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A Simplified Stabilizer ZX-calculus

Miriam Backens, Simon Perdrix, Quanlong Wang
2017 Electronic Proceedings in Theoretical Computer Science  
The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics.The language is sound and complete: a stabilizer ZX-diagram can be transformed into another one if and  ...  We show that the stabilizer ZX-calculus can be simplified, removing unnecessary equations while keeping only the essential axioms which potentially capture fundamental structures of quantum mechanics.  ...  Simplified Rules In this section, we give a new system of rules for the stabilizer ZX-calculus, shown in Figure 2 .  ... 
doi:10.4204/eptcs.236.1 fatcat:esbg4zsi4vcdbgy5pqc3nncioy

Towards a Minimal Stabilizer ZX-calculus [article]

Miriam Backens, Simon Perdrix, Quanlong Wang
2020 arXiv   pre-print
The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics.  ...  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  ...  A Simplified Stabilizer zx-calculus The zx-calculus is a graphical language based on categorical quantum mechanics.  ... 
arXiv:1709.08903v6 fatcat:q2gm674snval7bkohtlxh2qayi

Towards a Minimal Stabilizer ZX-calculus

Miriam Backens, Simon Perdrix, Quanlong Wang
2017 Logical Methods in Computer Science  
The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics.  ...  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  ...  A Simplified Stabilizer zx-calculus The zx-calculus is a graphical language based on categorical quantum mechanics.  ... 
doi:10.23638/lmcs-16(4:19)2020 fatcat:qg62mqnybrhpjl3yaldlsqgsbe

Qutrit ZX-calculus is Complete for Stabilizer Quantum Mechanics

Quanlong Wang
2018 Electronic Proceedings in Theoretical Computer Science  
In this paper, we show that a qutrit version of ZX-calculus, with rules significantly different from that of the qubit version, is complete for pure qutrit stabilizer quantum mechanics, where state preparations  ...  In contrast to the qubit case, the situation here is more complicated due to the richer structure of this qutrit ZX-calculus.  ...  : Theorem 5.6 The ZX-calculus is complete for qutrit stabilizer quantum mechanics.  ... 
doi:10.4204/eptcs.266.3 fatcat:iyqy25la55hf3cj4k6r4q6uj6u

Pivoting makes the ZX-calculus complete for real stabilizers

Ross Duncan, Simon Perdrix
2014 Electronic Proceedings in Theoretical Computer Science  
We derive an angle-free version of the ZX-calculus and show that it is complete for real stabilizer quantum mechanics.  ...  Therefore the ZX-calculus augmented with pivoting is strictly weaker than the calculus augmented with the Euler decomposition of the Hadamard gate.  ...  Pivoting makes the ZX-calculus complete for real stabilizers Proof.  ... 
doi:10.4204/eptcs.171.5 fatcat:jxubdy4nnbbzhh4pimdnav6bda

The ZX-calculus is complete for stabilizer quantum mechanics

Miriam Backens
2014 New Journal of Physics  
The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes.  ...  Here, we show that the ZX-calculus is complete for pure qubit stabilizer quantum mechanics, meaning any equality that can be derived using matrices can also be derived pictorially.  ...  Also, the ZX-calculus is indeed universal for stabilizer QM.  ... 
doi:10.1088/1367-2630/16/9/093021 fatcat:adnoquyv6vbppkdj4bsj5vlane

Making the stabilizer ZX-calculus complete for scalars

Miriam Backens
2015 Electronic Proceedings in Theoretical Computer Science  
The ZX-calculus is a graphical language for quantum processes with built-in rewrite rules.  ...  The ZX-calculus is known to be complete for scalar-free pure qubit stabilizer quantum mechanics, meaning any equality between two pure stabilizer operators that is true up to a non-zero scalar factor can  ...  Elements of the stabilizer ZX-calculus and the interpretation of diagrams A stabilizer ZX-calculus diagram consists of different types of nodes connected by edges.  ... 
doi:10.4204/eptcs.195.2 fatcat:vymqlbkcrjglxlke7im44cloz4

Simplification Strategies for the Qutrit ZX-Calculus [article]

Alex Townsend-Teague, Konstantinos Meichanetzidis
2022 arXiv   pre-print
A key notion is the stabiliser fragment of the ZX-calculus, a subfamily of ZX-diagrams for which rewriting can be done efficiently in terms of derived simplifying rewrites.  ...  The ZX-calculus is a graphical language for suitably represented tensor networks, called ZX-diagrams. Calculations are performed by transforming ZX-diagrams with rewrite rules.  ...  KM acknowledges financial support from the Royal Commission for the Exhibition of 1851 through a postdoctoral research fellowship, while ATT thanks the Einstein Foundation (Einstein Research Unit on Quantum  ... 
arXiv:2103.06914v2 fatcat:uscwwite35al5oomuar6alou3e

