A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond

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1 A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond Renaud Vilmart Université de Lorraine, CNRS, Inria, LORIA, F 5000 Nancy, France renaud.vilmart@loria.fr We consider a ZX-calculus augmented with triangle nodes which is well-suited to reason on the so-called Toffoli-Hadamard fragment of quantum mechanics. We precisely show the form of the matrices it represents, and we provide an axiomatisation which makes the language complete for the Toffoli-Hadamard quantum mechanics. We extend the language with arbitrary angles and show that any true equation involving linear diagrams which constant angles are multiple of are derivable. We show that a single axiom is then necessary and sufficient to make the language equivalent to the ZX-calculus which is known to be complete for Clifford+T quantum mechanics. As a by-product, it leads to a new and simple complete axiomatisation for Clifford+T quantum mechanics. 1 Introduction The ZX-Calculus is a powerful graphical language for quantum computing [9]. It has been introduced in 008 by Coecke and Duncan [7], and has several applications in quantum information processing (e.g. MBQC [1], quantum codes [6, 5, 11, 17])., The language manipulates diagrams, generated by roughly three kinds of vertices: and ; and which represent quantum evolutions thanks to the standard interpretation. It comes with a set of allowed transformations that preserve the represented process. These rules can be implemented in the interactive proof assistant Quantomatic [1, ]. The property that they preserve the semantics is called soundness, and is easily verifiable. Its converse, completeness, is obtained when any two diagrams that represent the same process can be transformed into one-another with only the rules of the ZX-Calculus. The first completeness result was for a restriction of the language called Clifford []. This result was adapted to a smaller restriction of the language, the so-called real stabilizer [13]. Problem was, these two restrictions are not universal: their diagrams cannot approximate all arbitrary evolution, and they are efficiently simulable on a classical computer. The easiest restriction of the language with approximate universality is the Clifford+T fragment. A first step was made when completeness was discovered for the 1-qubit operations of the restriction [3]. A complete axiomatisation for the many-qubits Clifford+T fragment was finally discovered in [18], and lead to a complete axiomatisation in the general case, first in [3, 16] where two new generators were added, and then in [19] with the original syntax. Thanks to its proximity to quantum circuitry, one of the potential uses of the ZX-Calculus is circuit simplification. The usual ZX-Calculus [7] can be seen as an generalisation of quantum circuits built with the gate set (CNot, H, R Z ()) where may take any value in R. Then, the real stabilizer ZX-Calculus corresponds to the set of gates when the angles are multiples of, the Clifford restriction to when angles are multiples of, and the Clifford+T restriction to when angles are multiples of. While the Clifford+T fragment (where T R Z (/)) of quantum mechanics is preeminent in quantum information processing, Toffoli-Hadamard quantum mechanics is also well known and widely used. The Toffoli-Hadamard quantum mechanics is the real counter-part of the Clifford+T quantum mechanics: Giulio Chiribella, Peter Selinger (Eds.): 15th International Conference on Quantum Physics and Logic (QPL 018). EPTCS 87, 019, pp , doi:10.0/eptcs c R. Vilmart This work is licensed under the Creative Commons Attribution License.

2 31 A ZX-Calculus for Toffoli-Hadamard the matrices of the Toffoli+Hadamard fragment are exactly the real matrices of the Clifford+T fragment. Toffoli-Hadamard quantum mechanics is also known to be approximately universal for quantum computing [1, 7]. The Toffoli gate is a 3-qubit unitary evolution which is nothing but a controlled CNot: it maps x,y,z to x,y,x y z. The Toffoli gate can be considered a generator of the considered language [10], or can be decomposed into CNot, H, and T gates [5, 6] and hence can be represented in the -fragment of the ZX-calculus. However, as pointed out in [18, ], the Toffoli gate admits simpler representations using the triangle node: Toffoli : [18] or Toffoli : [] and the triangle itself can be recovered from the Toffoli gate: Toffoli The triangle itself was introduced in [18] as a notation for a diagram of the Clifford+T restriction, and was used as one of the additional generators in [3,, 16]. Since the CNOT gate is depicted in ZX as the composition of two of its generators, we propose to add the triangle node as a generator for a version of the ZX-Calculus devoted to represent specifically Toffoli, without having to decompose it as a Clifford+T diagram. We first present the diagrams of the ZX-Calculus augmented with the triangle node, and give an axiomatisation for the Toffoli-Hadamard quantum mechanics in Section. We prove it is universal, sound and complete in Section 3. In Section, we try and adapt a theorem from [19] which gives a completeness result on a broader restriction of the language. We finally give in Section 5 a simple axiomatisation for the Clifford+T fragment of the ZX-Calculus with triangles (that corresponds to the (Toffoli, H, R Z (/)) gate set in circuitry), and prove it is complete. ZX-Calculus: A ZX-Calculus with Triangles.1 Diagrams and Standard Interpretation A ZX-diagram D : k l with k inputs and l outputs is generated by: n Z () : n m m R (n,m) n X () : n m m R (n,m) H : 1 1 e : 0 0 I : 1 1 σ : ε : 0 η : 0 : 1 1 where n, m N and R. The generator e is the empty diagram.

