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A Simplified Stabilizer ZX-calculus

2017
*
Electronic Proceedings in Theoretical Computer Science
*

The

doi:10.4204/eptcs.236.1
fatcat:esbg4zsi4vcdbgy5pqc3nncioy
*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 . ...##
###
Towards a Minimal Stabilizer ZX-calculus
[article]

2020
*
arXiv
*
pre-print

The

arXiv:1709.08903v6
fatcat:q2gm674snval7bkohtlxh2qayi
*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. ...##
###
Towards a Minimal Stabilizer ZX-calculus

2017
*
Logical Methods in Computer Science
*

The

doi:10.23638/lmcs-16(4:19)2020
fatcat:qg62mqnybrhpjl3yaldlsqgsbe
*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. ...##
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Qutrit ZX-calculus is Complete for Stabilizer Quantum Mechanics

2018
*
Electronic Proceedings in Theoretical Computer Science
*

In this paper, we show that

doi:10.4204/eptcs.266.3
fatcat:iyqy25la55hf3cj4k6r4q6uj6u
*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. ...##
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Pivoting makes the ZX-calculus complete for real stabilizers

2014
*
Electronic Proceedings in Theoretical Computer Science
*

We derive an angle-free version of the

doi:10.4204/eptcs.171.5
fatcat:jxubdy4nnbbzhh4pimdnav6bda
*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. ...##
###
The ZX-calculus is complete for stabilizer quantum mechanics

2014
*
New Journal of Physics
*

The

doi:10.1088/1367-2630/16/9/093021
fatcat:adnoquyv6vbppkdj4bsj5vlane
*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. ...##
###
Making the stabilizer ZX-calculus complete for scalars

2015
*
Electronic Proceedings in Theoretical Computer Science
*

The

doi:10.4204/eptcs.195.2
fatcat:vymqlbkcrjglxlke7im44cloz4
*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. ...##
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Simplification Strategies for the Qutrit ZX-Calculus
[article]

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 ...

##
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Completeness and the ZX-calculus
[article]

2016
*
arXiv
*
pre-print

I develop

arXiv:1602.08954v1
fatcat:tgiwn32u5bggpao3wfzen2didm
*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. ...##
###
A Complete Graphical Calculus for Spekkens' Toy Bit Theory

2015
*
Foundations of physics
*

Our language is inspired by

doi:10.1007/s10701-015-9957-7
fatcat:vtnfcnci6bdkdniwdlznmx6ymu
*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] . ...##
###
Circuit Relations for Real Stabilizers: Towards TOF+H
[article]

2019
*
arXiv
*
pre-print

The real

arXiv:1904.10614v2
fatcat:5osaxj3wnvcptbw6w34rdmflqq
*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 ...##
###
Complete set of circuit equations for stabilizer quantum mechanics

2014
*
Physical Review A. Atomic, Molecular, and Optical Physics
*

We find

doi:10.1103/physreva.90.012109
fatcat:zblaxov435h2hhkeiokxgpufum
*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] . ...##
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Kindergarden quantum mechanics graduates (...or how I learned to stop gluing LEGO together and love the ZX-calculus)
[article]

2021
*
arXiv
*
pre-print

We will focus mainly on what has become the Swiss army knife of the pictorial formalism: the

arXiv:2102.10984v1
fatcat:fh6z344vxbfabdcspdlbcaeh7i
*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] . ...##
###
Verifying the Smallest Interesting Colour Code with Quantomatic

2018
*
Electronic Proceedings in Theoretical Computer Science
*

In this paper we present

doi:10.4204/eptcs.266.10
fatcat:nkgbju4bbzf5teckp32ndeikfe
*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. ...##
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The ZX calculus is a language for surface code lattice surgery
[article]

2019
*
arXiv
*
pre-print

We give

arXiv:1704.08670v3
fatcat:bw7xqnfc6rattpcuqkf5dw55ga
*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. ...
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