A Numerical Analyst Looks at the "Cutoff Phenomenon" in Card Shuffling and Other Markov Chains

Diaconis and others have shown that certain Markov chains exhibit a "cutoff phenomenon" in which, after an initial period of seemingly little progress, convergence to the steady state occurs suddenly. Since Markov chains are just powers of matrices, how can such effects be explained in the language of applied linear algebra? We attempt to do this, focusing on two examples: random walk on a hypercube, which is essentially the same as the problem of Ehrenfest urns, and the celebrated case of riffle shuffling of a deck of cards. As is typical with transient phenomena in matrix processes, the reason for the cutoff is not readily apparent from an examination of eigenvalues or eigenvectors, but it is reflected strongly in pseudosprectra - provided they are measured in the 1-norm, not the 2-norm. We illustrate and explain the cutoff phenomenon with Matlab computations based in part on a new explicit formula for the entries of the $n \times n$ "riffle shuffle matrix", and note that while the normwise cutoff may occur at one point, such as $\frac{3}{2} \log_{2} n$ for the riffle shuffle, weak convergence may occur at an equally precise earlier point such as $\log_{2} n$

Unlacing the lace expansion: a survey to hypercube percolation

The purpose of this note is twofold. First, we survey the study of the percolation phase transition on the Hamming hypercube {0,1}^m obtained in the series of papers [9,10,11,24]. Secondly, we explain how this study can be performed without the use of the so-called "lace-expansion" technique. To that aim, we provide a novel simple proof that the triangle condition holds at the critical probability. We hope that some of these techniques will be useful to obtain non-perturbative proofs in the analogous, yet much more difficult study on high-dimensional tori.Comment: 19 pages. arXiv admin note: text overlap with arXiv:1201.395

Search on a Hypercubic Lattice using a Quantum Random Walk: I. d>2

Random walks describe diffusion processes, where movement at every time step is restricted to only the neighbouring locations. We construct a quantum random walk algorithm, based on discretisation of the Dirac evolution operator inspired by staggered lattice fermions. We use it to investigate the spatial search problem, i.e. finding a marked vertex on a $d$-dimensional hypercubic lattice. The restriction on movement hardly matters for $d>2$, and scaling behaviour close to Grover's optimal algorithm (which has no restriction on movement) can be achieved. Using numerical simulations, we optimise the proportionality constants of the scaling behaviour, and demonstrate the approach to that for Grover's algorithm (equivalent to the mean field theory or the $d\to\infty$ limit). In particular, the scaling behaviour for $d=3$ is only about 25% higher than the optimal $d\to\infty$ value.Comment: 11 pages, Revtex (v2) Introduction and references expanded. Published versio

Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa

Convergence Speed of the Consensus Algorithm with Interference and Sparse Long-Range Connectivity

We analyze the effect of interference on the convergence rate of average consensus algorithms, which iteratively compute the measurement average by message passing among nodes. It is usually assumed that these algorithms converge faster with a greater exchange of information (i.e., by increased network connectivity) in every iteration. However, when interference is taken into account, it is no longer clear if the rate of convergence increases with network connectivity. We study this problem for randomly-placed consensus-seeking nodes connected through an interference-limited network. We investigate the following questions: (a) How does the rate of convergence vary with increasing communication range of each node? and (b) How does this result change when each node is allowed to communicate with a few selected far-off nodes? When nodes schedule their transmissions to avoid interference, we show that the convergence speed scales with $r^{2-d}$, where $r$ is the communication range and $d$ is the number of dimensions. This scaling is the result of two competing effects when increasing $r$: Increased schedule length for interference-free transmission vs. the speed gain due to improved connectivity. Hence, although one-dimensional networks can converge faster from a greater communication range despite increased interference, the two effects exactly offset one another in two-dimensions. In higher dimensions, increasing the communication range can actually degrade the rate of convergence. Our results thus underline the importance of factoring in the effect of interference in the design of distributed estimation algorithms.Comment: 27 pages, 4 figure