3 R. Vilmart 315 and the two compositions: Spatial Composition: for any D 1 : a band D : c d, D 1 D : a+c b+d consists in placing D 1 and D side by side, D on the right of D 1. Sequential Composition: for any D 1 : a band D : b c, D D 1 : a cconsists in placing D 1 on the top of D, connecting the outputs of D 1 to the inputs of D. The standard interpretation of the ZX-diagrams associates to any diagram D : n m a linear map D :C n C m inductively defined as follows:. D 1 D : D 1 D D D 1 : D D 1 : ( 1 ) ( ) 1 0 : : 1 ( ) ( ) : : ( ) : For any R, : ( 1+e i), and for any n,m 0 such that n+m>0: { }} { : m e i n m n : ( where M 0 ( 1 ) and M k M M k 1 for any k N ). To simplify, the red and green nodes will be represented empty when holding a 0 angle:. Calculus : 0 and ZX represents the general ZX-Calculus with triangle, i.e. where the angles can take any value in R. In the following, we will focus on some restrictions of the language. We call the q -fragment of the ZX, the restriction of the ZX-Calculus with triangles, where all the angles are multiples of q, and we denote it ZX. q Two different diagrams may represent the same quantum evolution i.e. there exist two different diagrams D 1 and D such that D 1 D. This issue is addressed by giving a set of axioms: a set of permitted local diagram transformations that preserve the semantics. We give in Figure 1 a set of axioms for the general ZX-Calculus with triangles, and we define the ZX as the same set of axioms where the angles are restricted to{0, }. : 0

4 316 A ZX-Calculus for Toffoli-Hadamard β +β (S g ) (S r ) (Z) (T0) (BW) (HT) (TCX) (TW) Figure 1: Set of rules ZX for the ZX-Calculus with triangles. The right-hand side of is an empty diagram. (...) denote zero or more wires, while () denote one or more wires.,β R. ZX is obtained when restricting the angles, β {0, }. Remark. In the following, we will freely use the notation ZX q to denote either the set of diagrams in the q -fragment or the set of rules given for these specific diagrams. This set of axioms consists of the rules for the real stabiliser ZX-Calculus given in [13] except the so-called H-Loop:, augmented with 5 rules that include the node. However, this missing axiom can be retrieved (Lemma 7), so all the equations of the -fragment that do not involve -nodes are derivable. Additionally to these rules, the paradigm Only Topology Matters ensures we can be as lax as we want when manipulating diagrams, in the sense that what only matters is if two nodes are connected or not (and how many times). A distinction has to be made for where the orientation of the node matters. For instance: :

