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

Jeandel, Emmanuel; Perdrix, Simon; Vilmart, Renaud

doi:10.1109/lics.2019.8785754

Cited by 17 publications

(26 citation statements)

References 22 publications

(34 reference statements)

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“…The core rules of the ZX-calculus give a sound and complete [4] theory for Clifford circuits, a well-known class of circuits that can be efficiently classically simulated. More surprisingly, it was shown in 2018 that modest extensions to the ZX-calculus suffice to give completeness for families of circuits that are approximately universal [24,25] and exactly universal [9,20,23,39] for quantum computation.…”

Section: Introductionmentioning

confidence: 99%

Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus

Duncan

Kissinger

Perdrix

et al. 2020

Quantum

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121

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We present a completely new approach to quantum circuit optimisation, based on the ZX-calculus. We first interpret quantum circuits as ZX-diagrams, which provide a flexible, lower-level language for describing quantum computations graphically. Then, using the rules of the ZX-calculus, we give a simplification strategy for ZX-diagrams based on the two graph transformations of local complementation and pivoting and show that the resulting reduced diagram can be transformed back into a quantum circuit. While little is known about extracting circuits from arbitrary ZX-diagrams, we show that the underlying graph of our simplified ZX-diagram always has a graph-theoretic property called generalised flow, which in turn yields a deterministic circuit extraction procedure. For Clifford circuits, this extraction procedure yields a new normal form that is both asymptotically optimal in size and gives a new, smaller upper bound on gate depth for nearest-neighbour architectures. For Clifford+T and more general circuits, our technique enables us to to `see around' gates that obstruct the Clifford structure and produce smaller circuits than naïve `cut-and-resynthesise' methods.

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Section: Introductionmentioning

confidence: 99%

Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus

Duncan

Kissinger

Perdrix

et al. 2020

Quantum

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121

173

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“…The second diagram is the ZH version of the gadget used in the normal forms of [20], and it can be understood as follows:…”

Section: :=mentioning

confidence: 99%

“…This all depends on the provided description. It was already shown how to efficiently get a diagram from quantum circuits [4], from a measurement-based process [15], from a sequence of lattice surgery operations [13], from "sums-over-paths" [23], or even from the whole matrix representation of the process [20]. Although this last translation is efficient in the size of the matrix, the size of the matrix itself grows exponentially in the number of qubits, so few processes will actually be given in terms of their whole matrix.…”

Section: Introductionmentioning

confidence: 99%

“…The form of the ZX-diagram obtained from a quantum state by [20] is that of a binary tree: a branching in the tree corresponds to a cut in half of the represented vector, while the leaves of the tree exactly correspond to the entries in the vector. It is however possible to exploit redundancies in the entries of the vector, by merging similar subtrees.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Multiple-Valued Decision Diagrams in Graphical Calculi

Vilmart

2021

Preprint

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Graphical calculi such as the ZH-calculus are powerful tools in the study and analysis of quantum processes, with links to other models of quantum computation such as quantum circuits, measurementbased computing, etc. A somewhat compact but systematic way to describe a quantum process is through the use of quantum multiple-valued decision diagrams (QMDDs), which have already been used for the synthesis of quantum circuits as well as for verification. We show in this paper how to turn a QMDD into an equivalent ZHdiagram, and vice-versa, and show how reducing a QMDD translates in the ZH-Calculus, hence allowing tools from one formalism to be used into the other.

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“…A subset of these rules is also complete for the single-qubit Clifford+T group [Bac14b]. Other fragments of the ZX-calculus have recently been completed, these include the full Clifford+T fragment [JPV18a] as well as the full ZX-calculus [HNW18,JPV18b,JPV19,Vil19]. The language can also be extended to capture mixed-state quantum mechanics [CJPV19].…”

Section: Introductionmentioning

confidence: 99%

A Simplified Stabilizer ZX-calculus

Backens¹,

Perdrix²,

Wang³

2017

Electron. Proc. Theor. Comput. Sci.

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The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics. The language is sound and complete: a stabilizer ZX-diagram can be transformed into another one if and only if these two diagrams represent the same quantum evolution or quantum state. We show that the stabilizer ZX-calculus can be simplified, removing unnecessary equations while keeping only the essential axioms which potentially capture fundamental structures of quantum mechanics. We thus give a significantly smaller set of axioms and prove that meta-rules like 'colour symmetry' and 'upside-down symmetry', which were considered as axioms in previous versions of the language, can in fact be derived. In particular, we show that the additional symbol and one of the rules which had been recently introduced to keep track of scalars (diagrams with no inputs or outputs) are not necessary.

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

Cited by 17 publications

References 22 publications

Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus

Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus

Quantum Multiple-Valued Decision Diagrams in Graphical Calculi

A Simplified Stabilizer ZX-calculus

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