1,780 research outputs found
Towards a Minimal Stabilizer ZX-calculus
The stabilizer ZX-calculus is a rigorous graphical language for reasoning
about quantum mechanics. The language is sound and complete: one can transform
a stabilizer ZX-diagram into another one using the graphical rewrite rules if
and only if these two diagrams represent the same quantum evolution or quantum
state. We previously showed that the stabilizer ZX-calculus can be simplified
by reducing the number of rewrite rules, without losing the property of
completeness [Backens, Perdrix & Wang, EPTCS 236:1--20, 2017]. Here, we show
that most of the remaining rules of the language are indeed necessary. We do
however leave as an open question the necessity of two rules. These include,
surprisingly, the bialgebra rule, which is an axiomatisation of
complementarity, the cornerstone of the ZX-calculus. Furthermore, we show that
a weaker ambient category -- a braided autonomous category instead of the usual
compact closed category -- is sufficient to recover the meta rule 'only
connectivity matters', even without assuming any symmetries of the generators.Comment: 29 pages, minor updates for v
The GHZ/W-calculus contains rational arithmetic
Graphical calculi for representing interacting quantum systems serve a number
of purposes: compositionally, intuitive graphical reasoning, and a logical
underpinning for automation. The power of these calculi stems from the fact
that they embody generalized symmetries of the structure of quantum operations,
which, for example, stretch well beyond the Choi-Jamiolkowski isomorphism. One
such calculus takes the GHZ and W states as its basic generators. Here we show
that this language allows one to encode standard rational calculus, with the
GHZ state as multiplication, the W state as addition, the Pauli X gate as
multiplicative inversion, and the Pauli Z gate as additive inversion.Comment: In Proceedings HPC 2010, arXiv:1103.226
- …