5 R. Vilmart 317 If we can transform a diagram D 1 into a diagram D using the rules of ZX or the previous paradigm, we write ZX D 1 D. We said earlier that the rules can be locally applied. This means that for any three diagrams, D 1,D, and D, if ZX D 1 D, then: ZX D 1 DD D ZX D D 1 D D ZX D 1 DD D ZX D D 1 D D An important property of the usual ZX-Calculus is that colour-swapping the diagrams (transforming green nodes in red nodes and vice-versa) preserves the equality. We made the triangle red, and we can define a green triangle as : so that this property is preserved. 3 Universality, Soundness, Completeness The Toffoli+Hadamard gate set in circuitry is (approximately) universal [1, 7]: any quantum evolution can be approximated with arbitrary precision by a circuit with this set of gates. It is expected that the result extends to the adequate fragment of the ZX-Calculus. It does: Theorem 1. For any matrix M 1 NM m, n(z), there exists a ZX -diagram D : n m such that M D. In other words, the fragment represents exactly the integer matrices of dimensions powers of two and 1 multiplied by an arbitrary power of. One inclusion is easy to notice: the diagrams represent matrices in 1 NM m,n(z). Indeed, the standard interpretation of all the generators is an integer matrix, multiplied by 1 in the case of the Hadamard gate, and the two compositions preserve this structure. The second inclusion, the fact that any matrix can be represented as a ZX -diagram is less trivial, and will be shown in the following. The language is not only universal, it is sound: the transformations do preserve the represented quantum evolution i.e. ZX D 1 D D 1 D. This is a routine check. The converse, however is harder to prove. Completeness is obtained when two diagrams can be transformed into one another whenever they represent the same evolution. This is our main theorem: Theorem. The set of rules in Figure 1 makes the -fragment of the ZX-Calculus complete. For any diagrams D 1 and D of the fragment, D 1 D ZX D 1 D The proof uses the now usual method of a back and forth interpretation from the ZX-Calculus to the ZW-Calculus, another graphical language suited for quantum processes. 3.1 ZW-Calculus The ZW-Calculus, introduced in 010 as the GHZ/W-calculus by Coecke and Kissinger [8], was completed by Hadzihasanovic in 015 [1]. This first complete language is only universal for matrices over Z. Even though it has been extended to represent any complex matrix by the last author [15], the first version, closer to the expressive power of the -fragment of the ZX-Calculus, is the one we will use in the following of the paper. The ZW-diagrams are generated by: { } T e,,,,,,,,,

6 318 A ZX-Calculus for Toffoli-Hadamard The generators are then composed using the two same sequential and spatial compositions. Again, the diagrams represent quantum evolutions, so the language comes with a standard interpretation, inductively defined as: D 1 D : D 1 D : : ( ) 0 1 : 1 0. D D 1 : D D : : ( 1 ) ( ) 1 0 : 0 1 t 1 1 : 0 0 : ( ) ( ) : : The language also comes with its own set of axioms, given in Figure. Here again, the paradigm Only Topology Matters is respected (except for where the ordering of inputs and outputs is not to be messed with). Using this, we define 1-legged black and white dots as 3-legged dots with a loop: : and :, and we can build any -legged or 3-legged white and black dots. For instance: : : The language is of course sound, and as previously said, complete. 3. ZW 1 -Calculus 1 The -fragment of the ZX-Calculus represents elements of NM m,n(z), while the ZW-Calculus represents elements in M m,n(z). If the two languages had the same expressive power, it would greatly simplify the interpretation from one another, by avoiding complicated encodings such as in [18]. 1 What is missing from the ZW-Calculus is merely a global scalar, which can be any power of. Intuitively, adding a generator, which only represents the scalar 1 should be enough, for tensoring it to a diagram D should yield 1 D, and the different powers should be obtained by putting side by side enough occurrences of the new scalar. This should deal with the expressive power of the language. It remains then to bind the new generator to the others in the axiomatisation. Definition 3. We define the ZW 1 -Calculus as the extension of the ZW-Calculus such as: T 1 T e { } ZW 1 ZW { iv, z } The standard interpretation of a diagram D : n m is now a matrix D 1 NM m,n(z) that is an integer matrix times a power of 1, and is given by the standard interpretation of the ZW-Calculus extended with : 1.

7 R. Vilmart 319 0a 0b 0b 1a 1b 0c 0d 0d 1c 1d a b 3a 3b 5a 5b 5c 5d 6a 6b 6c X 7b 7a R R3 Figure : Set of rules for the ZW-Calculus. By building on top of the already complete ZW-Calculus, some results are preserved or easily extendable. For instance, knowing that the ZW-Calculus is sound, showing that the rules iv and z are sound is enough to conclude that the ZW 1 -Calculus is sound. For the completeness: Proposition. The ZW 1 is sound and complete: For two diagrams D 1,D of the ZW 1 -calculus, D 1 D ZW 1 D 1 D We now have a complete version of the ZW-Calculus, with the same yet to be shown expressive power as the -fragment of the ZX-Calculus. 3.3 From ZW 1 to ZX, and Expressive Power of the Latter We define here an interpretation [.] X that transforms any diagram of the ZW 1 -Calculus into a ZXdiagram:

