Decoherence in quantum walks – a review

The development of quantum walks in the context of quantum computation, as generalisations of random walk techniques, has led rapidly to several new quantum algorithms. These all follow a unitary quantum evolution, apart from the final measurement. Since logical qubits in a quantum computer must be protected from decoherence by error correction, there is no need to consider decoherence at the level of algorithms. Nonetheless, enlarging the range of quantum dynamics to include non-unitary evolution provides a wider range of possibilities for tuning the properties of quantum walks. For example, small amounts of decoherence in a quantum walk on the line can produce more uniform spreading (a top-hat distribution), without losing the quantum speed up. This paper reviews the work on decoherence, and more generally on non-unitary evolution, in quantum walks and suggests what future questions might prove interesting to pursue in this area.

[1]  Viv Kendon,et al.  Entanglement in coined quantum walks on regular graphs , 2005 .

[2]  Mark Hillery,et al.  Quantum walks with random phase shifts , 2006, quant-ph/0607092.

[3]  P. Knight,et al.  Propagating quantum walks: The origin of interference structures , 2003, quant-ph/0312133.

[4]  K. Birgitta Whaley,et al.  Quantum random-walk search algorithm , 2002, quant-ph/0210064.

[5]  Alexander Yu. Vlasov,et al.  On Quantum Cellular Automata , 2004, ArXiv.

[6]  Todd A. Brun,et al.  Quantum walks with infinite hitting times , 2006 .

[7]  Wojciech Tadej,et al.  A Concise Guide to Complex Hadamard Matrices , 2006, Open Syst. Inf. Dyn..

[8]  Viv Kendon,et al.  Decoherence can be useful in quantum walks , 2002, quant-ph/0209005.

[9]  Leonid Fedichkin,et al.  Nonunitary quantum walks on hypercycles , 2006 .

[10]  E. Farhi,et al.  Quantum computation and decision trees , 1997, quant-ph/9706062.

[11]  Alex D. Gottlieb,et al.  Two examples of discrete-time quantum walks taking continuous steps , 2003 .

[12]  Ofer Biham,et al.  One-dimensional quantum walk with unitary noise , 2003 .

[13]  P. Ribeiro,et al.  Aperiodic quantum random walks. , 2004, Physical review letters.

[14]  Leonid Fedichkin,et al.  Continuous-time quantum walks on a cycle graph (5 pages) , 2006 .

[15]  Christino Tamon,et al.  Mixing of quantum walk on circulant bunkbeds , 2006, Quantum Inf. Comput..

[16]  Christino Tamon,et al.  NON-UNIFORM MIXING OF QUANTUM WALK ON CYCLES , 2007, 0708.2096.

[17]  A. Gottlieb,et al.  Convergence of continuous-time quantum walks on the line. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Aharonov,et al.  Quantum random walks. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[19]  Norio Konno,et al.  Symmetry of Distribution for the One-Dimensional Hadamard Walk , 2002 .

[20]  Julia Kempe,et al.  Discrete Quantum Walks Hit Exponentially Faster , 2003, RANDOM-APPROX.

[21]  Andris Ambainis,et al.  Quantum random walks with decoherent coins , 2003 .

[22]  Peter C. Richter.  Quantum speedup of classical mixing processes , 2006, quant-ph/0609204.

[23]  Norio Konno,et al.  Localization of two-dimensional quantum walks , 2004 .

[24]  Norio Konno,et al.  Quantum Random Walks in One Dimension , 2002, Quantum Inf. Process..

[25]  Ashwin Nayak,et al.  Quantum Walk on the Line , 2000 .

[26]  A. Russell,et al.  Decoherence in quantum walks on the hypercube , 2005, quant-ph/0501169.

[27]  D. Meyer.  From quantum cellular automata to quantum lattice gases , 1996, quant-ph/9604003.

[28]  Simone Severini.  On the Digraph of a Unitary Matrix , 2003, SIAM J. Matrix Anal. Appl..

[29]  S. Lloyd.  Quantum-Mechanical Computers , 1995 .

[30]  Eric Vigoda,et al.  A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries , 2001, STOC '01.

[31]  P. Anderson.  Absence of Diffusion in Certain Random Lattices , 1958 .

[32]  H. Krovi,et al.  Hitting time for quantum walks on the hypercube (8 pages) , 2005, quant-ph/0510136.

[33]  Svante Janson,et al.  Convergence of coined quantum walks on R^d , 2005 .

