Quantum Walks with Entangled Coins

We present a mathematical formalism for the description of unrestricted quantum walks with entangled coins and one walker. The numerical behaviour of such walks is examined when using a Bell state as the initial coin state, two different coin operators, two different shift operators, and one walker. We compare and contrast the performance of these quantum walks with that of a classical random walk consisting of one walker and two maximally correlated coins as well as quantum walks with coins sharing different degrees of entanglement. We illustrate that the behaviour of our walk with entangled coins can be very different in comparison to the usual quantum walk with a single coin. We also demonstrate that simply by changing the shift operator, we can generate widely different distributions. We also compare the behaviour of quantum walks with maximally entangled coins with that of quantum walks with non-entangled coins. Finally, we show that the use of different shift operators on 2 and 3 qubit coins leads to different position probability distributions in 1 and 2 dimensional graphs.Comment: Two new sections and several changes from referees' comments. 12 pages and 12 (colour) figure

Controlling discrete quantum walks: coins and intitial states

In discrete time, coined quantum walks, the coin degrees of freedom offer the potential for a wider range of controls over the evolution of the walk than are available in the continuous time quantum walk. This paper explores some of the possibilities on regular graphs, and also reports periodic behaviour on small cyclic graphs.Comment: 10 (+epsilon) pages, 10 embedded eps figures, typos corrected, references added and updated, corresponds to published version (except figs 5-9 optimised for b&w printing here

Hitting time for quantum walks on the hypercube

Hitting times for discrete quantum walks on graphs give an average time before the walk reaches an ending condition. To be analogous to the hitting time for a classical walk, the quantum hitting time must involve repeated measurements as well as unitary evolution. We derive an expression for hitting time using superoperators, and numerically evaluate it for the discrete walk on the hypercube. The values found are compared to other analogues of hitting time suggested in earlier work. The dependence of hitting times on the type of unitary ``coin'' is examined, and we give an example of an initial state and coin which gives an infinite hitting time for a quantum walk. Such infinite hitting times require destructive interference, and are not observed classically. Finally, we look at distortions of the hypercube, and observe that a loss of symmetry in the hypercube increases the hitting time. Symmetry seems to play an important role in both dramatic speed-ups and slow-downs of quantum walks.Comment: 8 pages in RevTeX format, four figures in EPS forma

Experimental realization of a momentum-space quantum walk

We report on a discrete-time quantum walk that uses the momentum of ultra-cold rubidium-87 atoms as the walk space and two internal atomic states as the coin degree of freedom. Each step of the walk consists of a coin toss (a microwave pulse) followed by a unitary shift operator (a resonant ratchet pulse). We carry out a comprehensive experimental study on the effects of various parameters, including the strength of the shift operation, coin parameters, noise, and initialization of the system on the behavior of the walk. The walk dynamics can be well controlled in our experiment; potential applications include atom interferometry and engineering asymmetric walks.Comment: 11 pages, 11 figure

Decoherence in quantum walks - a review

The development of quantum walks in the context of quantum computation, as generalisations of random walk techniques, led rapidly to several new quantum algorithms. These all follow 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.Comment: 52 pages, invited review, v2 & v3 updates to include significant work since first posted and corrections from comments received; some non-trivial typos fixed. Comments now limited to changes that can be applied at proof stag

Kvantu automātu un meklēšanas algoritmu iespējas un ierobežojumi

Kvantu skaitļošana ir nozare, kas pēta uz kvantu mehānikas likumiem balstīto skaitļošanas modeļu īpašības. Disertācija ir veltīta kvantu skaitļošanas algoritmiskiem aspektiem. Piedāvāti rezultāti trijos virzienos: Kvantu galīgi automāti Analizēta stāvokļu efektivitāte kvantu vienvirziena galīgam automātam. Uzlabota labāka zināmā eksponenciālā atšķirība [AF98] starp kvantu un klasiskajiem galīgajiem automātiem. Grovera algoritma analīze Pētīta Grovera algoritma noturība pret kļūdām. Vispārināts [RS08] loģisko kļūdu modelis un piedāvāti vairāki jauni rezultāti. Kvantu klejošana Pētīta meklēšana 2D režģī izmantojot kvantu klejošanu. Paātrināts [AKR05] kvantu klejošanas meklēšanas algoritms. Atslēgas vārdi: Kvantu galīgi automāti, eksponenciālā atšķirība, Grovera algoritms, noturība pret kļūdām, kvantu klejošana LITERATŪRA [AF98] A. Ambainis, R. Freivalds. 1-way quantum finite automata: strengths, weaknesses and generalizations. Proceedings of the 39th IEEE Conference on Foundations of Computer Science, 332-341, 1998. arXiv:quant-ph/9802062v3 [AKR05] A. Ambainis, J. Kempe, A. Rivosh. Coins make quantum walks faster. Proceedings of SODA’05, 1099-1108, 2005. [RS08] O. Regev, L. Schiff. Impossibility of a Quantum Speed-up with a Faulty Oracle. Proceedings of ICALP’2008, Lecture Notes in Computer Science, 5125:773-781, 2008.Quantum computation is the eld that investigates properties of models of computation based on the laws of the quantum mechanics. The thesis is ded- icated to algorithmic aspects of quantum computation and provides results in three directions: Quantum nite automata We study space-eciency of one-way quantum nite automata. We improve best known exponential separation [AF98] between quantum and classical one-way nite automata. Analysis of Grover's algorithm We study fault-tolerance of Grover's algorithm. We generalize the model of logical faults by [RS08] and present several new results. Quantum walks We study search by quantum walks on two-dimensional grid. We im- prove (speed-up) quantum walk search algorithm by [AKR05]. Keywords: Quantum nite automata, exponential separation, Grover's al- gorithm, fault-tolerance, quantum walks BIBLIOGRAPHY [AF98] A. Ambainis, R. Freivalds. 1-way quantum nite automata: strengths, weaknesses and gen- eralizations. Proceedings of the 39th IEEE Conference on Foundations of Computer Science, 332-341, 1998. arXiv:quant-ph/9802062v3 [AKR05] A. Ambainis, J. Kempe, A. Rivosh. Coins make quantum walks faster. Proceedings of SODA'05, 1099-1108, 2005. [RS08] O. Regev, L. Schi. Impossibility of a Quantum Speed-up with a Faulty Oracle. Proceedings of ICALP'2008, Lecture Notes in Computer Science, 5125:773-781, 2008