8 30 A ZX-Calculus for Toffoli-Hadamard [.] X D 1 D [D 1 ] X [D ] X D 1 D [D 1 ] X [D ] X This interpretation preserves the semantics of the languages: Proposition 5. Let D be a ZW 1 -diagram. Then [D] X D. This is a routine check. The existence of this interpretation combined with the previous proposition is enough to prove Theorem 1: Theorem 1. Let M 1 NM m, n(z). There exist n N and M M m,n(z) 1 such that M n M. M is an integer matrix, so there exists D W a ZW-diagram such that D W M. We can then build the ZW 1 -diagram D W : D W ( ) n and notice that D W M. Finally, we build the ZX -diagram D X :[D W ] X and D X M by Proposition 5. Another very important result is that the ZX-Calculus proves the interpretation of all the rules of the ZW 1 -Calculus. More specifically: Proposition 6. For any ZW 1 -diagrams D 1 and D : The proof is in appendix. ZW 1 D 1 D ZX [D 1 ] X [D ] X 3. From ZX to ZW 1, and Completeness We now define an interpretation from the -fragment of the ZX-Calculus to the ZW 1 -Calculus. [.] W

9 R. Vilmart 31 p ( ) p {0,}, [ ] m [ ] [ ] n W W W D 1 D [D 1 ] W [D ] W D 1 D [D 1 ] W [D ] W p Here, ( ) either represents the identity if p 0, or the 1 1 white dot if p 1. The complexity of the interpretation of the green dot (and subsequently of the red dot), is merely due to the fact that we chose to present the expanded version of the ZW-Calculus [1], where spiders are exploded as compositions of - and 3-legged dots. Again, the interpretation preserves the semantics: Proposition 7. Let D be a ZX -diagram. Then [D] W D. And again, this is a routine check. Now, when composing the two interpretation, we can easily retrieve the initial diagram: Proposition 8. For any diagram D of the -fragment of the ZX-Calculus, ZX [[D] W ] X D. We can now prove the completeness theorem: Theorem. Let D 1 and D be two diagrams of the ZX such that D 1 D. By Proposition 7, [D 1 ] W [D ] W. By completeness of the ZW 1 (Proposition ), ZW 1 [D 1 ] W [D ] W. By Proposition 6, ZX [[D 1 ] W ] X [[D ] W ] X. Finally, by Proposition 8, ZX D 1 D, hence proving the completeness of the ZX -fragment. Beyond Toffoli+Hadamard The set of axioms in Figure 1 features two rules and, with parameters. In ZX, they are limited to {0,}, but not in ZX. We want to know what we can prove without the limitation to,β {0,} in the set of rules. To do so, we are going to adapt a result that was developed in [19], and see what is missing to prove equalities that are valid for linear diagrams with constants in {0, }. The following of the section will give a sketch of the proof but will not dive into its details. For more, see [19]. We call a linear diagram, a diagram where some angles are treated as variables, and where the parameters of the green and red dots are affine combinations of the variables, where coefficients are in Z:

10 3 A ZX-Calculus for Toffoli-Hadamard Definition 9. A ZX-diagram is linear in 1,..., k with constants in C R, if it is generated by R (n,m) Z (E), R (n,m) X (E), H,, e, I, σ, ε, η, and the spatial and sequential compositions, where n,m N, and E is of the form i n i i + c, with n i Zand c C. For instance: + β +β + is a well-formed equation on linear diagrams, with constants in {0,}. It is even sound, for it is sound for any values of and β. The idea of the proof is to separate the different occurrences of the variables from the rest of the diagram using the rule, change their colour if they are in red nodes using, and change the sign in front of the variables if needs be using the equation: - (K) - Then, the scalars with - can be remove in favour of scalars with on the other side of the equation, for, thanks to and Lemma 9: ZX D - 1 D ZX D 1 D. First considering the case of a unique variable, it results that for any pair of diagrams D 1 () and D () with variable, there exists a pair of variable-free diagrams D 1 and D such that: D 1 () D () D 1 D ( ) We then have to find a diagram that is precisely a projector onto the span of r. We give the family (P r ) of diagrams, inductively defined as: P 1 : P : P r : P P P r 1 P P P P P P P P P One can check that P from which, thanks to [19], we can immediately deduce that { ( ) } r P r is a projector onto span / R. It results that:

