arXiv subjects

Ashley Montanaro

Publications and source records attributed to Ashley Montanaro.

96 records · Page 6Linked to original sources

arXiv2007

Unbounded Error Quantum Query Complexity

This work studies the quantum query complexity of Boolean functions in a scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded error quantum query complexity is exactly half of its classical counterpart for any (partial or total) Boolean function. Moreover, we show that the "black-box" approach to convert quantum query algorithms into communication protocols by Buhrman-Cleve-Wigderson [STOC'98] is optimal even in the unbounded error setting. We also study a setting related to the unbounded error model, called the weakly unbounded error setting, where the cost of a query algorithm is given by q+log(1/2(p-1/2)), where q is the number of queries made and p>1/2 is the success probability of the algorithm. In contrast to the case of communication complexity, we show a tight Theta(log n) separation between quantum and classical query complexity in the weakly unbounded error setting for a partial Boolean function. We also show the asymptotic equivalence between them for some well-studied total Boolean functions.

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arXiv2007

Quantum search of partially ordered sets

We investigate the generalisation of quantum search of unstructured and totally ordered sets to search of partially ordered sets (posets). Two models for poset search are considered. In both models, we show that quantum algorithms can achieve at most a quadratic improvement in query complexity over classical algorithms, up to logarithmic factors; we also give quantum algorithms that almost achieve this optimal reduction in complexity. In one model, we give an improved quantum algorithm for searching forest-like posets; in the other, we give an optimal O(sqrt(m))-query quantum algorithm for searching posets derived from m*m arrays sorted by rows and columns. This leads to a quantum algorithm that finds the intersection of two sorted lists of n integers in O(sqrt(n)) time, which is optimal.

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arXiv2006

A lower bound on entanglement-assisted quantum communication complexity

We prove a general lower bound on the bounded-error entanglement-assisted quantum communication complexity of Boolean functions. The bound is based on the concept that any classical or quantum protocol to evaluate a function on distributed inputs can be turned into a quantum communication protocol. As an application of this bound, we give a very simple proof of the statement that almost all Boolean functions on n+n bits have linear communication complexity, even in the presence of unlimited entanglement.

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arXiv2006

On the quantum chromatic number of a graph

We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince an interrogator with certainty that they have a colouring of the graph. After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is 2, nor if it is 3 in a restricted quantum model; on the other hand, we exhibit a graph on 18 vertices and 44 edges with chromatic number 5 and quantum chromatic number 4.

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arXiv2006

On the distinguishability of random quantum states

We develop two analytic lower bounds on the probability of success p of identifying a state picked from a known ensemble of pure states: a bound based on the pairwise inner products of the states, and a bound based on the eigenvalues of their Gram matrix. We use the latter to lower bound the asymptotic distinguishability of ensembles of n random quantum states in d dimensions, where n/d approaches a constant. In particular, for almost all ensembles of n states in n dimensions, p>0.72. An application to distinguishing Boolean functions (the "oracle identification problem") in quantum computation is given.

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arXiv2005

Quantum walks on directed graphs

We consider the definition of quantum walks on directed graphs. Call a directed graph reversible if, for each pair of vertices (i, j), if i is connected to j then there is a path from j to i. We show that reversibility is a necessary and sufficient condition for a directed graph to allow the notion of a discrete-time quantum walk, and discuss some implications of this condition. We present a method for defining a "partially quantum" walk on directed graphs that are not reversible.

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