Magic states were originally introduced as a resource that enables universal quantum computation using classically simulable Clifford gates. This concept has been extended to matchgate circuits (MGCs) which are made of two-qubit nearest-neighbour quantum gates defined by a set of algebraic constraints. In our work, we study the Gaussian rank of a quantum state — defined as the minimum number of terms in any decomposition of that state into Gaussian states — and associated quantities: the Gaussian Fidelity and the Gaussian Extent.
First, we will give a review of rank-based simulation methods in the stabilizer setting, and describe the operational relevance of the Rank, Fidelity & Extent. We will then give some background on MGCs and their physical relevance. Then, we will move onto the new results from our work.
The primary result is a description of the algebraic structure of Gaussian states, including the independent sets of constraints upper-bounding the dimension of the manifold of Gaussian states. Using these equations, we can derive results including the form of linearly dependent triples of Gaussian states.
We then discuss properties of the Gaussian Fidelity and Extent, making progress towards resolving the question of multiplicativity of the Gaussian Fidelity.
Finally, we investigate the Gaussian Rank. we show that the rank of two copies of our canonical magic state is 4 for symmetry-restricted decompositions. Numerical investigation suggests that no low-rank decompositions exist of either 2 or 3 copies of the magic state. Finally, we consider approximate Gaussian rank and present approximate decompositions for selected magic states.
Paper available at