Preparation of Hadamard Gate for Open Quantum Systems by the Lyapunov Control Method

In this paper, the control laws based on the Lyapunov stability theorem are designed for a two-level open quantum system to prepare the Hadamard gate, which is an important basic gate for the quantum computers. First, the density matrix interested in quantum system is transferred to vector formation. Then, in order to obtain a controller with higher accuracy and faster convergence rate, a Lyapunov function based on the matrix logarithm function is designed. After that, a procedure for the controller design is derived based on the Lyapunov stability theorem. Finally, the numerical simulation experiments for an amplitude damping Markovian open quantum system are performed to prepare the desired quantum gate. The simulation results show that the preparation of Hadamard gate based on the proposed control laws can achieve the fidelity up to 0.9985 for the different coupling strengths.

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

[2]  Joel G. Broida,et al.  Essential Linear Algebra , 2009 .

[3]  Yu. V. Nazarov,et al.  Coherent and incoherent pumping of electrons in double quantum dots , 2001 .

[4]  K. Blum.  Density Matrix Theory and Applications , 1981 .

[5]  Shuang Cong,et al.  Preparation of Quantum Gates for Open Quantum Systems by Lyapunov Control Method , 2016, Open Syst. Inf. Dyn..

[6]  Vladimir Privman,et al.  Quantum computing with spin qubits in semiconductor structures , 2002 .

[7]  A. Varon,et al.  Protecting conditional quantum gates by robust dynamical decoupling. , 2012, Physical review letters.

[8]  L. C. Wang,et al.  Realization of quantum gates by Lyapunov control , 2014, 1401.3195.

[9]  Domenico D'Alessandro,et al.  Time-Optimal Control of a Two Level Quantum System via Interaction With an Auxiliary System , 2014, IEEE Transactions on Automatic Control.

[10]  S. Cong.  Control of Quantum Systems: Theory and Methods , 2014 .

[11]  G. Lindblad.  On the generators of quantum dynamical semigroups , 1976 .

[12]  Domenico D'Alessandro,et al.  Optimal control of two-level quantum systems , 2001, IEEE Trans. Autom. Control..

[13]  F. Petruccione,et al.  Parametrizations of density matrices , 2011, 1103.4542.

[14]  S. Mohammad Nejad,et al.  Realization of quantum Hadamard gate by applying optimal control fields to a spin qubit , 2010, 2010 2nd International Conference on Mechanical and Electronics Engineering.

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

[16]  I. Chuang,et al.  Quantum Computation and Quantum Information: Introduction to the Tenth Anniversary Edition , 2010 .

[17]  D. V. Treshchev,et al.  Loss of stability in Hamiltonian systems that depend on parameters , 1992 .

[18]  Tai-Yin Chiu,et al.  Optimal control of two-qubit quantum gates in a non-Markovian open system , 2016, 2016 12th IEEE International Conference on Control and Automation (ICCA).

[19]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[20]  Haidong Yuan,et al.  Reachable set of open quantum dynamics for a single spin in Markovian environment , 2013, Autom..

[21]  Elica Kyoseva,et al.  Optimal quantum control by composite pulses , 2014, 2014 Conference on Lasers and Electro-Optics (CLEO) - Laser Science to Photonic Applications.

[22]  Pierre de Fouquieres,et al.  Implementing quantum gates by optimal control with doubly exponential convergence. , 2012, Physical review letters.

[23]  Shuang Cong,et al.  Realization of quantum hadamard gate based on Lyapunov method , 2012, Proceedings of the 10th World Congress on Intelligent Control and Automation.

[24]  Joydip Ghosh,et al.  A note on the measures of process fidelity for non-unitary quantum operations , 2011, 1111.2478.

[25]  Matthew D. Grace,et al.  Combining dynamical-decoupling pulses with optimal control theory for improved quantum gates , 2011 .

[26]  Angela Gamouras,et al.  Optimal two-qubit quantum control in InAs quantum dots , 2013, CLEO: 2013.