Finding Optimal Flows Efficiently

Abstract

Among the models of quantum computation, the One-way Quantum Computer [11,12] is one of the most promising proposals of physical realisation [13], and opens new perspectives for parallelisation by taking advantage of quantum entanglement [2]. Since a One-way QC is based on quantum measurement, which is a fundamentally nondeterministic evolution, a sufficient condition of global determinism has been introduced in [4] as the existence of a causal flow in a graph that underlies the computation. A O(n 3)-algorithm has been introduced [6] for finding such a causal flow when the numbers of output and input vertices in the graph are equal, otherwise no polynomial time algorithm was known for deciding whether a graph has a causal flow or not. Our main contribution is to introduce a O(m)-algorithm for finding a causal flow (where m is the number of edges of the graph), if any, whatever the numbers of input and output vertices are. This answers the open question stated by Danos and Kashefi [4] and by de Beaudrap [6]. Moreover, we prove that our algorithm produces a flow of minimal depth.Whereas the existence of a causal flow is a sufficient condition for determinism, it is not a necessary condition. A weaker version of the causal flow, called gflow (generalised flow) has been introduced in [3] and has been proved to be a necessary and sufficient condition for a family of deterministic computations. Moreover the depth of the quantum computation is upper bounded by the depth of the gflow. However the existence of a polynomial time algorithm that finds a gflow has been stated as an open question in [3]. In this paper we answer this positively with a polynomial time algorithm that outputs an optimal gflow of a given graph and thus finds an optimal correction strategy to the nondeterministic evolution due to measurements.KeywordsPolynomial Time AlgorithmGraph StateCorrection StrategyQuantum CircuitRecursive CallThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.