Adiabatic evolution under quantum control

Abstract One difficulty with adiabatic quantum computation is the limit on the computation time. Here we propose two schemes to speed-up the adiabatic evolution. To apply this controlled adiabatic evolution to adiabatic quantum computation, we design one of the schemes without any explicit knowledge of the instantaneous eigenstates of the final Hamiltonian. Whereas in another scheme, we assume that the ground state of the Hamiltonian is known, and this information can be used to design the control. By these techniques, a linear speed-up proportional to the nonlinearity can be predicted. As an illustration, we study a two-level system driven by a time-dependent magnetic field under the control. The problem of finding an item in an unsorted database by adiabatic evolution is also examined. The physics behind the control scheme is interpreted.

[1]  Xiaoting Wang,et al.  Analysis of Lyapunov Method for Control of Quantum States , 2010, IEEE Transactions on Automatic Control.

[2]  Weidong Li,et al.  Continuous measurements enhanced self-trapping of degenerate ultra-cold atoms: Nonlinear Quantum Zeno Effect , 2006, cond-mat/0609338.

[3]  E. Knill,et al.  DYNAMICAL DECOUPLING OF OPEN QUANTUM SYSTEMS , 1998, quant-ph/9809071.

[4]  E. Farhi,et al.  A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem , 2001, Science.

[5]  Self-trapping, quantum tunneling, and decay rates for a Bose gas with attractive nonlocal interaction , 1999, cond-mat/9909070.

[6]  Daniel A. Lidar,et al.  Adiabatic approximation with exponential accuracy for many-body systems and quantum computation , 2008, 0808.2697.

[7]  Todd A. Brun,et al.  A simple model of quantum trajectories , 2002 .

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

[9]  Gerard J. Milburn,et al.  Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential , 1997 .

[10]  S. Lloyd,et al.  DYNAMICAL SUPPRESSION OF DECOHERENCE IN TWO-STATE QUANTUM SYSTEMS , 1998, quant-ph/9803057.

[11]  Wiseman,et al.  Adaptive phase measurements of optical modes: Going beyond the marginal Q distribution. , 1995, Physical review letters.

[12]  Effect of feedback on the control of a two-level dissipative quantum system , 2008, 0806.1643.

[13]  Kaveh Khodjasteh,et al.  Dynamically error-corrected gates for universal quantum computation. , 2008, Physical review letters.

[14]  Wiseman,et al.  Quantum theory of continuous feedback. , 1994, Physical review. A, Atomic, molecular, and optical physics.

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

[16]  JM Geremia.  Distinguishing between optical coherent states with imperfect detection (9 pages) , 2004 .

[17]  Andrew J. Landahl,et al.  Continuous quantum error correction via quantum feedback control , 2002 .

[18]  Lorenza Viola,et al.  Quantum Markovian Subsystems: Invariance, Attractivity, and Control , 2007, IEEE Transactions on Automatic Control.

[19]  L. Fu,et al.  Quantum Entanglement Manifestation of Transition to Nonlinear Self-trapping for Bose-Einstein Condensates in a Symmetric Double-Well , 2006, cond-mat/0609337.

[20]  K. Jacobs,et al.  FEEDBACK CONTROL OF QUANTUM SYSTEMS USING CONTINUOUS STATE ESTIMATION , 1999 .

[21]  A. Smerzi,et al.  Quantum Coherent Atomic Tunneling between Two Trapped Bose-Einstein Condensates , 1997, cond-mat/9706221.

[22]  Daniel A. Lidar,et al.  Quantum adiabatic brachistochrone. , 2009, Physical review letters.

[23]  Lorenza Viola,et al.  Analysis and synthesis of attractive quantum Markovian dynamics , 2008, Autom..

[24]  Kurt Jacobs,et al.  Feedback cooling of a nanomechanical resonator , 2003 .

[25]  V. M. Kenkre,et al.  TRANSITIONS IN COHERENT OSCILLATIONS BETWEEN TWO TRAPPED BOSE-EINSTEIN CONDENSATES , 1999 .

[26]  Gernot Schaller,et al.  General error estimate for adiabatic quantum computing , 2006 .

[27]  Howard Mark Wiseman,et al.  Optimal input states and feedback for interferometric phase estimation , 2001 .

[28]  Claudio Altafini.  Feedback Control of Spin Systems , 2007, Quantum Inf. Process..

[29]  J. J. Hope,et al.  Stabilizing entanglement by quantum-jump-based feedback , 2007 .

[30]  Gerard J. Milburn,et al.  Practical scheme for error control using feedback , 2004 .

[31]  A. Smerzi,et al.  Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects, π oscillations, and macroscopic quantum self-trapping , 1997 .

[32]  V. P. Belavkin,et al.  Measurement, filtering and control in quantum open dynamical systems , 1999 .

[33]  X. Yi,et al.  Feedback control on geometric phase in dissipative two-level systems , 2008, 0812.4621.

[34]  N. Cerf,et al.  Quantum search by local adiabatic evolution , 2001, quant-ph/0107015.

[35]  Stefano Mancini,et al.  Optomechanical Cooling of a Macroscopic Oscillator by Homodyne Feedback , 1998 .

[36]  S. Lloyd,et al.  Coherent quantum feedback , 2000 .

[37]  Ramakrishna,et al.  Controllability of molecular systems. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[38]  John K. Stockton,et al.  Adaptive homodyne measurement of optical phase. , 2002, Physical review letters.

[39]  A. I. Solomon,et al.  Complete controllability of quantum systems , 2000, quant-ph/0010031.

[40]  Daniel A. Lidar,et al.  Accuracy versus run time in an adiabatic quantum search , 2010, 1008.0863.

[41]  Kurt Jacobs.  Feedback control for communication with non-orthogonal states , 2007, Quantum Inf. Comput..

[42]  Chunfeng Wu,et al.  Driving quantum systems into decoherence-free subspaces by Lyapunov control , 2009, 0908.1048.

[43]  Kurt Jacobs,et al.  A straightforward introduction to continuous quantum measurement , 2006, quant-ph/0611067.

[44]  Kaveh Khodjasteh,et al.  Dynamical Quantum Error Correction of Unitary Operations with Bounded Controls , 2009, 0906.0525.

[45]  Hideo Mabuchi,et al.  Quantum feedback control of atomic motion in an optical cavity. , 2003, Physical Review Letters.