Miriam Backens ; Simon Perdrix ; Quanlong Wang - Towards a Minimal Stabilizer ZX-calculus

lmcs:3961 - Logical Methods in Computer Science, December 22, 2020, Volume 16, Issue 4 - https://doi.org/10.23638/LMCS-16(4:19)2020
Towards a Minimal Stabilizer ZX-calculus

Authors: Miriam Backens ; Simon Perdrix ; Quanlong Wang

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.

Volume: Volume 16, Issue 4
Published on: December 22, 2020
Accepted on: October 28, 2020
Submitted on: September 27, 2017
Keywords: Quantum Physics,Computer Science - Logic in Computer Science,F.1.1,F.3.2


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