Theory of initialization-free decoherence-free subspaces and subsystems (14 pages)

We introduce a generalized theory of decoherence-free subspaces and subsystems (DFSs), which do not require accurate initialization. We derive a set of conditions for the existence of DFSs within this generalized framework. By relaxing the initialization requirement we show that a DFS can tolerate arbitrarily large preparation errors. This has potentially significant implications for experiments involving DFSs, in particular for the experimental implementation, over DFSs, of the large class of quantum algorithms which can function with arbitrary input states.

[1]  E. Sudarshan,et al.  Completely Positive Dynamical Semigroups of N Level Systems , 1976 .

[2]  G. Guo,et al.  Reducing decoherence in quantum-computer memory with all quantum bits coupling to the same environment , 1996, quant-ph/9612003.

[3]  U. Toronto,et al.  Analysis of generalized Grover quantum search algorithms using recursion equations , 2000, quant-ph/0010077.

[4]  D. Kribs.  QUANTUM CHANNELS, WAVELETS, DILATIONS AND REPRESENTATIONS OF $\mathcal{O}_{n}$ , 2003, Proceedings of the Edinburgh Mathematical Society.

[5]  C. Monroe,et al.  Experimental Issues in Coherent Quantum-State Manipulation of Trapped Atomic Ions , 1997, Journal of research of the National Institute of Standards and Technology.

[6]  Completely positive post-Markovian master equation via a measurement approach , 2004, quant-ph/0404077.

[7]  Viola,et al.  Theory of quantum error correction for general noise , 2000, Physical review letters.

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

[9]  Dissipation and decoherence in a quantum register , 1998 .

[10]  M B Plenio,et al.  Efficient factorization with a single pure qubit and logN mixed qubits. , 2000, Physical review letters.

[11]  K. Mølmer,et al.  Wave-function approach to dissipative processes in quantum optics. , 1992, Physical review letters.

[12]  Jeong San Kim,et al.  Quantum algorithms without initializing the auxiliary qubits. , 2005, Physical review letters.

[13]  A. G. White,et al.  Experimental verification of decoherence-free subspaces. , 2000, Science.

[14]  K. Kraus,et al.  States, effects, and operations : fundamental notions of quantum theory : lectures in mathematical physics at the University of Texas at Austin , 1983 .

[15]  Edward Davies.  Quantum dynamical semigroups and the neutron diffusion equation , 1977 .

[16]  Guang-Can Guo,et al.  Preserving Coherence in Quantum Computation by Pairing Quantum Bits , 1997 .

[17]  K. Lendi,et al.  Quantum Dynamical Semigroups and Applications , 1987 .

[18]  J. Gea-Banacloche,et al.  Three-qubit quantum error-correction scheme for collective decoherence , 2001 .

[19]  D. Deutsch,et al.  Rapid solution of problems by quantum computation , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[20]  Unruh.  Maintaining coherence in quantum computers. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[21]  Daniel A. Lidar,et al.  Decoherence-Free Subspaces for Quantum Computation , 1998, quant-ph/9807004.

[22]  R Laflamme,et al.  Experimental Realization of Noiseless Subsystems for Quantum Information Processing , 2001, Science.

[23]  S. Nakajima.  On Quantum Theory of Transport Phenomena Steady Diffusion , 1958 .

[24]  P. Zanardi,et al.  Noiseless Quantum Codes , 1997, quant-ph/9705044.

[25]  P. Zanardi,et al.  Error avoiding quantum codes , 1997, quant-ph/9710041.

[26]  Daniel A. Lidar,et al.  CONCATENATING DECOHERENCE-FREE SUBSPACES WITH QUANTUM ERROR CORRECTING CODES , 1998, quant-ph/9809081.

[27]  R. Zwanzig.  Ensemble Method in the Theory of Irreversibility , 1960 .

[28]  Daniel A Lidar,et al.  Magnetic resonance realization of decoherence-free quantum computation. , 2003, Physical review letters.

[29]  Paolo Zanardi.  Stabilizing quantum information , 2000 .

[30]  D A Lidar,et al.  Efficient universal leakage elimination for physical and encoded qubits. , 2002, Physical review letters.

[31]  K. B. Whaley,et al.  Decoherence-free subspaces for multiple-qubit errors. I. Characterization , 1999, quant-ph/9908064.

[32]  Markus Grassl,et al.  Grover's quantum search algorithm for an arbitrary initial amplitude distribution , 1999 .

[33]  Soojoon Lee,et al.  Quantum functional oracles , 2002 .

[34]  Daniel A. Lidar,et al.  From completely positive maps to the quantum Markovian semigroup master equation , 2001 .

[35]  Paolo Zanardi,et al.  QUANTUM INFORMATION IN SEMICONDUCTORS : NOISELESS ENCODING IN A QUANTUM-DOT ARRAY , 1998 .

[36]  K. B. Whaley,et al.  Theory of decoherence-free fault-tolerant universal quantum computation , 2000, quant-ph/0004064.

[37]  Artur Ekert,et al.  Quantum computers and dissipation , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[38]  P. Knight,et al.  The Quantum jump approach to dissipative dynamics in quantum optics , 1997, quant-ph/9702007.

[39]  S. Filippo.  Quantum computation using decoherence-free states of the physical operator algebra , 1999, quant-ph/9910005.

[40]  Lorenza Viola,et al.  Implementation of universal control on a decoherence-free qubit , 2002 .

[41]  J. Raimond,et al.  Quantum Computing: Dream or Nightmare? , 1996 .

[42]  Krzysztof Wódkiewicz,et al.  Depolarizing channel as a completely positive map with memory , 2004 .

[43]  Dong Pyo Chi,et al.  Initialization-free generalized Deutsch-Jozsa algorithm , 2001 .

[44]  Hoi-Kwong Lo,et al.  Introduction to Quantum Computation Information , 2002 .

[45]  N. Gisin,et al.  The quantum-state diffusion model applied to open systems , 1992 .

[46]  Eli Biham,et al.  Grover's quantum search algorithm for an arbitrary initial mixed state , 2002 .

[47]  Francesco Petruccione,et al.  The Theory of Open Quantum Systems , 2002 .

[48]  M. A. Rowe,et al.  A Decoherence-Free Quantum Memory Using Trapped Ions , 2001, Science.

[49]  E. Knill,et al.  Power of One Bit of Quantum Information , 1998, quant-ph/9802037.

[50]  P. Zanardi,et al.  Purity and state fidelity of quantum channels , 2004, quant-ph/0403074.

[51]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[52]  Daniel A. Lidar,et al.  Universal Leakage Elimination , 2004, quant-ph/0409049.

[53]  K J Resch,et al.  Experimental application of decoherence-free subspaces in an optical quantum-computing algorithm. , 2003, Physical review letters.

[54]  Lorenza Viola,et al.  Verification procedures for quantum noiseless subsystems , 2003 .

[55]  David Poulin,et al.  Unified and generalized approach to quantum error correction. , 2004, Physical review letters.