11 R. Vilmart 33 Lemma 10. For any r 1 and any -free diagrams D 1,D : r n, R, D 1 D P r P r D 1 D P r P r Hence, R, D 1 () D () D 1 D. Since D 1, D and P r are ZX -diagrams, the equality of diagrams on the right hand-side is provable: P r P r P r P r D 1 D D 1 D All we need now to finish adapting the result on linear diagrams is the following property: P r This property is naturally obtained by induction in the general case provided it is true for the base cases r {1,}. When r 1, the result is obvious. When r, it deduces easily from: (P) In this case: D 1 P r P r D D () D 1 D so finally: R, D 1 () D 1 () D () The result extends naturally to the multi-variable-case. In conclusion, provided we have the two equations (K) and (P) as axioms, we can prove any sound equation on linear diagrams with constants in {0,}. With ZX K,P denoting the axiomatisation ZX augmented with the rules (K) and (P): Theorem 11. For any ZX-diagrams D 1 ( ) and D ( ) linear in 1,..., k with constants in {0,}: R k,d 1 ( )D ( ) R k, ZX K,P D 1 ( )D ( ) We give two corollaries of this theorem:

12 3 A ZX-Calculus for Toffoli-Hadamard Corollary 1. Corollary 13. ZX K,P + + ZX K,P -γ γ β β β β -γ γ 5 Axiomatisations for Clifford+T and the General ZX With the addition of the rules (K) and (P), and the result on linear diagrams (Theorem 11), we feel like we are not far from having a complete axiomatisation for the -fragment of the ZX-Calculus. However: Lemma 1 ([0]). ZX K,P (E) The language here is different than in [0], but the argument is easily adaptable. We give a larger axiomatisation denoted ZX K,P,E in Figure 3. This axiomatisation condenses the β +β (S g ) (S r ) (E) (K) - (BW) (HT) (TCX) (TW) (P) Figure 3: Set of rules ZX K,P,E. The right-hand side of (E) is an empty diagram. (...) denote zero or more wires, while () denote one or more wires.

13 R. Vilmart (A) β 1 β 0 0 e i 3 (1+e i (1+e i 1 (1+e i 0 ))) e iβ 1 (1+e iβ 0 ) Figure : Rule for the completeness of the general ZX-Calculus initial one ( ZX) and the rules (K) and (P), but also replaces by (E) and gets rid of (Z), though it is easy to show that the new axiomatisation proves the initial one: Proposition 15. ZX K,P,E ZX. As a consequence, this axiomatisation is complete for the -fragment of the ZX-Calculus. It turns out, it is also complete for the -fragment of the ZX-Calculus: Theorem 16. The set of rules ZX K,P,E makes the -fragment of the ZX-Calculus complete. For any D 1 and D diagrams of this fragment: D 1 D ZX K,P,E D 1 D In contrast to the axiomatisations for Clifford+T in [18, ], ZX K,P,E got rid of the axioms that were specific to particular angles outside {0, }, except for (E), which is necessary. We end up with an axiomatisation for an augmented ZX-Calculus that is complete for the -fragment, as in []. However, the method used to achieve completeness is different: in [], the authors start from an axiomatisation which is complete in general, and adapt it to the -fragment, whereas in this paper, we began with the easiest fragment of ZX, and gradually built upon it. As a result, the axiomatisation ZX K,P,E uses fewer axioms (1 against 30) and one fewer generator. The completeness result on linear diagrams with constants in {0,} (Theorem 11) can be seen as a generalisation of the completeness of ZX (Theorem ). Similarly, the completeness result for ZX K,P,E (Theorem 16) can be generalised to linear diagrams: Theorem 17. For any ZX-diagrams D 1 ( ) and D ( ) linear in 1,..., k with constants in Z: R k,d 1 ( )D ( ) R k, ZX K,P,E D 1 ( )D ( ) In [19], it has been shown that in the classical ZX-Calculus, completeness from Clifford+T to the general language could be obtained by adding only one rule. Adding precisely the same rule in ZX would produce the same result. However, we prefer to give a version of the axiom which is more relevant to the current language, in Figure. Notice that here again, the axiom is not angle-specific, and that it is non-linear. This is enough to have the completeness: Theorem 18. The set of rules ZX A,K,P,E makes the general ZX-Calculus complete. For any diagrams D 1 and D : D 1 D ZX A,K,P,E D 1 D

