Abstract
In this paper we survey the geometric method in quantum control. By presenting a geometric representation of nonlocal two-qubit quantum operation, we show that the control of two-qubit quantum operations can be reduced to a steering problem in a tetrahedron. Two physical examples are given to illustrate this method. We also provide analytic approaches to construct universal quantum circuit from any arbitrary quantum gate.
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References
Mabuchi H, Khaneja N. Principles and applications of control in quantum systems. Int J Robust Nonlin Control, 2005, 15: 647–667
Tarn T J, Huang G, Clark J W. Modelling of quantum mechanical control systems. Math Model, 1980, 1: 109
Huang G M, Tarn T J, Clark J W. On the controllability of quantummechanical systems. J Math Phys, 1983, 24: 2608
Peirce A, Dahleh M, Rabitz H. Optimal control of quantum mechanical systems: Existence, numerical approximations, and applications. Phys Rev A, 1988, 37: 4950
Rabitz H, Hsieh M, Rosenthal C. Quantum optimally controlled transition landscapes. Science, 2004, 303: 998
Shor P W. Algorithms for quantum computation: Discrete logarithms and factoring. In: Goldwasser S, ed. Proceedings of 35nd Annual Symposium on Foundations of Computer Science, 1994. 124–134
Braunstein S L, van Loock P. Quantum information with continuous variables. Rev Mod Phys 2005, 77: 513–577
Nielsen M, Chuang I. Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2000
Barenco A, Bennett C H, Cleve R, et al. Elementary gates for quantum computation. Phys Rev A, 1995, 52: 3457
Murray R M, Li Z, Sastry S S. A Mathematical Introduction to Robotic Manipulation. Boca Raton: CRC Press, 1994
D’Alessandro D, Dahleh M. Optimal control of two-level quantum systems. IEEE Trans Automatic Control, 2001, 46: 866–876
Helgason S. Differential Geometry, Lie groups, and Symmetric Spaces Academic. New York: Academic, 1978
Makhlin Y. Nonlocal properties of two-qubit gates and mixed states, and the optimization of quantum computations. Quant inf Process, 2002, 1: 243
Zhang J, Vala J, Sastry S, et al. Geometric theory of nonlocal two-qubit operations. Phys Rev A, 2003, 67: 042313
Loss D, DiVincenzo D P. Quantum computation with quantum dots. Phys Rev A, 1998, 57: 120
Ouyang M, Awschalom D D. Coherent spin transfer between molecularly bridged quantum dots. Science, 2003, 301: 1074–1078
Kempe J, Bacon D, Lidar D, et al. Theory of decoherence-free faulttolerant universal quantum computation. Phys Rev A, 2001, 63: 042307
Zhang J, Vala J, Sastry S, et al. Exact two-qubit universal quantum circuit. Phys Rev Lett, 2003, 91: 027903
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Zhang, J. Geometric method in quantum control. Chin. Sci. Bull. 57, 2223–2227 (2012). https://doi.org/10.1007/s11434-012-5186-z
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DOI: https://doi.org/10.1007/s11434-012-5186-z