When Only Topology Matters

Carette, Titouan

doi:10.48550/arxiv.2102.03178

Cited by 6 publications

(13 citation statements)

References 31 publications

(53 reference statements)

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“…This part of the ruleset is the direct generalisation of the rules of the qubit ZX-calculus. One of the most differentiating features of our calculus is that it is not flexsymmetric [14], which means that we cannot see the ZXW diagrams as open graphs. Another interesting difference is that there are two ways to change the colour of spiders based on where the H and H † boxes are located.…”

Section: Discussion Of the Qudit Zx-part Of The Rulesmentioning

confidence: 99%

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Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus

Poór¹,

Wang²,

Shaikh³

et al. 2023

Preprint

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The ZX-calculus is a universal graphical language for qubit quantum computation, meaning that every linear map between qubits can be expressed in the ZX-calculus. Furthermore, it is a complete graphical rewrite system: any equation involving linear maps that is derivable in the Hilbert space formalism for quantum theory can also be derived in the calculus by rewriting. It has widespread usage within quantum industry and academia for a variety of tasks such as quantum circuit optimisation, error-correction, and education.The ZW-calculus is an alternative universal graphical language that is also complete for qubit quantum computing. In fact, its completeness was used to prove that the ZX-calculus is universally complete. This calculus has advanced how quantum circuits are compiled into photonic hardware architectures in the industry.Recently, by combining these two calculi, a new calculus has emerged for qubit quantum computation, the ZXW-calculus. Using this calculus, graphical-differentiation, -integration, and -exponentiation were made possible, thus enabling the development of novel techniques in the domains of quantum machine learning and quantum chemistry.Here, we generalise the ZXW-calculus to arbitrary finite dimensions, that is, to qudits.

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Section: Discussion Of the Qudit Zx-part Of The Rulesmentioning

confidence: 99%

“…A flexsymmetric [14] version of the qudit ZXW-calculus can be obtained by defining the X-spider and the W node differently:…”

Section: Conclusion and Further Workmentioning

confidence: 99%

Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus

Poór¹,

Wang²,

Shaikh³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…This requires some explanation, because this does not look symmetric in the inputs and outputs. However, note that ω| = (|ω These symmetries mean our spiders are flexsymmetric, as defined by Carette [15], and as a result we may treat our ZX-diagrams as undirected graphs with the spiders as vertices. Note that here the cups and caps are defined with respect to the Z basis:…”

Section: The Qutrit Zx-calculusmentioning

confidence: 99%

“…Additionally, the X-spider is not really a spider any more in the sense that it doesn't satisfy the standard spider-fusion equation. Instead it satisfies the 'harvestman equation' [15] that also holds for for instance the W-spider [32] and H-box [4]:…”

Section: The Qutrit Zx-calculusmentioning

confidence: 99%

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Phase gadget compilation for diagonal qutrit gates

Wetering¹,

Yeh²

2022

Preprint

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Phase gadgets have proved to be an indispensable tool for reasoning about ZX-diagrams, being used in optimisation and simulation of quantum circuits and the theory of measurement-based quantum computation. In this paper we study phase gadgets for qutrits. We present the flexsymmetric variant of the original qutrit ZX-calculus, which allows for rewriting that is closer in spirit to the original (qubit) ZX-calculus. In this calculus phase gadgets look as you would expect, but there are non-trivial differences in their properties. We devise new qutrit-specific tricks to extend the graphical Fourier theory of qubits, resulting in a translation between the 'additive' phase gadgets and a 'multiplicative' counterpart we dub phase multipliers. This enables us to build different types of qutrit multiplecontrolled phase gates.As an application of these results we find a construction for emulating arbitrary qubit diagonal unitaries, and specifically find an emulation for the qubit CCZ gate that only requires three singlequtrit non-Clifford gates to implement -provably lower than the four T gates needed using just qubit gates.

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Completeness of the ZH-calculus

et al. 2023

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There are various gate sets used for describing quantum computation. A particularly popular one consists of Clifford gates and arbitrary single-qubit phase gates. Computations in this gate set can be elegantly described by the ZX-calculus, a graphical language for a class of string diagrams describing linear maps between qubits. The ZX-calculus has proven useful in a variety of areas of quantum information, but is less suitable for reasoning about operations outside its natural gate set such as multi-linear Boolean operations like the Toffoli gate. In this paper we study the ZH-calculus, an alternative graphical language of string diagrams that does allow straightforward encoding of Toffoli gates and other more complicated Boolean logic circuits. We find a set of simple rewrite rules for this calculus and show it is complete with respect to matrices over Z[12], which correspond to the approximately universal Toffoli+Hadamard gateset. Furthermore, we construct an extended version of the ZH-calculus that is complete with respect to matrices over any ring R where 1+1 is not a zero-divisor.

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When Only Topology Matters

Cited by 6 publications

References 31 publications

Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus

Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus

Phase gadget compilation for diagonal qutrit gates

Completeness of the ZH-calculus

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