SZX-Calculus: Scalable Graphical Quantum Reasoning

@inproceedings{Carette2019SZXCalculusSG,
  title={SZX-Calculus: Scalable Graphical Quantum Reasoning},
  author={Titouan Carette and Dominic C. Horsman and Simon Perdrix},
  booktitle={MFCS},
  year={2019}
}
We introduce the Scalable ZX-calculus (SZX-calculus for short), a formal and compact graphical language for the design and verification of quantum computations. The SZX-calculus is an extension of the ZX-calculus, a powerful framework that captures graphically the fundamental properties of quantum mechanics through its complete set of rewrite rules. The ZX-calculus is, however, a low level language, with each wire representing a single qubit. This limits its ability to handle large and… 

Figures from this paper

Quantum Algorithms and Oracles with the Scalable ZX-calculus
TLDR
This work considers the standard oracle-based quantum algorithms: Deutsch-Jozsa, Bernstein-Vazirani, Simon, and Grover, and shows they can be described and proved graphically.
ZX-calculus for the working quantum computer scientist
TLDR
This review discusses Clifford computation and graphically prove the Gottesman-Knill theorem, a recently introduced extension of the ZX-calculus that allows for convenient reasoning about Toffoli gates, and the recent completeness theorems that show that, in principle, all reasoning about quantum computation can be done using Zx-diagrams.
LOv-Calculus: A Graphical Language for Linear Optical Quantum Circuits
TLDR
The LO v -calculus is introduced, a graphical language for reasoning about linear optical quantum circuits with so-called vacuum state auxiliary inputs, and the axiomatics of the language are presented and its soundness and completeness are proved.
Complete ZX-calculi for the stabiliser fragment in odd prime dimensions
We introduce a family of ZX-calculi which axiomatise the stabiliser fragment of quantum theory in odd prime dimensions. These calculi recover many of the nice features of the qubit ZX-calculus which
Graphical Calculi and their Conjecture Synthesis
TLDR
This work continues the exploration of graphical calculi, inside and outside of the quantum computing setting, by investigating the algebraic structures with which the authors label diagrams, and introduces two important new calculi here.
The ZX calculus is a language for surface code lattice surgery
TLDR
The operations of the ZX calculus -- a form of quantum diagrammatic reasoning based on bialgebras -- match exactly the operations of lattice surgery, and ZX diagram re-write rules give lattice operations for these operations that are novel, efficient, and highly configurable.
DisCoPy for the quantum computer scientist
TLDR
This report gives a short presentation of DisCoPy as a toolbox for the quantum computer scientist and a review of its recent developments.
A Superposition-Based Calculus for Diagrammatic Reasoning
TLDR
A class of rooted graphs which are expressive enough to encode various kinds of classical or quantum circuits are introduced and a new Superposition calculus is proposed to check the unsatisfiability of formulas consisting of equations or disequations over these graphs.
The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford
We show that the formalism of “Sum-Over-Path” (SOP), used for symbolically representing linear maps or quantum operators, together with a proper rewrite system, has the structure of a dagger-compact
An Automated Deductive Verification Framework for Circuit-building Quantum Programs
TLDR
Qbricks is proposed, a formal verification environment for circuit-building quantum programs, featuring both parametric specifications and a high degree of proof automation, and the main tool is developed, PPS, a parametric extension of the recently developed path sum semantics.
...
1
2
3
...

References

SHOWING 1-10 OF 50 REFERENCES
The ZX−calculus is complete for stabilizer quantum mechanics
The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics (QM), meaning any pure state, unitary
Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus
TLDR
A simplification strategy for ZX-diagrams is given based on the two graph transformations of local complementation and pivoting and it is shown that the resulting reduced diagram can be transformed back into a quantum circuit.
Completeness of the ZX-Calculus
TLDR
This work improves on the known-to-be-complete presentation for the so-called Clifford fragment of the ZX-Calculus, and provides a complete axiomatisation for an altered version of the language which involves an additional generator, making the presentation simpler.
A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics
TLDR
The ZX-Calculus is made complete for the so-called Clifford+T quantum mechanics by adding two new axioms to the language, and it is proved that the π/4-fragment of the ZX -Calculus represents exactly all the matrices over some finite dimensional extension of the ring of dyadic rationals.
ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity
TLDR
A new graphical calculus is presented that is sound and complete for universal quantum computation by demonstrating the reduction of any diagram to an easily describable normal form, which suggests that this calculus will be significantly more convenient for reasoning about the interplay between classical non-linear behaviour and purely quantum operations.
A Near-Minimal Axiomatisation of ZX-Calculus for Pure Qubit Quantum Mechanics
  • R. Vilmart
  • Mathematics
    2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
  • 2019
TLDR
This paper introduces the singular-value decomposition of a ZX-diagram, and uses it to show that all the rules of the former axiomatisation are provable with the new one.
Two complete axiomatisations of pure-state qubit quantum computing
TLDR
Extended versions of ZW and ZX calculus are presented, and their completeness for pure-state qubit theory is proved by a strategy that rewrites all diagrams into a normal form, thus solving two major open problems in categorical quantum mechanics.
Completeness of Graphical Languages for Mixed States Quantum Mechanics
TLDR
A new construction, the discard construction, is introduced, which transforms any †-symmetric monoidal category into a symmetric Monoidal category equipped with a discard map, which provides an extension for several graphical languages that are proved to be complete for general quantum operations.
Interacting Quantum Observables: Categorical Algebra and Diagrammatics
TLDR
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.
The ZX calculus is a language for surface code lattice surgery
TLDR
The operations of the ZX calculus -- a form of quantum diagrammatic reasoning based on bialgebras -- match exactly the operations of lattice surgery, and ZX diagram re-write rules give lattice operations for these operations that are novel, efficient, and highly configurable.
...
1
2
3
4
5
...