An extremal result for geometries in the one-way measurement model

  • Authors:

  • Niel de Beaudrap;Martin Pei

  • Affiliations:

  • Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada;Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada

  • Venue:
  • Quantum Information & Computation
  • Year:
  • 2008

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Abstract

We present an extremal result for the class of graphs G which (together with some specified sets of input and output vertices, I and O) have a certain "flow" property introduced by Danos and Kashefi for the one-way measurement model of quantum computation. The existence of a flow for a triple (G, I,O) allows a unitary embedding to be derived from any choice of measurement bases allowed in the one-way measurement model. We prove an upper bound on the number of edges that a graph G may have, in order for a triple (G, I,O) to have a flow for some I,O ⊆ V (G), in terms of the number of vertices in G and O. This implies that finding a flow for a triple (G, I,O) when |I| = |O| = k (corresponding to unitary transformations in the measurement model) and |V (G)| = n can be performed in time O(k2n), improving the earlier known bound of O(km) given in [8], where m = |E(G)|.