Graphical Structures for Design and Verification of Quantum Error Correction

Chancellor, Nicholas; Kissinger, Aleks; Roffe, Joschka; Zohren, Stefan; Horsman, Dominic

doi:10.48550/arxiv.1611.08012

Cited by 20 publications

(26 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One of the best-studied graphical calculi is the ZX-calculus [9,10,12], which has been applied extensively in the study of quantum circuits and related structures in quantum computation (e.g. measurement-based quantum computing [6,17,28], fault-tolerant quantum computations [16,25], error-correcting codes [8], and circuit optimisation [18,19,29]). ZX-diagrams consist of spiders, a type of well-behaved linear map that is depicted by coloured dots:…”

Section: Introductionmentioning

confidence: 99%

Completeness of the ZH-calculus

Backens¹,

Kissinger²,

Miller-Bakewell³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

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[ 1 2 ], 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.

show abstract

Section: Introductionmentioning

confidence: 99%

Completeness of the ZH-calculus

Backens¹,

Kissinger²,

Miller-Bakewell³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…This culminated in the ZX-calculus [6], a graphical language that provides a complete set of rules for qubit quantum computing [7,8]. ZX diagrams have recently been used for state-of-the-art quantum circuit optimisation [9,10,11], compilation [12,13], extraction [14] and error correction [15,16]. In recent work, ZX diagrams have been used to study quantum machine learning [17,18] and its application to quantum natural language processing [19,20].…”

Section: Introductionmentioning

confidence: 99%

Diagrammatic Differentiation for Quantum Machine Learning

Toumi,

Yeung,

de Felice

2021

Preprint

View full text Add to dashboard Cite

We introduce diagrammatic differentiation for tensor calculus by generalising the dual number construction from rigs to monoidal categories. Applying this to ZX diagrams, we show how to calculate diagrammatically the gradient of a linear map with respect to a phase parameter. For diagrams of parametrised quantum circuits, we get the well-known parameter-shift rule at the basis of many variational quantum algorithms. We then extend our method to the automatic differentation of hybrid classical-quantum circuits, using diagrams with bubbles to encode arbitrary non-linear operators. Moreover, diagrammatic differentiation comes with an open-source implementation in DisCoPy, the Python library for monoidal categories. Diagrammatic gradients of classical-quantum circuits can then be simplified using the PyZX library and executed on quantum hardware via the tket compiler. This opens the door to many practical applications harnessing both the structure of string diagrams and the computational power of quantum machine learning.

show abstract

“…The most important tool used is the ZX-calculus, a graphical language for describing and reasoning quantum processes. ZXcalculus was developed by Coecke and Duncan in [23,24], which has various applications including quantum circuit synthesis [25][26][27][28], measurement-based quantum computing [29,30], quantum error correction [31,32], condensed matter physics [33], and quantum natural language processing [34]. In the ZX-calculus, the objects to deal with are ZX-diagrams, which consist of two kinds of tensors, Z-spiders and X-spiders.…”

Section: Introductionmentioning

confidence: 99%

Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus

Zhao,

Gao

2021

Preprint

View full text Add to dashboard Cite

In this paper, we propose a general scheme to analyze the barren plateau phenomenon in training quantum neural networks with the ZX-calculus. More precisely, we extend the barren plateaus theorem from unitary 2-design circuits to any parameterized quantum circuits under certain reasonable assumptions. The main technical contribution of this paper is representing certain integrations as ZX-diagrams and computing them with the ZX-calculus. The method is used to analyze four concrete quantum neural networks with different structures. It is shown that, for the hardware efficient ansatz and the MPS-inspired ansatz, there exist barren plateaus, while for the QCNN and the tree tensor network ansatz, there exists no barren plateau.

show abstract

Graphical Structures for Design and Verification of Quantum Error Correction

Cited by 20 publications

References 29 publications

Completeness of the ZH-calculus

Completeness of the ZH-calculus

Diagrammatic Differentiation for Quantum Machine Learning

Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus

Contact Info

Product

Resources

About