[34]  E. Sudarshan,et al.  Zeno's paradox in quantum theory , 1976 .

[35]  Julia Kempe,et al.  Quantum Random Walks Hit Exponentially Faster , 2002, ArXiv.

[36]  R. Renner,et al.  Information-theoretic security proof for quantum-key-distribution protocols , 2005, quant-ph/0502064.

[37]  Hiroshi Imai,et al.  An Analysis of Absorbing Times of Quantum Walks , 2002, UMC.

[38]  Norio Konno,et al.  Symmetricity of Distribution for One-Dimensional Hadamard Walk , 2002 .

[39]  Christino Tamon,et al.  A note on graphs resistant to quantum uniform mixing , 2003 .

[40]  Andris Ambainis,et al.  QUANTUM WALKS AND THEIR ALGORITHMIC APPLICATIONS , 2003, quant-ph/0403120.

[41]  Andris Ambainis,et al.  Quantum walks on graphs , 2000, STOC '01.

[42]  M. Lavagna.  Quantum Phase Transitions , 2001, cond-mat/0102119.

[43]  Eric Bach,et al.  One-dimensional quantum walks with absorbing boundaries , 2004, J. Comput. Syst. Sci..

[44]  Dirk Bouwmeester,et al.  Optical Galton board , 1999 .

[45]  B Kraus,et al.  Lower and upper bounds on the secret-key rate for quantum key distribution protocols using one-way classical communication. , 2004, Physical review letters.

[46]  W. Dur,et al.  Quantum walks in optical lattices , 2002, quant-ph/0207137.

[47]  Andrew M. Childs,et al.  Spatial search and the Dirac equation , 2004 .

[48]  Andrew M. Childs,et al.  Spatial search by quantum walk , 2003, quant-ph/0306054.

[49]  Frédéric Magniez,et al.  Quantum algorithms for the triangle problem , 2005, SODA '05.

[50]  P. Knight,et al.  Quantum walk on the line as an interference phenomenon , 2003, quant-ph/0304201.

[51]  DyerMartin,et al.  A random polynomial-time algorithm for approximating the volume of convex bodies , 1991 .

[52]  Barry C. Sanders,et al.  Quantum walks in higher dimensions , 2002 .

[53]  Mario Szegedy,et al.  Spectra of Quantized Walks and a $\sqrt{\delta\epsilon}$ rule , 2004, quant-ph/0401053.

[54]  Vivien M. Kendon,et al.  Optimal computation with non-unitary quantum walks , 2008, Theor. Comput. Sci..

[55]  Norio Konno,et al.  A new type of limit theorems for the one-dimensional quantum random walk , 2002, quant-ph/0206103.

[56]  D. Meyer.  On the absence of homogeneous scalar unitary cellular automata , 1996, quant-ph/9604011.

[57]  J. Paz,et al.  Phase-space approach to the study of decoherence in quantum walks , 2003 .

[58]  S. A. Gurvitz.  Measurements with a noninvasive detector and dephasing mechanism , 1997 .

[59]  Christino Tamon,et al.  On mixing in continuous-time quantum walks on some circulant graphs , 2003, Quantum Inf. Comput..

[60]  Tomasz Luczak,et al.  Quantum walks on cycles , 2003 .

[61]  R. Siri,et al.  Decoherence in the quantum walk on the line , 2004, quant-ph/0403192.

[62]  Russ Bubley,et al.  Randomized algorithms , 1995, CSUR.

[63]  Barry C. Sanders,et al.  Complementarity and quantum walks , 2005 .

[64]  R. Toral,et al.  Fluctuation and Noise Letters , 2007 .

[65]  R. Siri,et al.  Quantum random walk on the line as a Markovian process , 2004 .

[66]  Mark Hillery,et al.  Scattering theory and discrete-time quantum walks , 2003, quant-ph/0312062.

[67]  Andrew M. Childs,et al.  Quantum algorithms for subset finding , 2005, Quantum Inf. Comput..

[68]  Norio Konno,et al.  A Path Integral Approach for Disordered Quantum Walks in One Dimension , 2004 .

[69]  M. Szegedy,et al.  Quantum Walk Based Search Algorithms , 2008, TAMC.

[70]  Andris Ambainis,et al.  One-dimensional quantum walks , 2001, STOC '01.

[71]  Peter C. Richter.  Almost uniform sampling via quantum walks , 2006, quant-ph/0606202.

