Mathematical Theory and Computational Practice

Hoyrup, Mathieu; Rojas, Cristóbal

doi:10.1007/978-3-642-03073-4

Cited by 8 publications

(4 citation statements)

References 5 publications

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“…In [8] we proved that the Euler decomposition cannot be derived from the axioms of the ZX-calculus; however, in that paper we considered slightly weaker axioms. It is straight-forward to give a countermodel for the ZX-calculus of today.…”

Section: Definition 12mentioning

confidence: 99%

Pivoting makes the ZX-calculus complete for real stabilizers

Duncan

Perdrix

2014

Electron. Proc. Theor. Comput. Sci.

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We show that pivoting property of graph states cannot be derived from the axioms of the ZX-calculus, and that pivoting does not imply local complementation of graph states. Therefore the ZX-calculus augmented with pivoting is strictly weaker than the calculus augmented with the Euler decomposition of the Hadamard gate. We derive an angle-free version of the ZX-calculus and show that it is complete for real stabilizer quantum mechanics.The ZX-calculus is a formal theory for reasoning about quantum computational systems [3]. It consists of a graphical language based on the Pauli Z and X observables, and a collection of axioms expressed as graph rewrite rules. The ZX-calculus is expressive enough to represent any quantum circuit, and its equations are complete for the stabilizer fragment of quantum mechanics [1]. Due to its graphical nature, and its close relationship to the Z and X observables, the ZX-calculus is particularly well adapted to the study of graph states and measurement-based quantum computation [9,7].In addition to the two observables, the ZX-calculus also contains an operator for the Hadamard map: this is the map which exchanges the Z and X bases, and thus provides a duality principle for the graphical language. In previous work [8] the authors showed that if the Hadamard can be expressed in terms of Z and X rotations-that is, as an Euler decomposition-then Van Den Nest's theorem [14] about local complementation of graph states follows, and vice versa. Furthermore, these results cannot be derived from the original axioms, hence the theory "ZX-calculus + Euler" is strictly stronger than the plain ZXcalculus.In this paper we find a theory intermediate between the two, albeit having a similar flavour. We consider an operation on graph states called pivoting and show that its defining property is equivalent to the possibility to express (one of) the Pauli matrices in terms of the Hadamard. Since pivoting can be done via local complementation, "ZX-calculus + Euler" is stronger than "ZX-calculus + Pivot". However, we will show that, once again, these equations cannot be derived from the plain ZX-calculus.The theory "ZX-calculus + Euler" is known to be complete for the stabiliser fragment of quantum mechanics [1]: the stabiliser fragment corresponds to the sub-calculus where all angles are multiples of π/2. We show that the intermediate calculus "ZX-calculus + Pivot" is complete for the real stabiliser fragment of quantum mechanics, and that this fragment admits an angle-free axiomatisation.Real quantum mechanics is sufficient for quantum computing [2], in the sense that any unitary evolution on n-qubits can be simulated (using a simple encoding) by a real unitary evolution acting on n + 1 qubits. As a consequence, while not complete for (complex) quantum mechanics, the intermediate calculus "ZX-calculus + Pivot" might be useful and simpler than the the whole "ZX-calculus + Euler" for proving properties of quantum systems, for example via rewriting.Remark. There is some variation about which axioms compris...

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Section: Definition 12mentioning

confidence: 99%

Pivoting makes the ZX-calculus complete for real stabilizers

Duncan

Perdrix

2014

Electron. Proc. Theor. Comput. Sci.

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“…All the axioms of Figure 1, but (EU), are standard in the ZX-calculus. Roughly speaking: (S) and (I) correspond to the axiomatisation of orthonormal basis [9], each color being associated with an orthonormal basis; (CP) and (B) capture the fact that the two bases are strongly complementary [7]; (H) means that Hadamard can be used to exchange the colours and (HD) means that Hadamard can be decomposed using π 2 -rotations [15]; (E) states that some particular scalars (ZX-diagram with no input/output) can vanish, which means that their interpretation is one [25]. In the following we investigate the properties of (EU).…”

Section: Calculusmentioning

confidence: 99%

“…Indeed, in [1], nearly all the rules for Clifford -i.e. all of the axioms in Figure 1 except (E) and (EU)-are proven to be necessary, and all arguments stand here: -(S): It is the only axiom that can transform a node of degree four or higher into a diagram containing lower-degree nodes -(I g ) or (I r ): These are the only two axioms that can transform a diagram with nodes connected to a boundary to a node-free diagram -(CP): It is the only axiom that can transform a diagram with two connected outputs into one with two disconnected outputs -(HD): The necessity of this axiom requires a non-trivial interpretation given in [15,17], and given again in the Appendix at page 15. -(H): It is the only axiom that matches red nodes with 4+ degree to green nodes of the same degree However, (E) and (EU) can also be proven to be necessary:…”

Section: On Minimalitymentioning

confidence: 99%

A Near-Minimal Axiomatisation of ZX-Calculus for Pure Qubit Quantum Mechanics

Vilmart

2019

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

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Recent developments in the ZX-Calculus have resulted in complete axiomatisations first for an approximately universal restriction of the language, and then for the whole language. The main drawbacks were that the axioms that were added to achieve completeness were numerous, tedious to manipulate and lacked a physical interpretation. We present in this paper two complete axiomatisations for the general ZX-Calculus, that we believe are optimal, in that all their equations are necessary and moreover have a nice physical interpretation.

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“…Since its introduction in [11], the zx-calculus has undergone various refinements, and various changes to the axioms have been considered; for examples see [16,17,4,27]. In this paper, we use the unconditional calculus, without scalars 2 , without supplementarity, and we take the Hadamard gate as a primitive element.…”

Section: The Zx-calculusmentioning

confidence: 99%