14 36 A ZX-Calculus for Toffoli-Hadamard Acknowledgement The author acknowledges support from the projects ANR-17-CE SoftQPro, ANR-17-CE VanQuTe, PIA-GDN/Quantex, and STIC-AmSud 16-STIC-05 FoQCoSS. All diagrams were written with the help of TikZit. The author would like to thank Emmanuel Jeandel and Simon Perdrix for fruitful discussions and suggestions. References [1] Dorit Aharonov (003): A Simple Proof that Toffoli and Hadamard are Quantum Universal. Arxiv:quant-ph/ [] Miriam Backens (01): The ZX-calculus is complete for stabilizer quantum mechanics. New Journal of Physics 16(9), p , doi: / /16/9/ [3] Miriam Backens (01): The ZX-calculus is complete for the single-qubit Clifford+T group. Electronic Proceedings in Theoretical Computer Science 17, pp , doi:10.0/eptcs [] Miriam Backens, Simon Perdrix & Quanlong Wang (016): A Simplified Stabilizer ZX-calculus. In: QPL 016, Electronic Proceedings in Theoretical Computer Science, pp. 1 0, doi:10.0/eptcs [5] Niel de Beaudrap & Dominic Horsman (017): The ZX calculus is a language for surface code lattice surgery. CoRR [6] Nicholas Chancellor, Aleks Kissinger, Joschka Roffe, Stefan Zohren & Dominic Horsman (016): Graphical Structures for Design and Verification of Quantum Error Correction. Arxiv: Last revised Jan [7] Bob Coecke & Ross Duncan (011): Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics 13(), p , doi: / /13// [8] Bob Coecke & Aleks Kissinger (010): The Compositional Structure of Multipartite Quantum Entanglement. In: Automata, Languages and Programming, Springer Berlin Heidelberg, pp , doi: / _5. [9] Bob Coecke & Aleks Kissinger (017): Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, doi: / [10] Cole Comfort & J. Robin B. Cockett (018): The category TOF. CoRR [11] Ross Duncan & Maxime Lucas (01): Verifying the Steane code with Quantomatic. Electronic Proceedings in Theoretical Computer Science 171, pp. 33 9, doi:10.0/eptcs [1] Ross Duncan & Simon Perdrix (010): Rewriting measurement-based quantum computations with generalised flow. Lecture Notes in Computer Science 6199, pp , doi: / _. [13] Ross Duncan & Simon Perdrix (013): Pivoting makes the ZX-calculus complete for real stabilizers. In: QPL 013, Electronic Proceedings in Theoretical Computer Science, pp. 50 6, doi:10.0/eptcs [1] Amar Hadzihasanovic (015): A Diagrammatic Axiomatisation for Qubit Entanglement. In: th Annual ACM/IEEE Symposium on Logic in Computer Science, pp , doi: /lics [15] Amar Hadzihasanovic (017): The algebra of entanglement and the geometry of composition. Ph.D. thesis, University of Oxford. Arxiv: [16] Amar Hadzihasanovic, Kang Feng Ng & Quanlong Wang (018): Two Complete Axiomatisations of Purestate Qubit Quantum Computing. In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 18, ACM, New York, NY, USA, pp , doi:10.115/

15 R. Vilmart 37 [17] Clare Horsman (011): Quantum picturalism for topological cluster-state computing. New Journal of Physics 13(9), p , doi: / /13/9/ [18] Emmanuel Jeandel, Simon Perdrix & Renaud Vilmart (018): A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics. In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 18, ACM, New York, NY, USA, pp , doi:10.115/ [19] Emmanuel Jeandel, Simon Perdrix & Renaud Vilmart (018): Diagrammatic Reasoning Beyond Clifford+T Quantum Mechanics. In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 18, ACM, New York, NY, USA, pp , doi:10.115/ [0] Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart & Quanlong Wang (017): ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics. In Kim G. Larsen, Hans L. Bodlaender & Jean-Francois Raskin, editors: nd International Symposium on Mathematical Foundations of Computer Science (MFCS 017), Leibniz International Proceedings in Informatics (LIPIcs) 83, Schloss Dagstuhl Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, pp. 11:1 11:13, doi:10.30/lipics. MFCS [1] A. Kissinger, L. Dixon, R. Duncan, B. Frot, A. Merry, D. Quick, M. Soloviev & V. Zamdzhiev (011): Quantomatic. Available athttp://quantomatic.github.io/. [] Aleks Kissinger & Vladimir Zamdzhiev (015): Quantomatic: A Proof Assistant for Diagrammatic Reasoning. In Amy P. Felty & Aart Middeldorp, editors: Automated Deduction - CADE-5, Springer International Publishing, Cham, pp , doi: / _. [3] Kang Feng Ng & Quanlong Wang (017): A universal completion of the ZX-calculus. Arxiv: [] Kang Feng Ng & Quanlong Wang (018): Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics. Arxiv: [5] Michael A. Nielsen & Isaac L. Chuang (010): Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, doi: /cbo [6] Peter Selinger (013): Quantum circuits of T-depth one. Phys. Rev. A 87, p. 030, doi: / PhysRevA [7] Yaoyun Shi (003): Both Toffoli and controlled-not need little help to do universal quantum computing. Quantum Information & Computation 3(1), pp Available at cfm?id A Appendix A.1 Lemmas Lemma 19. Lemma. Lemma 5. Lemma 0. Lemma 3. Lemma. Lemma 6. Lemma 1. Lemma 7.

