Finding Optimal Flows Efficiently

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.