Energy control in a quantum oscillator using coherent control and engineered environment

Pechen, Alexander; Borisenok, Sergey; Fradkov, Alexander L.

doi:10.1016/j.chaos.2022.112687

Cited by 10 publications

(6 citation statements)

References 46 publications

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“…Theorem 4. In the representation of the density matrix by vector q evolving according to (49), Hessian of the objective functional F[ f ] = I(q f T ) takes the form:…”

Section: Robustness Of the Optimal Controlsmentioning

confidence: 99%

“…Sometimes a solution for the optimal shape of the control can be obtained analytically. However, generally it is not the case and various numerical optimization methods are used, including GRadient Ascent Pulse Engineering (GRAPE) numerical optimization algorithm [30], gradient flows [31], Krotov method [32,33], genetic algorithms for coherent control of closed systems [34] and incoherent control of open quantum systems in [13], gradient free CRAB optimization algorithm [35], Hessian based optimization as in the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm and combined approaches [36,37], machine learning such as quantum reinforcement learning with incoherent control [38], deep reinforcement learning [39], autoencoders [40], speed gradient algorithm [41], Lyapunov control [42] and various schemes [43,44,45,46,47]. Monotonically convergent optimization in quantum control using Krotov's method was obtained for a large class of quantum control problems [48].…”

Section: Introductionmentioning

confidence: 99%

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GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controls

Petruhanov¹,

Pechen²

2023

J. Phys. A: Math. Theor.

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The GRadient Ascent Pulse Engineering (GRAPE) method is widely used for optimization in quantum control. GRAPE is gradient search method based on exact expressions for gradient of the control objective. It has been applied to various coherently controlled closed and open quantum systems. In this work, we adopt the GRAPE method for optimizing objective functionals in open quantum systems driven by both coherent and incoherent controls. In our case, a tailored or engineered environment acts on the controlled system as control via time-dependent decoherence rates γ i ( t ) or, equivalently, via spectral density n ω ( t ) of the environment. To develop the GRAPE approach for this problem, we compute gradient of various objectives for general N-level open quantum systems for the piecewise constant class of control. The case of a single qubit is considered in details and solved analytically. For this case, an explicit analytical expression for evolution and objective gradient is obtained via diagonalization of a 3 × 3 matrix determining the system’s dynamics in the Bloch ball. The diagonalization is obtained by solving a cubic equation via Cardano’s method. The efficiency of the algorithm is demonstrated through numerical simulations for the state-to-state transition problem and its complexity is estimated. Robustness of the optimal controls is also studied.

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“…Theorem 4. In the representation of the density matrix by vector q evolving according to (49), Hessian of the objective functional F[ f ] = I(q f T ) takes the form:…”

Section: Robustness Of the Optimal Controlsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controls

Petruhanov¹,

Pechen²

2023

J. Phys. A: Math. Theor.

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“…Various optimization methods are used to find controls for quantum systems including Krotov type approaches ( [30,31,32], Zhu-Rabitz [33] method, GRadient Ascent Pulse Engineering (GRAPE) [34], Hessian based optimization in the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm and combined approaches [35,36], speed gradient method [37], gradient free Chopped RAndom Basis (CRAB) optimization [38], genetic algorithms [39,6], dual annealing [11], machine learning [40], etc. Quantum speed limit for a controlled qubit moving inside a leaky cavity has been studied [41].…”

Section: Introductionmentioning

confidence: 99%

Quantum Gate Generation in Two-Level Open Quantum Systems by Coherent and Incoherent Photons Found with Gradient Search

Petruhanov,

Pechen

2023

Preprint

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In this work, we consider an environment formed by incoherent photons as a resource for controlling open quantum systems via an incoherent control. We exploit a coherent control in the Hamiltonian and an incoherent control in the dissipator which induces the time-dependent decoherence rates γ k (t) (via time-dependent spectral density of incoherent photons) for generation of single-qubit gates for a two-level open quantum system which evolves according to the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation with time-dependent coefficients determined by these coherent and incoherent controls. The control problem is formulated as minimization of the objective functional, which is the sum of Hilbert-Schmidt norms between four fixed basis states evolved under the GKSL master equation with controls and the same four states evolved under the ideal gate transformation. The exact expression for the gradient of the objective functional with respect to piecewise constant controls is obtained. Subsequent optimization is performed using a gradient type algorithm with an adaptive step size that leads to oscillating behaviour of the gradient norm vs iterations. Optimal trajectories in the Bloch ball for various initial states are computed. A relation of quantum gate generation with optimization on complex Stiefel manifolds is discussed. We develop methodology and apply it here for unitary gates as a testing example. The next step is to apply the method for generation of non-unitary processes and to multi-level quantum systems.

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“…For finding both coherent and incoherent controls, genetic evolutionary algorithms were initially used [19]. Recently, the speed gradient method [59], gradient projection methods [60], the Krotov method, and stochastic free-gradient optimization methods [61] were adapted.…”

Section: Introductionmentioning

confidence: 99%

Quantum Control Landscapes for Generation of H and T Gates in an Open Qubit with Both Coherent and Environmental Drive

Petruhanov,

Pechen

2023

Photonics

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An important problem in quantum computation is the generation of single-qubit quantum gates such as Hadamard (H) and π/8 (T) gates, which are components of a universal set of gates. Qubits in experimental realizations of quantum computing devices are interacting with their environment. While the environment is often considered as an obstacle leading to a decrease in the gate fidelity, in some cases, it can be used as a resource. Here, we consider the problem of the optimal generation of H and T gates using coherent control and the environment as a resource acting on the qubit via incoherent control. For this problem, we studied the quantum control landscape, which represents the behavior of the infidelity as a functional of the controls. We considered three landscapes, with infidelities defined by steering between two, three (via Goerz–Reich–Koch approach), and four matrices in the qubit Hilbert space. We observed that, for the H gate, which is a Clifford gate, for all three infidelities, the distributions of minimal values obtained with a gradient search have a simple form with just one peak. However, for the T gate, which is a non-Clifford gate, the situation is surprisingly different—this distribution for the infidelity defined by two matrices also has one peak, whereas distributions for the infidelities defined by three and four matrices have two peaks, which might indicate the possible existence of two isolated minima in the control landscape. It is important that, among these three infidelities, only those defined with three and four matrices guarantee the closeness of the generated gate to a target and can be used as a good measure of closeness. We studied sets of optimized solutions for the most general and previously unexplored case of coherent and incoherent controls acting together and discovered that they form sub-manifolds in the control space, and unexpectedly, in some cases, two isolated sub-manifolds.

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Energy control in a quantum oscillator using coherent control and engineered environment

Cited by 10 publications

References 46 publications

GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controls

GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controls

Quantum Gate Generation in Two-Level Open Quantum Systems by Coherent and Incoherent Photons Found with Gradient Search

Quantum Control Landscapes for Generation of H and T Gates in an Open Qubit with Both Coherent and Environmental Drive

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