Jordan-Wigner formalism for classical simulation beyond binary matchgates
Richard Jozsa, Akimasa Miyake · arXiv (Cornell University) · 2013
The unitary matchgate circuits introduced by Valiant provide an interesting class of quantum circuits that are classically efficiently simulatable. They were shown by Terhal & DiVincenzo and Knill to be related to the physics of non-interacting fermions. The Jordan-Wigner (JW) formalism provides an efficient classical simulation of the latter, which turns out to be equivalent to the restricted case of circuits of binary (2-qubit) matchgates. Valiant's formalism allows further unitary gates: in particular we may include arbitrary 1-qubit gates on the first qubit line at any stage within a binary matchgate circuit. In this note we show how the JW formalism may be extended to provide an efficient classical simulation of such extended circuits, and we show how the simulability also follows from some elementary Lie algebra theory. The essential ingredients have been indicated previously by Knill in a condensed and abstract form, and our purpose is to make these results explicit and transparent.