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arXiv:0708.2118 · Physical Review A

Optimal control theory for continuous-variable quantum gates

Re-Bing Wu, Raj Chakrabarti, HERSCHEL A. RABITZ

Published May 1, 2008• 34 citations
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Abstract

The methodology of optimal control theory is applied to the problem of implementing quantum gates in continuous-variable (CV) systems with quadratic Hamiltonians. We demonstrate that it is possible to define a fidelity measure for CV gate optimization that is devoid of traps, such that the search for optimal control fields using local algorithms will not be hindered. The optimal control of several quantum computing gates, as well as that of algorithms composed of these primitives, is investigated using several typical physical models and compared for discrete-variable and continuous-variable quantum systems. Numerical simulations indicate that the optimization of generic CV quantum gates is inherently more expensive than that of generic discrete variable quantum gates, but can be routinely achieved for all the major classes of computing primitives. The exact-time controllability of CV systems, hitherto largely ignored in the design of information processing models, is shown to play an important role in determining the maximal achievable gate fidelity. Moreover, the ability to control interactions between qunits can be exploited to delimit the total control fluence. Future experimental model systems should carefully tune these parameters so as to enable the implementation of CV quantum information processing with optimal fidelity.

DOI
10.1103/physreva.77.052303

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The methodology of optimal control theory is applied to the problem of implementing quantum gates in continuous-variable (CV) systems with quadratic Hamiltonians. We demonstrate that it is possible to define a fidelity measure for CV gate optimization that is devoid of traps, such that the search for optimal control fields using local algorithms will not be hindered. The optimal control of several quantum computing gates, as well as that of algorithms composed of these primitives, is investigated using several typical physical models and compared for discrete-variable and continuous-variable quantum systems. Numerical simulations indicate that the optimization of generic CV quantum gates is inherently more expensive than that of generic discrete variable quantum gates, but can be routinely achieved for all the major classes of computing primitives. The exact-time controllability of CV systems, hitherto largely ignored in the design of information processing models, is shown to play an important role in determining the maximal achievable gate fidelity. Moreover, the ability to control interactions between qunits can be exploited to delimit the total control fluence. Future experimental model systems should carefully tune these parameters so as to enable the implementation of CV quantum information processing with optimal fidelity.
We study the Hamiltonian-independent contribution to the complexity of quantum optimal control problems. The optimization of controls that steer quantum systems to desired objectives can itself be considered a classical dynamical system that executes an analog computation. The system-independent component of the equations of motion of this dynamical system can be integrated analytically for various classes of discrete quantum control problems. For the maximization of observable expectation values from an initial pure state and the maximization of the fidelity of quantum gates, the time complexity of the corresponding computation belongs to the class continuous log (CLOG), the lowest analog complexity class, equivalent to the discrete complexity class NC. The simple scaling of the Hamiltonian-independent contribution to these problems with quantum system dimension indicates that with appropriately designed search algorithms, quantum optimal control can be rendered efficient even for large systems.
External controls are necessary to enact quantum logic operations, and the inevitable control noise will result in gate errors in a realistic quantum circuit. We investigate the robustness of quantum gates to the random noise in an optimal control field, by utilizing properties of the quantum control landscape that relates the physical objective (in the present case, the quantum gate fidelity) to the applied controls. An approximate result obtained for the statistical expectation value of the gate fidelity in the weak noise regime is shown to be in excellent agreement with direct Monte Carlo sampling over noise process realizations for fidelity values relevant for practical quantum information processing. Using this approximate result, we demonstrate that maximizing the robustness to additive/multiplicative white noise is equivalent to minimizing the total control time/fluence. Also, a genetic optimization algorithm is used to identify controls with improved robustness to a colored noise characterized by its autocorrelation function.
This paper reveals an important role that controllability plays in the complexity of optimizing quantum control dynamics. We show that the loss of controllability generally leads to multiple locally suboptimal controls when gate fidelity in a quantum control system is maximized, which does not happen if the system is controllable. Such local suboptimal controls may attract an optimization algorithm into a local trap when a global optimal solution is sought, even if the target gate can be perfectly realized. This conclusion results from an analysis of the critical topology of the corresponding quantum control landscape, which refers to the gate fidelity objective as a functional of the control fields. For uncontrollable systems, due to SU(2) and SU(3) dynamical symmetries, the control landscape corresponding to an implementable target gate is proven to possess multiple locally optimal critical points, and its ruggedness can be further increased if the target gate is not realizable. These results imply that the optimization of quantum dynamics can be seriously impeded when operating with local search algorithms under these conditions, and thus full controllability is demanded.
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.
The ability to engineer high-fidelity gates on quantum processors in the presence of systematic errors remains the primary barrier to achieving quantum advantage. Quantum optimal control methods have proven effective in experimentally realizing high-fidelity gates, but they require exquisite calibration to be performant. We apply robust trajectory optimization techniques to suppress gate errors arising from system parameter uncertainty. We propose a derivative-based approach that maintains computational efficiency by using forward-mode differentiation. Additionally, the effect of depolarization on a gate is typically modeled by integrating the Lindblad master equation, which is computationally expensive. We employ a computationally efficient model and utilize time-optimal control to achieve high-fidelity gates in the presence of depolarization. We apply these techniques to a fluxonium qubit and suppress simulated gate errors due to parameter uncertainty below ${10}^{\ensuremath{-}7}$ for static parameter deviations of the order of 1%.
Non-Gaussian gates remain a key bottleneck for universal continuous-variable (CV) quantum computation because the nonlinearities they require are difficult to engineer. To address this challenge, we develop an efficient qubit-oscillator Rabi synthesis scheme for polynomial phase gates, with a total interaction time that scales polylogarithmically with the inverse target error ε\varepsilonε. Specifically, for a class of readily preparable initial states, we show that a degree-RRR phase gate can be approximated by an analytically constructed Rabi sequence with total time O(log⁡(R−1)/2+o(1)(1/ε))O(\log^{(R-1)/2+o(1)}(1/\varepsilon))O(log(R−1)/2+o(1)(1/ε)). This construction requires no numerical optimization and therefore extends naturally to arbitrarily large multimode systems. We further establish a total-time lower bound of Ω(log⁡(R−1)/2(1/ε))Ω(\log^{(R-1)/2}(1/\varepsilon))Ω(log(R−1)/2(1/ε)), showing that the synthesis is near optimal. As applications, we use this scheme to simulate representative CV quantum dynamics and implement a CV quantum algorithm for solving linear partial differential equations. These results establish qubit-oscillator Rabi control as an efficient, analytically compilable, and near-optimal primitive for CV quantum information processing.