Mehdi Mhalla

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

Generalized flow and determinism in measurement-based quantum computation

\Theta(n^{3/2})

Finding Optimal Flows Efficiently

in the matrix model and in

Information Flow in Secret Sharing Protocols

\Theta(\sqrt{nm})

New Protocols and Lower Bounds for Quantum Secret Sharing with Graph States

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

Which Graph States are Useful for Quantum Information Processing

\Theta(n^{3/2})

On a modular domination game

in the matrix model but in

On the minimum degree up to local complementation: bounds and complexity

\Theta(n)

Quantum Secret Sharing with Graph States

in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions.

Resources required for preparing graph states

We extend the notion of quantum information flow defined by Danos and Kashefi (2006 Phys. Rev. A 74 052310) for the one-way model (Raussendorf and Briegel 2001 Phys. Rev. Lett. 86 910) and present a necessary and sufficient condition for the stepwise uniformly deterministic computation in this model. The generalized flow also applied in the extended model with measurements in the (X, Y), (X, Z) and (Y, Z) planes. We apply both measurement calculus and the stabiliser formalism to derive our main theorem which for the first time gives a full characterization of the stepwise uniformly deterministic computation in the one-way model. We present several examples to show how our result improves over the traditional notion of flow, such as geometries (entanglement graph with input and output) with no flow but having generalized flow and we discuss how they lead to an optimal implementation of the unitaries. More importantly one can also obtain a better quantum computation depth with the generalized flow rather than with flow. We believe our characterization result is particularly valuable for the study of the algorithms and complexity in the one-way model.