We prove that the ZX-calculus is incomplete for quantum mechanics. We suggest
the addition of a new 'color-swap' rule, of which currently no analytical
formulation is known and which we suspect may be necessary, but not sufficient
to make the ZX-calculus complete.Comment: In Proceedings QPL 2014, arXiv:1412.810

de Witt, Christian Schröder

Zamdzhiev, Vladimir

Electronic Proceedings in Theoretical Computer Science

English

arXiv

We prove that the ZX-calculus is incomplete for quantum mechanics. We suggest the addition of a new 'color-swap' rule, of which currently no analytical formulation is known and which we suspect may be necessary, but not sufficient to make the ZX-calculus complete

Christian Schröder de Witt

Vladimir Zamdzhiev

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B. Coecke, I. Hasuo & P. Panangaden (Eds.):Quantum Physics and Logic 2014 (QPL 2014).EPTCS 172, 2014, pp. 285–292, doi:10.4204/EPTCS.172.20© C. Schröder de Witt & V. ZamdzhievThis work is licensed under theCreative Commons Attribution License.The ZX-calculus is incomplete for quantum mechanicsChristian Schröder de WittScherenbergstr. 22, 10439 Berlincaschroeder@outlook.comVladimir ZamdzhievDepartment of Computer ScienceUniveristy of OxfordOxford, United Kingdomvladimir.zamdzhiev@cs.ox.ac.ukWe prove that the ZX-calculus is incomplete for quantum mechanics. We suggest the addition of anew ’color-swap’ rule, of which currently no analytical formulation is known and which we suspectmay be necessary, but not sufficient to make the ZX-calculus complete.1 IntroductionCoecke and Abramsky pioneered the field of categorical quantum mechanics in [1]. Later, from thisstudy, an intuitive graphical calculus (dubbed the ZX-calculus) was developed by Coecke and Dun-can [4][3], which can be used as an alternative to Dirac notation in a wide number of applications[5][11][6][8].Backens recently proved that the ZX-calculus is complete for an important subset of quantum mechanics,namely stabilizer quantum mechanics, i.e. that for stabilizer quantum mechanics, any equation that canbe shown to hold in the Dirac formalism can also be shown to hold within the ZX-calculus[2]. For herproof, she relied on operations on a special class of quantum states, namely graph states. This paperadresses the question of whether the ZX-calculus is complete for the whole of quantum mechanics, andthe answer is found to be negative.1.1 Syntax and Semantics of the ZX-calculusThe syntax and semantics of the ZX-calculus are presented below. The semantics are given in Hilbertspace. We begin with atomic diagrams. The inputs to the diagrams are located at the bottom and theoutputs are located at the top.uwwwv}~=(1 00 1)= Iuwwwv}~=1 0 0 00 0 1 00 1 0 00 0 0 1= σ286 The ZX-calculus is incomplete for quantum mechanicss {= 〈00|+ 〈11|s {= |00〉+ |11〉uwwwv H}~=1√2(1 11 −1)= Huwwwvα}~=|0m〉 7→ |0n〉|1m〉 7→ eiα |1n〉rest 7→ 0uwwwvα}~=|+m〉 7→ |+n〉|−m〉 7→ eiα |−n〉rest 7→ 0where in the last two diagrams m is the number of inputs and n is the number of outputs. The labels ofthe red and green dots form the circle group under addition. So, admissible values are α ∈ [0,2π). Wealso make the convention that we will not write a label for the points when α = 0.We can create compound diagrams from smaller diagrams in two ways - either placing two diagramsnext to each horizontally, or plugging the outputs of one diagram to the inputs of another. IfuwwwvΨ1}~= D1 anduwwwvΨ2}~= D2thenuwwwvΨ1 Ψ2}~= D1⊗D2and uwwwwwwwwwwwwwwvΨ1Ψ2}~= D1 ◦D2In the latter diagram, the number of outputs of Ψ2 has to be the same as the number of inputs of Ψ1.By following the above rules we can represent any pure state map on qubits as a diagram in the ZX-calculus [4].C. Schröder de Witt & V. Zamdzhiev 2871.2 ZX Equational RulesThe rules of the ZX-calculus are given by:"Only the topology matters" (T)αβ...............= α +β... ...... ...αβ...............= α +β... ...... ...(S1)= == = (S2)= == = (B1) = (B2)π...=π π...π...=π π...(K1)πα=−αππα=−απ(K2)α......