Universal MBQC with generalised parity-phase interactions and Pauli measurements

We introduce a new family of models for measurement-based quantum computation which are deterministic and approximately universal. The resource states which play the role of graph states are prepared via 2-qubit gates of the form $\exp(-i\frac{\pi}{2^{n}} Z\otimes Z)$. When $n = 2$, these are equivalent, up to local Clifford unitaries, to graph states. However, when $n > 2$, their behaviour diverges in two important ways. First, multiple applications of the entangling gate to a single pair of qubits produces non-trivial entanglement, and hence multiple parallel edges between nodes play an important role in these generalised graph states. Second, such a state can be used to realise deterministic, approximately universal computation using only Pauli $Z$ and $X$ measurements and feed-forward. Even though, for $n > 2$, the relevant resource states are no longer stabiliser states, they admit a straightforward, graphical representation using the ZX-calculus. Using this representation, we are able to provide a simple, graphical proof of universality. We furthermore show that for every $n > 2$ this family is capable of producing all Clifford gates and all diagonal gates in the $n$-th level of the Clifford hierarchy.Comment: 19 pages, accepted for publication in Quantum (quantum-journal.org). A previous version of this article had the title: "Universal MBQC with M{\o}lmer-S{\o}rensen interactions and two measurement bases

Twisted character of a small representation of GL(4)

We compute by a purely local method the (elliptic) twisted by transpose-inverse character \chi_{\pi_Y} of the representation \pi_Y=I_{(3,1)}(1_3x\chi_Y) of G=GL(4,F), where F is a p-adic field, p not 2, and Y is an unramified quadratic extension of F; \chi_Y is the nontrivial character of F^\x/N_{Y/F}Y^x. The representation \pi_Y is normalizedly induced from \pmatrix m_3&\ast 0&m_1\endpmatrix \mapsto\chi_Y(m_1), m_i in GL(i,F), on the maximal parabolic subgroup of type (3,1). We show that the twisted character \chi_{\pi_Y} of \pi_Y is an unstable function: its value at a twisted regular elliptic conjugacy class with norm in C_Y=``GL(2,Y)/F^x'' is minus its value at the other class within the twisted stable conjugacy class. It is zero at the classes without norm in C_Y. Moreover \pi_Y is the endoscopic lift of the trivial representation of C_Y. We deal only with unramified Y/F, as globally this case occurs almost everywhere. Naturally this computation plays a role in the theory of lifting of C_Y and GSp(2) to GL(4) using the trace formula. Our work extends -- to the context of nontrivial central characters -- the work of math.NT/0606262, where representations of PGL(4,F) are studied. In math.NT/0606262 a 4-dimensional analogue of the model of the small representation of PGL(3,F) introduced with Kazhdan in a 3-dimensional case is developed, and the local method of computation introduced by us in the 3-dimensional case is extended. As in math.NT/0606262 we use here the classification of twisted (stable) regular conjugacy classes in GL(4,F).Comment: Accepted for publication by the International Journal of Number Theor