[72]  Martin E. Dyer,et al.  A random polynomial-time algorithm for approximating the volume of convex bodies , 1991, JACM.

[73]  Alexander Russell,et al.  Quantum Walks on the Hypercube , 2002, RANDOM.

[74]  J. Douglas.  Aspects and applications of the random walk , 1995 .

[75]  Svante Janson,et al.  Weak limits for quantum random walks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[76]  Christino Tamon,et al.  Mixing and decoherence in continuous-time quantum walks on cycles , 2006, Quantum Inf. Comput..

[77]  Andris Ambainis,et al.  Quantum to classical transition for random walks. , 2003, Physical review letters.

[78]  Andris Ambainis,et al.  Quantum walks driven by many coins , 2002, quant-ph/0210161.

[79]  P. Knight,et al.  Optical Cavity Implementations of the Quantum Walk , 2003, quant-ph/0305165.

[80]  D Mozyrsky,et al.  Relaxation and the Zeno effect in qubit measurements. , 2003, Physical review letters.

[81]  J. P. Keating,et al.  Localization and its consequences for quantum walk algorithms and quantum communication , 2007 .

[82]  John Watrous.  Quantum Simulations of Classical Random Walks and Undirected Graph Connectivity , 2001, J. Comput. Syst. Sci..

[83]  Mourad E. H. Ismail,et al.  Three routes to the exact asymptotics for the one-dimensional quantum walk , 2003, quant-ph/0303105.

[84]  D. Abbott,et al.  Quantum walks with history dependence , 2003, quant-ph/0311009.

[85]  Barry C. Sanders,et al.  Quantum quincunx in cavity quantum electrodynamics , 2003 .

[86]  Nayak Ashwin,et al.  Quantum Walk on the Line , 2000 .

[87]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[88]  Frederick W. Strauch,et al.  Relativistic quantum walks , 2006 .

[89]  Daniel A. Spielman,et al.  Exponential algorithmic speedup by a quantum walk , 2002, STOC '03.

[90]  Viv Kendon,et al.  Decoherence vs entanglement in coined quantum walks , 2008 .

[91]  Ashley Montanaro,et al.  Quantum walks on directed graphs , 2005, Quantum Inf. Comput..

[92]  Uzy Smilansky,et al.  Quantum Chaos on Graphs , 1997 .

[93]  Viv Kendon,et al.  Decoherence versus entanglement in coined quantum walks , 2006, quant-ph/0612229.

[94]  C. Lomont.  THE HIDDEN SUBGROUP PROBLEM - REVIEW AND OPEN PROBLEMS , 2004, quant-ph/0411037.

[95]  R. Feynman.  Quantum mechanical computers , 1986 .

[96]  Uwe Schöning.  A Probabilistic Algorithm for k-SAT and Constraint Satisfaction Problems , 1999, FOCS.

[97]  G. J. Milburn,et al.  Implementing the quantum random walk , 2002 .

[98]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[99]  Gilles Brassard,et al.  Strengths and Weaknesses of Quantum Computing , 1997, SIAM J. Comput..

[100]  Christino Tamon,et al.  Universal mixing of quantum walk on graphs , 2006, Quantum Inf. Comput..

[101]  V. Kendon,et al.  A random walk approach to quantum algorithms , 2006, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[102]  Will Flanagan,et al.  Controlling discrete quantum walks: coins and initial states , 2003 .

[103]  Andris Ambainis.  Quantum Walk Algorithm for Element Distinctness , 2004, FOCS.

[104]  Gregor Tanner,et al.  Families of Line-Graphs and Their Quantization , 2001, nlin/0110043.

[105]  Viv Kendon,et al.  Quantum walks on general graphs , 2003, quant-ph/0306140.

[106]  Frederick W. Strauch,et al.  Connecting the discrete- and continuous-time quantum walks , 2006 .

[107]  Julia Kempe,et al.  Quantum random walks: An introductory overview , 2003, quant-ph/0303081.

[108]  Leonardo Ermann,et al.  Decoherence induced by a chaotic enviroment: A quantum walker with a complex coin , 2006 .

[109]  R. Laflamme,et al.  Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor , 2005, quant-ph/0507267.

[110]  R. Leighton,et al.  Feynman Lectures on Physics , 1971 .

[111]  R. Renner,et al.  An information-theoretic security proof for QKD protocols , 2005, quant-ph/0502064.