= α......H H H HHHHH(C) H =−π2−π2−π2 (EU)= (D1) = (D2)288 The ZX-calculus is incomplete for quantum mechanics1.3 Completeness, Soundness and UniversalityThe original rules of the ZX-calculus, as put forward by Coecke and Duncan [4] did not contain theEuler decomposition of the Hadamard gate, EU. Subsequently, Duncan and Perdrix proved [7] that therule EU was not derivable from within the original ZX-calculus. The original ZX-calculus was thereforeincomplete. Informally, incompleteness signifies that there are equations that can be proven to hold inDirac-von Neumann notation that cannot be proven in the ZX-calculus. This would reduce the power ofthe graphical calculus and possibly limit its applications in automated reasoning.Backens showed in [2] that the current ZX-calculus, which is simply the original ZX-calculus extendedby EU, is complete for an important fragment of quantum mechanics, namely stabilizer quantum me-chanics (SQM). Her proof relies on the fact that each SQM state is, under local Clifford operations [9],equivalent to a special entangled state, namely a graph state [10]. This allows one to abstract away frommatrix representations and instead decide equivalence between different SQM states by performing localcomplementations, a class of graph manipulations, between graph states. In this way, Backens showedthat SQM states may be represented by so-called rGS-LC diagrams, which are only equivalent iff theyare graphically identical. In this way, equivalence can be decided in the ZX-calculus.However, ideally, one would wish the ZX-calculus to be as physically expressive as the complete Dirac-von Neumann formalism. To this goal, three important properties of the calculus need to be established:universality, soundness and completeness.The ZX-calculus is sound [4]. That is, if ZX ` D1 = D2 then JD1K = eiφ JD2K. In other words, if twodiagrams are equal under the axioms of the ZX-calculus, then their Hilbert space interpretations are equalup to a global phase.Secondly, the ZX-calculus is universal, meaning that it can express any quantum state and gate. This iseasily proven by showing that the ZX-calculus can express any of the set of universal quantum gates [4].Finally, completeness is the converse of soundness. That is, if JD1K = JD2K then ZX ` D1 = D2. In thenext section, we will show that the ZX-calculus does not have this property.2 IncompletenessBefore we present the proof, let us recall a standard result from quantum mechanics, namely the Eulerdecomposition of single-qubit gates [12]. By this result, any single-qubit unitary gate can be expressed(up to a global phase) through just three consecutive rotations in appropriate bases. For the ZX-calculus,this means that there always exist real angles αi,βi,γi,φi such that for any ZX diagram D with one inputand one output, we have:uwwwwwvD}~= eiφ1uwwwwwvα1β1γ1}~= eiφ2uwwwwwvα2β2γ2}~C. Schröder de Witt & V. Zamdzhiev 289We prove the incompleteness of the ZX-calculus by using a similar argument to that of Duncan andPerdrix in [7], where they show that the Euler decomposition of the Hadamard gate is not derivablewithin the ZX-calculus.In particular, we can define alternative models for the ZX-calculus by setting:uwwwwwwwvα......}~k:=uwwwwwwwvkα......}~;uwwwwwwwvα......}~k:=uwwwwwwwvkα......}~J·Kk := J·K , otherwisewhere k ∈ Z and J·K is the standard interpretation functor for ZX diagrams in Hilbert space. In otherwords, we multiply all angles in our diagrams by an integer k and consider the corresponding interpreta-tion.These models are sound when k = 4p+1 for p ∈ Z. This can be easily verified by checking that each ofthe equational rules remains valid under this interpretation.Consider the following two ZX diagrams:D1 :=π3π32π3π3π3and D2 :=αβαφπ290 The ZX-calculus is incomplete for quantum mechanicswhereα :=−arccos(52√13)≈ 0.2561πβ :=−2arcsin(√34)≈−0.2851πφ := arcsin(√34)−α ≈ 0.3987πThen, we haveJD1K = JD2KThe two scalar factors are introduced so that the equality is exact, otherwise it would be true up to theglobal phase eiφ .Let us assume for contradiction that D1 and D2 are equal under the axioms of the ZX-calculus, i.e.ZX `D1 =D2. Since J·K−3 provides a sound model of the calculus, it must also be the case that JD1K−3 =λ JD2K−3, for some λ ∈ C.However, it is easy to check that this is not true (JD1K−3 is equal to a scalar times the identity, whereasJD2K−3 isn’t). Therefore the two diagrams D1 and D2 are not equal under the axioms of the ZX-calculus,even though they have equal Hilbert space interpretations. This means the ZX-calculus is incomplete.3 Conclusion and Future WorkThe primary contribution of this work is showing that the ZX-calculus is incomplete for quantum me-chanics. A natural question to ask is what additional rules can be added to the calculus in order to increaseits proving power. The proof that we have used doesn’t use any special properties of the presented dia-grams – it will be straightforward to apply the same proof to another pair of single-qubit unitary gateswhere one