16 38 A ZX-Calculus for Toffoli-Hadamard Lemma 8. Lemma 3. Lemma 39. Lemma 9. β Lemma 30. +β Lemma 35. Lemma 0. Lemma 31. Lemma 36. Lemma 1. Lemma 3. Lemma 37. Lemma 33. Lemma 38. Lemma. A. Proof of Lemmas Lemma 19. (S) Lemma 0. (S) (S)

17 R. Vilmart 39 Lemma 1. (S) (S) Lemma. (HT) (T0) (T0) 19 Lemma 3. (T0) (T0) (HT) Lemma. (S) (TCX) 3 (S) Lemma 5. (BW) (BW) Lemma

18 330 A ZX-Calculus for Toffoli-Hadamard Lemma 7. (HT) (TCX) 0 3 (S) 6 (HT) Lemmas 8, 30. By completeness of the -fragment of the ZX-Calculus. Lemma 9. This proof is classical and uses 8, and. The proof is the same whatever the values of and β are. Lemma (T0) Lemma 3. 6 (BW) (T0) Lemma 33. (S) 3 (TW) (S) 3

19 R. Vilmart 331 Lemma Lemma 35. (S) 33 3 Lemma (TW) 8 (TW) 3 (S) 35 Lemma 37. (S) (HT) 1 (S) Lemma

20 33 A ZX-Calculus for Toffoli-Hadamard Lemma 39. (S) (S) Lemma 0. First: (S) (BW) (S) (TW) Moreover, from 37, we can easily derive: (S) and (S) 5 37 Finally: 8 (TW) 3

21 R. Vilmart 333 Lemma (TCX) 8 5 Lemma. 1 (T0) 3 A.3 Additional Corollaries of Theorem 11 Corollary 3. ZX K,P Corollary. ZX K,P β β β β A. Proofs for Completeness Proposition. Soundness is obvious, since the rule iv is sound. Now let D 1 and D be two diagrams of the ZW 1 -Calculus such that D 1 D.We can rewrite D 1 and D as D i d i ( ) n i for some integers n i and diagrams d i of the ZW-Calculus that do not use the symbol. We first assume D i 0. Notice then that n 1 n mod. Indeed D 1 D d 1 n 1 d n. Since d i are matrices over Z, n 1 and n are either both odd or both even. ( ) n i First, assume n i 0 mod. From the new introduced rule, we get that ZW 1 d i D i. W.l.o.g. assume n 1 n. Then d 1 ( ) n n 1 n n 1 d 1 n D 1 d. Since d 1

22 33 A ZX-Calculus for Toffoli-Hadamard ( ) n n 1 and d are ZW-diagrams and have the same interpretation, thanks to the completeness of ( ) n n 1 the ZW-Calculus, ZW 1 d 1 d ( ) n i.e. ZW 1 D 1 D. ( ) n n 1 d, which implies ZW 1 d 1 ( ) n Now, we can easily show ZW 1 D 1 D ZW 1 D 1 D, proving the result when n i 1 mod : ZW 1 D 1 D ZW 1 D 1 D iv ZW 1 D 1 D ZW 1 D 1 D Finally, if D 1 D 0, then d i 0. By completeness of the ZW, ZW d 1 d and ZW d i d i. Hence, using iv n i times, ZW 1 d i d i d i ( ) n i d i ( ) n i D i, so ZW 1 D 1 d 1 d D. Proposition 6. We prove 1 here that all the rules of the ZW 1 -Calculus are preserved by [.] X. X: 39 (HL) 0a, 0c, 0d and 0d come directly from the paradigm Only Topology Matters. 0b: b : Using the result for rule 0b, The proof is strictly the same as in [18]. We give it again here for coherence and to guarantee that all the necessary lemmas have been proven.

