit fixed-frequency transmons simulated with two and three levels.
We study the implementation of one-, two- and three-qubit quantum gates for interacting qubits using optimal control. Markovian and non-Markovian environments are compared and efficient optimization algorithms utilizing analytic gradient expressions and quasi-Newton updates are given for both cases. The performance of the algorithms is analysed for a large set of problems in terms of the fidelities attained and the observed convergence behaviour. New notions of success rate and success speed are introduced and density plots are utilized to study the effects of key parameters, such as gate operation times, and random variables such as the initial fields required to start the iterative algorithm. Core characteristics of the optimal fields are analysed statistically. Substantial differences between Markovian and non-Markovian environments in terms of the possibilities for control and the control mechanisms are uncovered. In the non-Markovian case, gate fidelities improve substantially when the details of the system bath coupling are taken into account, although imperfections such as field leakage can be a significant problem. In the Markovian case, computation time is saved if the fields are pre-optimized neglecting the environment, while including the latter generally does not significantly improve gate fidelities.
We demonstrate the use of optimal control to design two entropy-manipulating quantum gates which are more complex than the corresponding, commonly used, gates, such as CNOT and Toffoli (CCNOT): A two-qubit gate called polarization exchange (PE) and a three-qubit gate called polarization compression (COMP) were designed using GRAPE, an optimal control algorithm. Both gates were designed for a three-spin system. Our design provided efficient and robust nuclear magnetic resonance (NMR) radio frequency (RF) pulses for 13 C 2-trichloroethylene (TCE), our chosen three-spin system. We then experimentally applied these two quantum gates onto TCE at the NMR lab. Such design of these gates and others could be relevant for near-future applications of quantum computing devices.
Methods of optimal control are applied to a model system of interacting two-level particles (e.g., spin-half atomic nuclei or electrons or two-level atoms) to produce high-fidelity quantum gates while simultaneously negating the detrimental effect of decoherence. One set of particles functions as the quantum information processor, whose evolution is controlled by a time-dependent external field. The other particles are not directly controlled and serve as an effective environment, coupling to which is the source of decoherence. The control objective is to generate target one- and two-qubit unitary gates in the presence of strong environmentally-induced decoherence and under physically motivated restrictions on the control field. The quantum-gate fidelity, expressed in terms of a novel state-independent distance measure, is maximized with respect to the control field using combined genetic and gradient algorithms. The resulting high-fidelity gates demonstrate the feasibility of precisely guiding the quantum evolution via optimal control, even when the system complexity is exacerbated by environmental coupling. It is found that the gate duration has an important effect on the control mechanism and resulting fidelity. An analysis of the sensitivity of the gate performance to random variations in the system parameters reveals a significant degree of robustness attained by the optimal control solutions.
Methods of optimal control are applied to a model system of interacting two-level particles (e.g., spin-half atomic nuclei or electrons or two-level atoms) to produce high-fidelity quantum gates while simultaneously negating the detrimental effect of decoherence. One set of particles functions as the quantum information processor, whose evolution is controlled by a time-dependent external field. The other particles are not directly controlled and serve as an effective environment, coupling to which is the source of decoherence. The control objective is to generate target one- and two-qubit unitary gates in the presence of strong environmentally-induced decoherence and under physically motivated restrictions on the control field. The quantum-gate fidelity, expressed in terms of a novel state-independent distance measure, is maximized with respect to the control field using combined genetic and gradient algorithms. The resulting high-fidelity gates demonstrate the feasibility of precisely guiding the quantum evolution via the optimal control, even when the system complexity is exacerbated by environmental coupling. It is found that the gate duration has an important effect on the control mechanism and resulting fidelity. An analysis of the sensitivity of the gate performance to random variations in the system parameters reveals a significant degree of robustness attained by the optimal control solutions.