Verifying the Smallest Interesting Colour Code with Quantomatic

@inproceedings{Garvie2017VerifyingTS,
  title={Verifying the Smallest Interesting Colour Code with Quantomatic},
  author={Liam Garvie and Ross Duncan},
  booktitle={QPL},
  year={2017}
}
In this paper we present a Quantomatic case study, verifying the basic properties of the Smallest Interesting Colour Code error detecting code. 

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References

SHOWING 1-10 OF 29 REFERENCES

Verifying the Steane code with Quantomatic

A partially mechanized proof of the correctness of Steane's 7-qubit error correcting code, using the tool Quantomatic, represents the largest and most complicated verification task yet carried out using Qu Phantomatic.

Coherent Parity Check Construction for Quantum Error Correction

A framework for constructing and analysing quantum error correction codes that gives simple and intuitive tools for designing codes based on device specifications and how automated reasoning allows us to perform computation between qubits in the same code-block without requiring multiple codeblocks and/or transversal gates is presented.

Graph States and the Necessity of Euler Decomposition

A new equation is introduced, for the Euler decomposition of the Hadamard gate, and it is demonstrated that Van den Nest's theorem--locally equivalent graphs represent the same entanglement--is equivalent to this new axiom.

Generalised compositional theories and diagrammatic reasoning

The use of diagrammatic calculus is illustrated in one particular case, namely the study of complementarity and non-locality, two fundamental concepts of quantum theory whose relationship is explored in later part of this chapter.

Pivoting makes the ZX-calculus complete for real stabilizers

An angle-free version of the ZX-calculus is derived and it is shown that it is complete for real stabilizer quantum mechanics and does not imply local complementation of graph states.

Quantomatic: A proof assistant for diagrammatic reasoning

This work briefly outlines the theoretical basis of Quantomatic's rewriting engine, then gives an overview of the core features and architecture and gives a simple example project that computes normal forms for commutative bialgebras.

Supplementarity is Necessary for Quantum Diagram Reasoning

It is proved that its π 4-fragment is not complete, in other words the ZX-calculus is notcomplete for the so called "Clifford+T quantum mechanics".

A Simplified Stabilizer ZX-calculus

It is proved that meta-rules like 'colour symmetry' and 'upside-down symmetry', which were considered as axioms in previous versions of the stabilizer ZX-calculus, can in fact be derived.

A software methodology for compiling quantum programs

This work presents a software architecture for compiling quantum programs from a high-level language program to hardware-specific instructions, and describes the necessary layers of abstraction and their differences and similarities to classical layers of a computer-aided design flow.

Interacting Quantum Observables: Categorical Algebra and Diagrammatics

The ZX-calculus is introduced, an intuitive and universal graphical calculus for multi-qubit systems, which greatly simplifies derivations in the area of quantum computation and information and axiomatize phase shifts within this framework.