| dc.description.abstract | In Valiant's matchgate theory, 2-input 2-output matchgates are 4 4 matrices that
satisfy ten so-called matchgate identities. We prove that the set of all such matchgates
(including non-unitary and non-invertible ones) coincides with the topological
closure of the set of all matrices obtained as exponentials of linear combinations of
the 2-qubit Jordan-Wigner (JW) operators and their quadratic products, extending a
previous result of Knill. In Valiant's theory, outputs of matchgate circuits can be classically
computed in poly-time. Via the JW formalism, Terhal & DiVincenzo and Knill
established a relation of a unitary class of these circuits to the e fficient simulation of
non-interacting fermions. We describe how the JW formalism may be used to give an
e cient simulation for all cases in Valiant's simulation theorem, which in particular includes
the case of non-interacting fermions generalised to allow arbitrary 1-qubit gates
on the first line at any stage in the circuit. Finally we give an exposition of how these
simulation results can be alternatively understood from some basic Lie algebra theory,
in terms of a formalism introduced by Somma et al. | |
