A family of unitaries for the quantum period finding algorithm

We use differentiable programming and gradient descent to find unitary matrices that can be used in the period finding algorithm to extract period information from the state of a quantum computer post-application of the oracle. The standard procedure is to use the inverse quantum Fourier transform. Our findings suggest that this is not the only unitary matrix appropriate for the period finding algorithm. There exists a family of unitary matrices that can affect out the same transformation and they are related by a symmetry. An analysis of this symmetry is presented. These unitary matrices can be learned by an algorithm which reveals the underlying symmetry. We also find simple neural networks are able to learn this symmetry and differentiate such unitary matrices from randomly generated ones.