Towards Normal Forms for GHZ∕W Calculus

Abstract

Recently, a novel GHZ/W graphical calculus has been established to study and reason more intuitively about interacting quantum systems. The compositional structure of this calculus was shown to be well-equipped to sufficiently express arbitrary mutlipartite quantum states equivalent under stochastic local operations and classical communication (SLOCC). However, it is still not clear how to explicitly identify which graphical properties lead to what states. This can be achieved if we have well-behaved normal forms for arbitrary graphs within this calculus. This article lays down a first attempt at realizing such normal forms for a restricted class of such graphs, namely simple and regular graphs. These results should pave the way for the most general cases as part of future work.

Quantum Circuits in Additive Hilbert Space

“…Quantum computer science and quantum mechanics in general have a rich history in diagrammatic representations [22,[33][34][35]. Our approach draws a lot in spirit to diagrammatic formalizations of quantum computation such as the ZX calculus and its variations [35][36][37][38]. Just like the circuit model and unlike our proposal, these calculi all rely on the multiplicative tensor ⊗-structure.…”