of them is the Euler decomposition of the other. To eliminate this class of counter-examples,we believe that a "color-swap" rule of the form:α1β1γ1=α2β2γ2C. Schröder de Witt & V. Zamdzhiev 291might be needed. This would require identifying functions f1, f2, f3, s.t. the above rule is valid andα1 = f1(α2,β2,γ2)β1 = f2(α2,β2,γ2)γ1 = f3(α2,β2,γ2)In other words, an analytic solution for converting from ZXZ to XZX Euler decompositions of single-qubit unitary gates is required.Whether the addition of such a ’color-swap’ would be sufficient to render the ZX-calculus complete iscurrently unknown. Some simple candidates for further possible non-derivable equalities are presentedhere:γβ1 βN...β2 =gb1 bN...b2β1 β2α2γ2γ1α1= b2b1c1a1 a2c2We suggest to conduct numerical investigations into the question of whether such non-derivable equali-ties between complex ZX-calculus diagrams exist.AcknowledgementsWe would like to thank Aleks Kissinger, Miriam Backens and Bob Coecke for constructive discussions.One of the authors also wishes to gratefully acknowledge support from the EPSRC and the ScatcherdEuropean Scholarship.References[1] S. Abramsky & B. Coecke (2004): A categorical semantics of quantum protocols. In: Logicin Computer Science, 2004. Proceedings of the 19th Annual IEEE Symposium on, pp. 415–425,doi:10.1109/LICS.2004.1319636.[2] Miriam Backens (2012): The ZX-calculus is complete for stabilizer quantum mechanics. In: Proceedings 9thInternational Workshop on Quantum Physics and Logic, Brussels, Belguim October 10-12, 2012. Availableat http://arxiv.org/abs/1307.7025.292 The ZX-calculus is incomplete for quantum mechanics[3] Bob Coecke & Ross Duncan (2008): Interacting Quantum Observables. In Luca Aceto, Ivan Damgård,LeslieAnn Goldberg, MagnúsM. Halldórsson, Anna Ingólfsdóttir & Igor Walukiewicz, editors: Automata,Languages and Programming, Lecture Notes in Computer Science 5126, Springer Berlin Heidelberg, pp.298–310, doi:10.1007/978-3-540-70583-3_25.[4] Bob Coecke & Ross Duncan (2011): Interacting quantum observables: categorical algebra and diagram-matics. New Journal of Physics 13(4), p. 043016, doi:10.1088/1367-2630/13/4/043016.[5] Bob Coecke, Ross Duncan, Aleks Kissinger & Quanlong Wang (2012): Strong Complementarity and Non-locality in Categorical Quantum Mechanics. In: Proceedings of the 2012 27th Annual IEEE/ACM Sym-posium on Logic in Computer Science, LICS ’12, IEEE Computer Society, Washington, DC, USA, pp.245–254, doi:10.1109/LICS.2012.35.[6] Ross Duncan & Maxime Lucas (2013): Verifying the Steane code with Quantomatic. In: Proceedings 10thInternational Workshop on Quantum Physics and Logic, Barcelona, Spain, July 17-19, 2013. Available athttp://arxiv.org/abs/1306.4532.[7] Ross Duncan & Simon Perdrix (2009): Graph States and the Necessity of Euler Decomposition. In KlausAmbos-Spies, Benedikt Löwe & Wolfgang Merkle, editors: Mathematical Theory and Computational Prac-tice, Lecture Notes in Computer Science 5635, Springer Berlin Heidelberg, pp. 167–177, doi:10.1007/978-3-642-03073-4_18.[8] Ross Duncan & Simon Perdrix (2010): Rewriting Measurement-Based Quantum Computations with Gen-eralised Flow. In Samson Abramsky, Cyril Gavoille, Claude Kirchner, Friedhelm Meyer auf der Heide &PaulG. Spirakis, editors: Automata, Languages and Programming, Lecture Notes in Computer Science 6199,Springer Berlin Heidelberg, pp. 285–296, doi:10.1007/978-3-642-14162-1_24.[9] D. Gottesman (1999): The Heisenberg Representation of Quantum Computers. Group22: Proceedings of theXXII International Colloquium on Group Theoretical Methods in Physics (eds. S. P. Corney, R. Delbourgo,and P. D. Jarvis), pp. 32–43.[10] M. Hein, J. Eisert & H. J. Briegel (2004): Multiparty entanglement in graph states. Physical Review A69(6), p. 062311, doi:10.1103/PhysRevA.69.062311. Available at http://link.aps.org/doi/10.1103/PhysRevA.69.062311.[11] Anne Hillebrand (2011): Superdense Coding with GHZ and Quantum Key Distribution with W in the ZX-calculus. In Bart Jacobs, Peter Selinger & Bas Spitters, editors: Proceedings 8th International Workshopon Quantum Physics and Logic, Nijmegen, Netherlands, October 27-29, 2011, Electronic Proceedings inTheoretical Computer Science 95, Open Publishing Association, pp. 103–121, doi:10.4204/EPTCS.95.10.[12] Michael A Nielsen & Isaac L Chuang (2010): Quantum computation and quantum information. Cambridgeuniversity press, doi:10.1017/CBO9780511976667.

The ZX-calculus is incomplete for quantum mechanics

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