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

Vilmart, Renaud

doi:10.4204/eptcs.287.18

Cited by 14 publications

(17 citation statements)

References 19 publications

(42 reference statements)

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“…Restrictions similar to the ones considered here were previously studied in the context of foundations [27], randomized benchmarking [18], and graphical languages for quantum computing [8,20,30]. Furthermore, our study fits within a larger program, initiated by Aaronson, Grier, and Schaeffer, which aims at classifying quantum operations.…”

Section: Introductionmentioning

confidence: 99%

Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits

Amy

Glaudell

Ross

2020

Quantum

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Kliuchnikov, Maslov, and Mosca proved in 2012 that a 2 × 2 unitary matrix V can be exactly represented by a single-qubit Clifford+T circuit if and only if the entries of V belong to the ring Z[1/ √ 2, i]. Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+T circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+T circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number-theoretic characterizations for certain restricted Clifford+T circuits by considering unitary matrices over subrings of Z[1/ √ 2, i]. We focus on the subrings Z[1/2], Z[1/√ 2], Z[1/ √ -2], and Z[1/2, i], and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates {X, CX, CCX} with an analogue of the Hadamard gate and an optional phase gate.

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Section: Introductionmentioning

confidence: 99%

Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits

Amy

Glaudell

Ross

2020

Quantum

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“…For instance, in the π 12 -fragment: = Finally, it is to be noticed that all the fragments considered in this paper contains the angle π 4 , as some axioms of 1 contains π 4 . However the results presented in this paper can be generalised to fragments which do not contain π 4 using the ∆ZX [28], where the triangle is part of the syntax.…”

Section: Discussionmentioning

confidence: 99%

“…For instance, the complete set of rules for the general ZX-Calculus is denoted ZX A . Introduced in [20] as a syntactic sugar and used as a generator in [25,26,28] is the so-called triangle:…”

Section: Calculusmentioning

confidence: 99%

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A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness

Jeandel

Perdrix

Vilmart

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Recent completeness results on the ZX-Calculus used a third-party language, namely the ZW-Calculus. As a consequence, these proofs are elegant, but sadly non-constructive. We address this issue in the following. To do so, we first describe a generic normal form for ZX-diagrams in any fragment that contains Clifford+T quantum mechanics. We give sufficient conditions for an axiomatisation to be complete, and an algorithm to reach the normal form. Finally, we apply these results to the Clifford+T fragment and the general ZX-Calculus -for which we already know the completeness-, but also for any fragment of rational angles: we show that the axiomatisation for Clifford+T is also complete for any fragment of dyadic angles, and that a simple new rule (called cancellation) is necessary and sufficient otherwise.

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“…We also allow for "gaps" in tof and cnot gates, as in CNOT. These gates must satisfy the identities given in Figure 2: 16], except for the 3-bit-controlled-not gate.…”

Section: The Category Tofmentioning

confidence: 99%

The Category TOF

Cockett

Comfort

2019

Electron. Proc. Theor. Comput. Sci.

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We provide a complete set of identities for the symmetric monoidal category, TOF, generated by the Toffoli gate and computational ancillary bits. We do so by demonstrating that the functor which evaluates circuits on total points, is an equivalence into the full subcategory of sets and partial isomorphisms with objects finite powers of the two element set. The structure of the proof builds -and follows the proof of Cockett et al. -which provided a full set of identities for the cnot gate with computational ancillary bits. Thus, first it is shown that TOF is a discrete inverse category in which all of the identities for the cnot gate hold; and then a normal form for the restriction idempotents is constructed which corresponds precisely to subobjects of the total points of TOF. This is then used to show that TOF is equivalent to FPinj 2 , the full subcategory of sets and partial isomorphisms in which objects have cardinality 2 n for some n ∈ N.

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

Cited by 14 publications

References 19 publications

Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits

Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits

A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness

The Category TOF

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