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Journal of Systems Engineering and Electronics

Comparison of time optimal control for two level quantum systems

Shuang Cong, Jie Wen, Xu‐Bo Zou

Published Feb 1, 2014• 2 citations
Publisher

Abstract

The time optimal problem for a two level quantum system is studied. We compare two different control strategies of bang-bang control and the geometric control, respectively, especially in the case of minimizing the time of steering the state from North Pole to South Pole on the Bloch sphere with bounded control. The time performances are compared for different parameters by the individual numerical simulation experiments, and the experimental results are analyzed. The results show that the geometric control spends less time than the bang-bang control does.

DOI
10.1109/jsee.2014.00011

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The time optimal problem for a two level quantum system is studied. We compare two different control strategies of bang-bang control and the geometric control, respectively, especially in the case of minimizing the time of steering the state from North Pole to South Pole on the Bloch sphere with bounded control. The time performances are compared for different parameters by the individual numerical simulation experiments, and the experimental results are analyzed. The results show that the geometric control spends less time than the bang-bang control does.
On a two-level quantum system driven by an external field, we consider the population transfer problem from the first to the second level, minimizing the time of transfer, with bounded field amplitude. On the Bloch sphere (i.e. after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on 2-D manifolds. Let (−E,E) be the two energy levels, and [equacion] the bound on the field amplitude. For each values of E and M, we provide the explicit expression of the time optimal trajectory steering the state one to the state two in terms of a parameter that should be computed numerically. For MR= 2E. On the other side, for M > E the time optimal trajectory steering the state one to the state two is bang-bang with exactly one switching. Fixed E we also prove that for M →∞ the time needed to reach the state two tends to zero. Finally we compare these results with some known results of Khaneja, Brockett and Glaser and with those obtained in the Rotating Wave Approximation.
We study time-minimum optimal control for a class of quantum two-dimensional dissipative systems whose dynamics are governed by the Lindblad equation and where control inputs acts only in the Hamiltonian. The dynamics of the control system are analyzed as a bi-linear control system on the Bloch ball after a decoupling of such dynamics into intra- and inter-unitary orbits. The (singular) control problem consists of finding a trajectory of the state variables solving a radial equation in the minimum amount of time, starting at the completely mixed state and ending at the state with the maximum achievable purity.The boundary value problem determined by the time-minimum singular optimal control problem is studied numerically. If controls are unbounded, simulations show that multiple local minimal solutions might exist. To find the unique globally minimal solution, we must repeat the algorithm for various initial conditions and find the best solution out of all of the candidates. If controls are bounded, optimal controls are given by bang-bang controls using the Pontryagin minimum principle. Using a switching map we construct optimal solutions consisting of singular arcs. If controls are bounded, the analysis of our model also implies classical analysis done previously for this problem.
The flip of quantum spin used as qubit has a significant effect on quantum computing and quantum information. Therefore, we study the dynamics of quantum spin in two-level spin-1/2 system by using Bang-Bang control protocol in this work. Here, system is driven by constant filed B0 along z-axis and controlled field B(t) along x or y-axis. when |B(t)| ≥ BC is satisfied, the optimal control time and switching time of spin, from state |1〉 to |0〉, of the system with and without amplitude damping or phase damping are given. Our results demonstrate that the fidelity of state of system with amplitude damping is still up to 0.9985 by Bang-Bang control protocol, even if the coupling strength, γ, of system with environment is equal to 0.3 and Bang-Bang control protocol is easier than oscillating filed to operate and analyze.
Abstract A systematic scheme is proposed to numerically estimate the quantum speed limit and temporal shape of optimal control in two-level and three-level quantum systems with bounded amplitude. For the two-level system, two quantum state transitions are studied as illustration. Comparisons between numerical and analytical results are made, and deviations are significantly small. For the three-level system, two critical time points are determined with high accuracy, and optimal controls are obtained for different durations. The shape of optimized control field is simple and does not switch frequently, thus are easy to implement in experiment. In addition, we compare our method with the chopped random basis (CRAB), and the performance of our method is much better than that of CRAB. Our scheme is of importance in estimating the quantum speed limit and optimal control for cases in which analytical solution is absent.