Completeness and the ZX-calculus [article]

Miriam Backens
2016 arXiv   pre-print
I develop a graphical calculus similar to the ZX-calculus that fully describes Spekkens' toy theory, and show that it is complete.  ...  In this thesis, I consider the ZX-calculus, a graphical language for pure state qubit quantum mechanics.  ...  Map-state duality in the zx-calculus Ignoring scalars in the stabilizer completeness proof simplifies matters significantly.  ... 
arXiv:1602.08954v1 fatcat:tgiwn32u5bggpao3wfzen2didm

A Complete Graphical Calculus for Spekkens' Toy Bit Theory

Miriam Backens, Ali Nabi Duman
2015 Foundations of physics  
Our language is inspired by a similar graphical language for quantum mechanics called the ZX-calculus.  ...  The toy theory for the simplest possible underlying system closely resembles stabilizer quantum mechanics, a fragment of quantum theory which is efficiently classically simulable but also non-local.  ...  The algorithm is adapted from a similar one for the stabilizer ZX-calculus [4] .  ... 
doi:10.1007/s10701-015-9957-7 fatcat:vtnfcnci6bdkdniwdlznmx6ymu

Circuit Relations for Real Stabilizers: Towards TOF+H [article]

Cole Comfort
2019 arXiv   pre-print
The real stabilizer fragment of quantum mechanics was shown to have a complete axiomatization in terms of the angle-free fragment of the ZX-calculus.  ...  We then construct translations to and from the angle-free fragment of the ZX-calculus, showing that they are inverses.  ...  However, we are interested in a simple fragment of the ZX-calculus, namely the angle-free calculus for real stabilizer circuits, ZX π , described in [15] (slightly modified to account for scalars): Definition  ... 
arXiv:1904.10614v2 fatcat:5osaxj3wnvcptbw6w34rdmflqq

Complete set of circuit equations for stabilizer quantum mechanics

André Ranchin, Bob Coecke
2014 Physical Review A. Atomic, Molecular, and Optical Physics  
We find a sufficient set of equations between quantum circuits from which we can derive any other equation between stabilizer quantum circuits.  ...  To establish this result, we rely upon existing work on the completeness of the graphical ZX language for quantum processes.  ...  The ZX network simplifies numerous quantum calculations. It allows us to study a number of fundamental aspects of quantum theory from a high-level mathematical point of view [25] [26] [27] .  ... 
doi:10.1103/physreva.90.012109 fatcat:zblaxov435h2hhkeiokxgpufum

Kindergarden quantum mechanics graduates (...or how I learned to stop gluing LEGO together and love the ZX-calculus) [article]

Bob Coecke, Dominic Horsman, Aleks Kissinger, Quanlong Wang
2021 arXiv   pre-print
We will focus mainly on what has become the Swiss army knife of the pictorial formalism: the ZX-calculus.  ...  First we look at some of the ideas behind the ZX-calculus, comparing and contrasting it with the usual quantum circuit formalism.  ...  The completeness of the real stabilizer ZX-calculus then followed in [52] .  ... 
arXiv:2102.10984v1 fatcat:fh6z344vxbfabdcspdlbcaeh7i

Verifying the Smallest Interesting Colour Code with Quantomatic

Liam Garvie, Ross Duncan
2018 Electronic Proceedings in Theoretical Computer Science  
In this paper we present a Quantomatic case study, verifying the basic properties of the Smallest Interesting Colour Code error detecting code.  ...  The terms of the zx-calculus naturally form a †-symmetric monoidal category [13] , or more precisely a †-PROP [25, 14] , which we call ZX.  ...  Therefore it is a particularly interesting test case for the use of zx-calculus and automated reasoning.  ... 
doi:10.4204/eptcs.266.10 fatcat:nkgbju4bbzf5teckp32ndeikfe

The ZX calculus is a language for surface code lattice surgery [article]

Niel de Beaudrap, Dominic Horsman
2019 arXiv   pre-print
We give a first taste of the power of the calculus as a language for lattice surgery by considering two operations (T gates and producing a CNOT ) and show how ZX diagram re-write rules give lattice surgery  ...  In this paper we demonstrate that the operations of the ZX calculus --- a form of quantum diagrammatic reasoning based on bialgebras --- match exactly the operations of lattice surgery.  ...  This also demonstrates a simple example of the usefulness of the ZX calculus to simplify quantum information processing procedures, as follows.  ... 
arXiv:1704.08670v3 fatcat:bw7xqnfc6rattpcuqkf5dw55ga
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