Peter Høyer

Consequences and limits of nonlocal strategies

This paper investigates various aspects of the nonlocal effects that can arise when entangled quantum information is shared between two parties. A natural framework for studying nonlocality is that of cooperative games with incomplete information, where two cooperating players may share entanglement. Here, nonlocality can be quantified in terms of the values of such games. We review some examples of non-locality and show that it can profoundly affect the soundness of two-prover interactive proof systems. We then establish limits on nonlocal behavior by upper-bounding the values of several of these games. These upper bounds can be regarded as generalizations of the so-called Tsirelson inequality. We also investigate the amount of entanglement required by optimal and nearly optimal quantum strategies.

An exact quantum polynomial-time algorithm for Simon's problem

We investigate the power of quantum computers when they are required to return an answer that is guaranteed to be correct after a time that is upper-bounded by a polynomial in the worst case. We show that a natural generalization of Simons problem can be solved in this way, whereas previous algorithms required quantum polynomial time in the expected sense only, without upper bounds on the worst-case running time. This is achieved by generalizing both Simons and Grovers algorithms and combining them in a novel way. It follows that there is a decision problem that can be solved in exact quantum polynomial time, which would require expected exponential time on any classical bounded-error probabilistic computer if the data is supplied as a black box.

Quantum Query Complexity of Some Graph Problems

Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example, we show that the query complexity of Minimum Spanning Tree is in

Quantum search on bounded-error inputs

\Theta(n^{3/2})

Quantum cryptanalysis of hash and claw-free functions

in the matrix model and in

Quantum algorithms for element distinctness

\Theta(\sqrt{nm})

Multiparty quantum communication complexity

in the array model, while the complexity of Connectivity is also in

The quantum query complexity of the hidden subgroup problem is polynomial

\Theta(n^{3/2})

Arbitrary phases in quantum amplitude amplification

in the matrix model but in

Quantum Cryptanalysis of Hash and Claw-Free Functions

\Theta(n)