Quadratic form expansions for unitaries

We introduce a technique to analyse unitary operations when they are presented in a form similar to a sum over paths - i.e. as a sum of phase contributions due to different paths between computational basis states - when the phase contributed by each path is described by a quadratic form over real numbers. The polynomial which describes the phase of each path can be used to obtain a diagonal unitary matrix describing a collection of possibly fractional controlled-Z operators and single qubit Z rotations on a network of qubits, akin to the preparation of the entangled resource in the one-way measurement model of quantum computing. Using this we show that if the obtain entangled state has a flow property then a quantum circuit of the original unitary can be found automatically, and that the time required to do so is polynomial in the size of the phase polynomial and precision of it’s parameters