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arXiv:1406.2907 · Physical Review A

Optimal control of quantum gates in an exactly solvable non-Markovian open quantum bit system

Jung‐Shen B. Tai, Kuan-Ting Lin, Hsi‐Sheng Goan

Published Jun 9, 2014• 23 citations
arXivHTMLPDF

Abstract

We apply quantum optimal control theory (QOCT) to an exactly solvable non-Markovian open quantum bit (qubit) system to achieve state-independent quantum control and construct high-fidelity quantum gates for moderate qubit decaying parameters. An important quantity, improvement $\mathcal{I}$, is proposed and defined to quantify the correction of gate errors due to the QOCT iteration when the environment effects are taken into account. With the help of the exact dynamics, we explore how the gate error is corrected in the open qubit system and determine the conditions for significant improvement. The model adopted in this paper can be implemented experimentally in realistic systems such as the circuit QED system.

DOI
10.1103/physreva.89.062310

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We apply quantum optimal control theory (QOCT) to an exactly solvable non-Markovian open quantum bit (qubit) system to achieve state-independent quantum control and construct high-fidelity quantum gates for moderate qubit decaying parameters. An important quantity, improvement I\mathcal{I}I, is proposed and defined to quantify the correction of gate errors due to the QOCT iteration when the environment effects are taken into account. With the help of the exact dynamics, we explore how the gate error is corrected in the open qubit system and determine the conditions for significant improvement. The model adopted in this paper can be implemented experimentally in realistic systems such as the circuit QED system.
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.
Based on the exact non-Markovian master equation for two interacting two-level atoms coupled to a dissipative zero-temperature bonsonic environment, we extend the results of Tai [Phys. Rev. A 89, 062310 (2014)], in which optimal control approach is used to implement one-qubit gates within the same environment, to two-qubit gate implementations with the help of Krotov's iterative method for optimal control. Furthermore, for practical applications, the realization of a quantum circuit comprising cascaded quantum gates is considered.
An efficient optimal-control theory based on the Krotov method is introduced for a non-Markovian open quantum system with a time-nonlocal master equation in which the control parameter and the bath correlation function are correlated. This optimal-control method is developed via a quantum dissipation formulation that transforms the time-nonlocal master equation to a set of coupled linear time-local equations of motion in an extended auxiliary Liouville space. As an illustration, the optimal-control method is applied to find the control sequences for high-fidelity ZZZ gates and identity gates of a qubit embedded in a non-Markovian bath. ZZZ gates and identity gates with errors less than ${10}^{\ensuremath{-}5}$ for a wide range of bath decoherence parameters can be achieved for the non-Markovian open qubit system with control over only the ${\ensuremath{\sigma}}_{z}$ term. The control-dissipation correlation and the memory effect of the bath are crucial in achieving the high-fidelity gates.
Quantum systems are inherently sensitive to environmental noise and imperfections in external control fields, which pose a significant challenge for the practical implementation of quantum technologies. These noise sources degrade the fidelity of quantum gates, making their mitigation a key requirement for realizing reliable quantum computing. In this study, we apply Optimal Control Theory (OCT) within a thermodynamically consistent Markovian framework to design high-fidelity quantum gates in the presence of thermal relaxation. Such a description is essential for realistic modeling and optimization of noisy quantum gates in near-term quantum technologies, where strong control fields and thermal environments act simultaneously. Our approach combines OCT with a control-dependent dissipative generator derived from the non-adiabatic master equation framework based on time-dependent invariants of the free evolution. As a result, the driving fields modify both the unitary and dissipative parts of the evolution. We implement the scheme for one- and two-qubit gates embedded in larger Hilbert spaces and compare direct-control and ancilla-assisted architectures. Using logical-subspace-resolved diagnostics, we quantify how the optimized dynamics redistributes the dissipative action between logical and ancillary sectors in the model systems studied here. In particular, we show that ancilla-assisted control can reduce the effective thermal-noise burden on the logical subspace in the relevant parameter regime, while direct control remains the most effective route when available. High-precision propagation of the full open-system dynamics reveals substantial fidelity improvements, in some cases by orders of magnitude, while clarifying the limits of mitigation at large relaxation rates and temperatures.
Constructing high-fidelity control fields that are robust to control, system, and/or surrounding environment uncertainties is a crucial objective for quantum information processing. Using the two-state Landau-Zener model for illustrative simulations of a controlled qubit, we generate optimal controls for $\ensuremath{\pi}/2$ and $\ensuremath{\pi}$ pulses and investigate their inherent robustness to uncertainty in the magnitude of the drift Hamiltonian. Next, we construct a quantum-control protocol to improve system-drift robustness by combining environment-decoupling pulse criteria and optimal control theory for unitary operations. By perturbatively expanding the unitary time-evolution operator for an open quantum system, previous analysis of environment-decoupling control pulses has calculated explicit control-field criteria to suppress environment-induced errors up to (but not including) third order from $\ensuremath{\pi}/2$ and $\ensuremath{\pi}$ pulses. We systematically integrate this criteria with optimal control theory, incorporating an estimate of the uncertain parameter to produce improvements in gate fidelity and robustness, demonstrated via a numerical example based on double quantum dot qubits. For the qubit model used in this work, postfacto analysis of the resulting controls suggests that realistic control-field fluctuations and noise may contribute just as significantly to gate errors as system and environment fluctuations.
A central challenge for implementing quantum computing in the solid state is decoupling the qubits from the intrinsic noise of the material. We investigate the implementation of quantum gates for a paradigmatic, non-Markovian model: a single-qubit coupled to a two-level system that is exposed to a heat bath. We systematically search for optimal pulses using a generalization of the novel open systems gradient ascent pulse engineering algorithm. Next to the known optimal bias point of this model, there are optimal pulses which lead to high-fidelity quantum operations for a wide range of decoherence parameters.