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arXiv:quant-ph/0203100 · Journal of Physics A Mathematical and General

Optimal control of one-qubit gates

K. M. Fonseca-Romero, G Useche Laverde, Federico Ardila

Published Jan 8, 2003• 3 citations
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Abstract

We consider the problem of carrying an initial Bloch vector to a final Bloch vector in a specified amount of time under the action of three control fields (a vector control field). We show that this control problem is solvable and therefore it is possible to optimize the control. We choose the physically motivated criteria of minimum energy expended in the control, minimum magnitude of the rate of change of the control and a combination of both. We find exact analytical solutions, determine the fields for a general one-qubit gate, and use the Y gate as an example. We argue that in the case of less than three controls, only the physical intuition does not provide a straight reasonable solution, and solve the problem in the case of a unique control minimizing the energy consumption.

DOI
10.1088/0305-4470/36/3/317

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Modelling optimal implementation of an arbitrary NNN-qubit quantum gate within the generalized Bloch vectors formalism
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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.
In quantum control theory, a question of fundamental and practical interest is how an arbitrary unitary transformation can be decomposed into a minimum number of elementary rotations for implementation, subject to various physical constraints. Examples include the singlet-triplet (ST) and exchange-only (EO) qubits in quantum-dot systems, and gate construction in the Solovay--Kitaev algorithm. For two important scenarios, we present complete solutions to the problems of optimal decomposition of single-qubit unitary gates with non-orthogonal rotations. For each unitary gate, the criteria for determining the minimal number of pieces is given, the explicit gate construction procedure, as well as a computer code for practical uses. Our results include an analytic explanation to the four-gate decomposition of EO qubits, previously determined numerically by Divincenzo et al. [Nature (London) 408, 339 (2000)]. Furthermore, compared with the approaches of Ramon sequence and its variant [Phys. Rev. Lett. 118, 216802 (2017)], our method can reduce about 50% of gate time for ST qubits. Finally, our approach can be extended to solve the problem of optimal control of topological qubits, where gate construction is achieved through the braiding operations.