Realization of quantum gates by Lyapunov control

Abstract We propose a Lyapunov control design to achieve specific (or a family of) unitary time-evolution operators, i.e., quantum gates in the Schrodinger picture by tracking control. Two examples are presented. In the first, we illustrate how to realize the Hadamard gate in a single-qubit system, while in the second, the controlled-NOT (CNOT) gate is implemented in two-qubit systems with the Ising and Heisenberg interactions. Furthermore, we demonstrate that the control can drive the time-evolution operator into the local equivalence class of the CNOT gate and the operator keeps in this class forever with the existence of Ising coupling.

[1]  D. D’Alessandro.  Introduction to Quantum Control and Dynamics , 2007 .

[2]  Johannes Kofler,et al.  Quantum Information and Randomness , 2010, European Review.

[3]  Ian R. Petersen,et al.  Quantum control theory and applications: A survey , 2009, IET Control Theory & Applications.

[4]  Franco Nori,et al.  Feedback control of Rabi oscillations in circuit QED , 2013 .

[5]  Tatsuhiko Koike,et al.  Time-optimal quantum evolution. , 2006, Physical review letters.

[6]  Mazyar Mirrahimi,et al.  Lyapunov control of bilinear Schrödinger equations , 2005, Autom..

[7]  I. Petersen,et al.  Optimal Lyapunov-based quantum control for quantum systems , 2012, 1203.5373.

[8]  Yuriy Makhlin.  Nonlocal Properties of Two-Qubit Gates and Mixed States, and the Optimization of Quantum Computations , 2002, Quantum Inf. Process..

[9]  Guoming Wang,et al.  Property testing of unitary operators , 2011, 1110.1133.

[10]  X. Yi,et al.  Adiabatic evolution under quantum control , 2009, 0910.5859.

[11]  K. Jacobs,et al.  Ultraefficient cooling of resonators: beating sideband cooling with quantum control. , 2011, Physical review letters.

[12]  J. Siewert,et al.  Natural two-qubit gate for quantum computation using the XY interaction , 2002, quant-ph/0209035.

[13]  V. Ramakrishna,et al.  Constructive control of quantum systems using factorization of unitary operators , 2002 .

[14]  Jun Li,et al.  Experimental demonstration of the Deutsch-Jozsa algorithm in homonuclear multispin systems , 2011 .

[15]  F. Verstraete,et al.  Quantum computation and quantum-state engineering driven by dissipation , 2009 .

[16]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[17]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[18]  C Langer,et al.  Spectroscopy Using Quantum Logic , 2005, Science.

[19]  Navin Khaneja,et al.  Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer , 2002 .

[20]  Xiaoting Wang,et al.  Entanglement generation between distant atoms by Lyapunov control , 2009, 0906.1830.

[21]  X. X. Yi,et al.  Lyapunov control on quantum open systems in decoherence-free subspaces , 2010 .

[22]  Timo O. Reiss,et al.  Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. , 2005, Journal of magnetic resonance.

[23]  Ian R. Petersen,et al.  Sliding mode control of two-level quantum systems , 2010, Autom..

[24]  Ronnie Kosloff,et al.  Quantum computing by an optimal control algorithm for unitary transformations. , 2002, Physical review letters.

[25]  Javad Sharifi,et al.  Lyapunov control of squeezed noise quantum trajectory , 2011 .

[26]  P. Hemmer,et al.  NMR Quantum Computing , 2009 .

[27]  Benjamin Rowland,et al.  Implementing quantum logic gates with gradient ascent pulse engineering: principles and practicalities , 2012, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[28]  Shlomo E. Sklarz,et al.  Quantum computation via local control theory : Direct sum vs. direct product Hilbert spaces , 2006 .

[29]  Mingzhen Tian,et al.  Robustness of single-qubit geometric gate against systematic error , 2011 .

[30]  K. B. Whaley,et al.  Optimizing entangling quantum gates for physical systems , 2011, 1104.2337.

[31]  C. F. Roos,et al.  ‘Designer atoms’ for quantum metrology , 2006, Nature.

[32]  Jitao Sun,et al.  One Lyapunov control for quantum systems and its application to entanglement generation , 2013 .

[33]  Franco Nori,et al.  Speed limits for quantum gates in multiqubit systems , 2012, 1202.5872.

[34]  Julien Salomon,et al.  Time-optimal monotonically convergent algorithm with an application to the control of spin systems , 2012 .

[35]  Milburn,et al.  Quantum theory of optical feedback via homodyne detection. , 1993, Physical review letters.

[36]  Michihiko Sugawara,et al.  General formulation of locally designed coherent control theory for quantum system , 2003 .

[37]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[38]  Sen Kuang,et al.  Lyapunov control methods of closed quantum systems , 2008, Autom..