Analysis of Absorbing Times of Quantum Walks

Quantum walks are expected to provide useful algorithmic tools for quantum computation. This paper introduces absorbing probability and time of quantum walks and gives both numerical simulation results and theoretical analyses on Hadamard walks on the line and symmetric walks on the hypercube from the viewpoint of absorbing probability and time.Comment: LaTeX2e, 14 pages, 6 figures, 1 table, figures revised, references added, to appear in Physical Review

Quantum Random Walks do not need a Coin Toss

Classical randomized algorithms use a coin toss instruction to explore different evolutionary branches of a problem. Quantum algorithms, on the other hand, can explore multiple evolutionary branches by mere superposition of states. Discrete quantum random walks, studied in the literature, have nonetheless used both superposition and a quantum coin toss instruction. This is not necessary, and a discrete quantum random walk without a quantum coin toss instruction is defined and analyzed here. Our construction eliminates quantum entanglement from the algorithm, and the results match those obtained with a quantum coin toss instruction.Comment: 6 pages, 4 figures, RevTeX (v2) Expanded to include relation to quantum walk with a coin. Connection with Dirac equation pointed out. Version to be published in Phys. Rev.

Quantum walks can find a marked element on any graph

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set $M$ consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time $HT(P,M)$ of any reversible random walk $P$ on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity $HT^+(\mathit{P,M})$ which we call extended hitting time. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the random walk $P$ and the absorbing walk $P'$, whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of this interpolation. Contrary to previous approaches, our results remain valid when the random walk $P$ is not state-transitive. We also provide algorithms in the cases when only approximations or bounds on parameters $p_M$ (the probability of picking a marked vertex from the stationary distribution) and $HT^+(\mathit{P,M})$ are known.Comment: 50 page

Absorption problems for quantum walks in one dimension

This paper treats absorption problems for the one-dimensional quantum walk determined by a 2 times 2 unitary matrix U on a state space {0,1,...,N} where N is finite or infinite by using a new path integral approach based on an orthonormal basis P, Q, R and S of the vector space of complex 2 times 2 matrices. Our method studied here is a natural extension of the approach in the classical random walk.Comment: 15 pages, small corrections, journal reference adde

Braiding Interactions in Anyonic Quantum Walks

The anyonic quantum walk is a dynamical model describing a single anyon propagating along a chain of stationary anyons and interacting via mutual braiding statistics. We review the recent results on the effects of braiding statistics in anyonic quantum walks in quasi-one dimensional ladder geometries. For anyons which correspond to spin-1/2 irreps of the quantum groups $SU(2)_k$, the non-Abelian species $(1<k<\infty)$ gives rise to entanglement between the walker and topological degrees of freedom which is quantified by quantum link invariants over the trajectories of the walk. The decoherence is strong enough to reduce the walk on the infinite ladder to classical like behaviour. We also present numerical results on mixing times of $SU(2)_2$ or Ising model anyon walks on cyclic graphs. Finally, the possible experimental simulation of the anyonic quantum walk in Fractional Quantum Hall systems is discussed.Comment: 13 pages, submitted to Proceedings of the 2nd International Conference on Theoretical Physics (ICTP 2012

Topological delocalization in the completely disordered two-dimensional quantum walk

We investigate numerically and theoretically the effect of spatial disorder on two-dimensional split-step discrete-time quantum walks with two internal "coin" states. Spatial disorder can lead to Anderson localization, inhibiting the spread of quantum walks, putting them at a disadvantage against their diffusively spreading classical counterparts. We find that spatial disorder of the most general type, i.e., position-dependent Haar random coin operators, does not lead to Anderson localization but to a diffusive spread instead. This is a delocalization, which happens because disorder places the quantum walk to a critical point between different anomalous Floquet-Anderson insulating topological phases. We base this explanation on the relationship of this general quantum walk to a simpler case more studied in the literature and for which disorder-induced delocalization of a topological origin has been observed. We review topological delocalization for the simpler quantum walk, using time evolution of the wave functions and level spacing statistics. We apply scattering theory to two-dimensional quantum walks and thus calculate the topological invariants of disordered quantum walks, substantiating the topological interpretation of the delocalization and finding signatures of the delocalization in the finite-size scaling of transmission. We show criticality of the Haar random quantum walk by calculating the critical exponent $\eta$ in three different ways and find $\eta$ $\approx$ 0.52 as in the integer quantum Hall effect. Our results showcase how theoretical ideas and numerical tools from solid-state physics can help us understand spatially random quantum walks.Comment: 18 pages, 18 figures. Similar to the published version. Comments are welcom