Gaussian decomposition of magic states for matchgate computations

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. We investigate the algebraic structure of Gaussian states and find and describe the independent sets of constraints upper-bounding the dimension of the manifold of Gaussian states. Furthermore, we describe the form of linearly dependent triples of Gaussian states and find the dimension of the manifold of solutions. By constructing the corresponding $\epsilon$-net for the Gaussian states, we are able to obtain upper bounds on the Gaussian fidelity. We identify a family of extreme points of the feasible set for the Dual Gaussian extent problem and show that Gaussian extent is multiplicative on systems of 4 qubits; and further that it is multiplicative on primal points whose optimal dual witness is in the above family. These extreme points turn out to be closely related to Extended Hamming Codes. We show that optimal dual witnesses are unique almost-surely, when the primal point lies in the interior of the normal cone of an extreme point. Furthermore, we show that the Gaussian 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.Comment: See also related works by Dias and Koenig and by Reardon-Smith et al. appearing in the same arXiv listin

Weakly nonlinear Holmboe waves

Holmboe waves are long-lived traveling waves commonly found in environmental stratified shear flows in which a relatively sharp, stable density interface is embedded within a more diffuse shear layer. Although previous research has focused on their linear properties (the Holmboe instability), and on their turbulent properties (Holmboe wave turbulence), little is known about their finite-amplitude properties in the nonlinear but non-turbulent regime. In this paper we tackle this problem with a weakly-nonlinear temporal stability analysis of Holmboe waves.Cambridge Mathematics Placement (CMP) Leverhulme Trust Isaac Newton Trus

Research data supporting "Weakly nonlinear Holmboe waves"

This dataset contains: - weakly nonlinear code.zip: all the MATLAB .m code files used to produce the results of the paper. This folder is organised in multiple folders and subfolders. It also contains a short screencast video showing how to run some tricky parts of the code. The Output subfolder contains the raw .mat data to reproduce figures 3,4,6,7,8,9 of the paper. - weakly nonlinear code README.pdf: all details about how to run the above code, its folder structure, details about inputs/outputs of individual subroutines, with flow charts to explain how to run successive parts of the code to produce useful outputs. - final plotting code.zip: the MATLAB .m code files used to plot figures 3,4,6,7,8,9, using the output data in the above folder. Further minor (manual) editing was required, but these codes make it at least clear how to access the non-trivial output data structures to plot useful results. The weakly nonlinear code folder contains multiple 'basic' plotting subroutines to plot raw results to guide research, but these were not used directly for the plots in the paper (except for figure 5, which is quite basic). These codes are sufficient to reproduce all results in the paper, and are suitable to serve as a basis for future work on weakly nonlinear Holmboe waves, as outlined in the conclusion of the paper.Cambridge Mathematics Placement (CMP) Leverhulme Trust Isaac Newton Trus