Some Upper and Lower Bounds regarding Query Complexity

Query complexity is one of the several notions of complexity de ned to measure the cost of algorithms. It plays an important role in demonstrating the quantum advantage, that quantum computing is faster than classical computation in solving some problems. Kempe showed that a discrete time quantum walk on hypercube hits the antipodal point with exponentially fewer queries than a simple random walk [K05]. Childs et al. showed that a continuous time quantum walk on \Glued Trees" detects the label of a special vertex with exponentially fewer queries than any classical algorithm [CCD03], and the result translates to discrete time quantum walk by an e cient simulation. Building on these works, we examine the query complexity of variations of the hyper- cube and Glued Tree problems. We rst show the gap between quantum and classical query algorithms for a modi ed hypercube problem is at most polynomial. We then strengthen the query complexity gap for the Glued Tree label detection problem by improving a classical lower bound technique; and we prove such a lower bound is nearly tight by giving a classical query algorithm whose query complexity matches the lower bound, up to a polylog factor

Decoherence in Discrete Quantum Walks

We present an introduction to coined quantum walks on regular graphs, which have been developed in the past few years as an alternative to quantum Fourier transforms for underpinning algorithms for quantum computation. We then describe our results on the effects of decoherence on these quantum walks on a line, cycle and hypercube. We find high sensitivity to decoherence, increasing with the number of steps in the walk, as the particle is becoming more delocalised with each step. However, the effect of a small amount of decoherence can be to enhance the properties of the quantum walk that are desirable for the development of quantum algorithms, such as fast mixing times to uniform distributions.Comment: 15 pages, Springer LNP latex style, submitted to Proceedings of DICE 200

Almost uniform sampling via quantum walks

Many classical randomized algorithms (e.g., approximation algorithms for #P-complete problems) utilize the following random walk algorithm for {\em almost uniform sampling} from a state space $S$ of cardinality $N$: run a symmetric ergodic Markov chain $P$ on $S$ for long enough to obtain a random state from within $\epsilon$ total variation distance of the uniform distribution over $S$. The running time of this algorithm, the so-called {\em mixing time} of $P$, is $O(\delta^{-1} (\log N + \log \epsilon^{-1}))$, where $\delta$ is the spectral gap of $P$. We present a natural quantum version of this algorithm based on repeated measurements of the {\em quantum walk} $U_t = e^{-iPt}$. We show that it samples almost uniformly from $S$ with logarithmic dependence on $\epsilon^{-1}$ just as the classical walk $P$ does; previously, no such quantum walk algorithm was known. We then outline a framework for analyzing its running time and formulate two plausible conjectures which together would imply that it runs in time $O(\delta^{-1/2} \log N \log \epsilon^{-1})$ when $P$ is the standard transition matrix of a constant-degree graph. We prove each conjecture for a subclass of Cayley graphs.Comment: 13 pages; v2 added NSF grant info; v3 incorporated feedbac