23 R. Vilmart 335 1a: (TW) 1b: 3 3 (S) (S) 1c, 1d, a and b come directly from the spider rules and (S). 3a is Lemma 8. 3b: 8 comes from the spider rule. 5a: We will need a few steps to prove this equality. i) 39 8 ii)

24 336 A ZX-Calculus for Toffoli-Hadamard iii) (BW) (TW) 8 iv) 8 iii) 8 v) vi) 8 v) 8

25 R. Vilmart 337 vii) i) 3 0 iii) 8 0 vi) iv) ii) viii) Finally, 30 viii) 0

26 338 A ZX-Calculus for Toffoli-Hadamard vii) viii) 5b: (T0) 5c: 19 5d: 0 (S) a: Thanks to the rule X we can get rid of induced by the crossing. Then, X

27 R. Vilmart 339 6b is exactly the copy rule. 6c: 0 (S) (T0) 7a: 7b: 8 R : 0 R 3 : iv: z: (S) 19 (Z)

28 30 A ZX-Calculus for Toffoli-Hadamard Proposition 15. All the equations in Figure 1 are present in the new axiomatisation, except, (Z) and (T0). Proving and Lemmas 19 and 0 from, (S),, and (E), is classical [0]. Notice also that we can easily derive the colour-swapped version of all the rules. Then: 19 (P) (HT) (1) - (K) - (S) - () (BW) (BW) (3) () (S) (3) (1) (K) (3) (BW) (S) (K) () (1) (T0) It only remains to prove (Z). Notice that it is not used in the proof of Lemmas 7 and 36, so they can be derived. Also, without (Z), is derivable (see []), so now: (P) (S) 36 0 () (E) 0 36 (P)

29 R. Vilmart 31 () () 19 (Z) Theorem 16. We remind the rules that make the -fragment complete [18]: ZX / β +β (S) + + (SUP) (EU) (E) (K ) - -γ γ β β (C) β β -γ γ (BW ) We will prove that they are derivable from ZX K,P,E. Some of the rules are already present in ZX K,P,E :, (S), (E),,,. Thanks to Proposition 15, we can use Theorem 11. This proves that rules (C) and (SUP) are derivable from ZX K,P,E (Corollaries 13 and 1), as well as (K ). There only remain (EU) and (BW ). (BW) appears in ZX K,P,E, but in its triangle-form. We need to prove that the triangle can be decomposed in the diagram of the -fragment used in [18]. The Euler decomposition of the Hadamard gate is derivable: ZX K,P,E

30 3 A ZX-Calculus for Toffoli-Hadamard 39 3 (S) We now have proven all the axioms that make the -fragment complete. We can use it in the following (denoted -C). The decomposition of the triangle node is derivable: ZX K,P,E (S) 5 -C 37 (EU) (TCX) (K) -C 3 -C 3 -C 0 (K) The rule (BW ) is now easily derivable from the decomposition of the triangle and the rule (BW). Theorem 18. The set of rules in [19] is complete in general. ZX A,K,P,E already proves all of them except

31 R. Vilmart 33 - β -β one: θ 1 θ -γ θ 3 γ. e iθ 3 cos(γ)e iθ 1 cos()+e iθ cos(β) By Theorem 17, ZX A,K,P,E proves: (5) (7) (6) - - (8) Now: - β -β - β -β - β -β - β -β θ 1 θ θ 1 θ (6) θ 1 θ θ 1 θ (8) - θ 1 β -β θ (7) β +θ 1 β+θ (7) (TW) +θ 1 -β-θ - β+θ -β (7) θ 1 +β-θ -β β+θ

32 3 A ZX-Calculus for Toffoli-Hadamard γ 0 γ 0 - γ 0 γ -γ (A) γ1 33 (7) γ 1 + γ 0 (8) θ 3 γ -γ θ 3 (6) -γ γ θ 3 where: e iθ 3 cos(γ)e i(γ 1+ γ 0 ) ( e i γ 0 + e i γ 0 ) e iγ 1 ( 1+e iγ 0 ) e i(β+θ ) (1+e iβ( 1+e i(+β+θ 1 θ ) ( 1+e i))) (A) ( 1+e iβ( )) 1+e i(β+θ 1 θ ) cos() ei(β+θ ) 1 ( e iθ ( e iβ + e iβ) + e iθ 1 cos() ) e iθ 1 cos()+e iθ cos(β)