# ZX-calculus for the working quantum computer scientist

@inproceedings{Wetering2020ZXcalculusFT, title={ZX-calculus for the working quantum computer scientist}, author={John van de Wetering}, year={2020} }

The ZX-calculus is a graphical language for reasoning about quantum computation that has recently seen an increased usage in a variety of areas such as quantum circuit optimisation, surface codes and lattice surgery, measurementbased quantum computation, and quantum foundations. The first half of this review gives a gentle introduction to the ZX-calculus suitable for those familiar with the basics of quantum computing. The aim here is to make the reader comfortable enough with the ZX-calculus…

## 38 Citations

### Equivalence Checking of Quantum Circuits with the ZX-Calculus

- Computer ScienceIEEE Journal on Emerging and Selected Topics in Circuits and Systems
- 2022

It is demonstrated how the ZX-calculus based approach for equivalence checking can be expanded in order to verify the results of compilation and optimizations on quantum circuits.

### Circuit Extraction for ZX-diagrams can be #P-hard

- Computer Science, MathematicsICALP
- 2022

This paper proves that any oracle that takes as input a ZX-diagram description of a unitary and produces samples of the output of the associated quantum computation enables eﬃcient probabilistic solutions to NP-complete problems.

### Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions

- Computer ScienceMFCS
- 2022

A family of ZX-calculi which axiomatise the stabiliser fragment of quantum theory in odd prime dimensions are introduced, and it is proved that these calculi are complete, i.e. provide a set of rewrite rules which can be used to prove any equality of stabiliser quantum operations.

### Hybrid Quantum-Classical Circuit Simplification with the ZX-Calculus

- Computer ScienceAPLAS
- 2021

This work uses an extension of the formal graphical ZX-calculus called ZX as an intermediary representation of the hybrid circuits to allow for granular optimizations below the quantum-gate level and derives a number of gFlow-preserving optimization rules for ZX diagrams that reduce the size of the graph.

### Spin-networks in the ZX-calculus

- Computer Science
- 2021

This paper writes the spin-networks of loop quantum gravity in the ZX-diagrammatic language of quantum computation by writing the Yutsis diagrams, a standard graphical calculus used in quantum chemistry and quantum gravity, which captures the main features of SU(2) representation theory.

### Diagrammatic Analysis for Parameterized Quantum Circuits

- Computer Science
- 2022

Extensions of the ZX-calculus especially suitable for parameterized quantum circuits, in particular for computing observable expectation values as functions of or for parameters, which are important algorithmic quantities in a variety of applications ranging from combinatorial optimization to quantum chemistry are described.

### Simplification Strategies for the Qutrit ZX-Calculus

- Computer Science
- 2021

The main contribution of this work is the derivation of efﬁcient rewrite strategies for the stabiliser fragment of the qutrit ZX-calculus, which constitutes a first non-trivial step towards the simplification ofqutrit quantum circuits.

### VyZX : A Vision for Verifying the ZX Calculus

- Computer ScienceArXiv
- 2022

Vy ZX is developed, a veriﬁed ZX-calculus in the Coq proof assistant that provides two distinct representations of ZX diagrams for ease of programming and proof.

### Addition and Differentiation of ZX-Diagrams

- Computer ScienceFSCD
- 2022

This work introduces a general, inductive definition of the addition of ZX-diagrams, relying on the construction of controlled diagrams, and provides an inductive differentiation of Zx-displays, based on the isolation of variables.

### Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions

- Computer ScienceQuantum Science and Technology
- 2022

We introduce an enhanced technique for strong classical simulation of quantum circuits which combines the ‘sum-of-stabilisers’ method with an automated simplification strategy based on the…

## References

SHOWING 1-10 OF 125 REFERENCES

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

- Computer ScienceQuantum
- 2020

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.

### Graphical Fourier Theory and the Cost of Quantum Addition

- Physics
- 2019

The ZX-calculus is a convenient formalism for expressing and reasoning about quantum circuits at a low level, whereas the recently-proposed ZH-calculus yields convenient expressions of mid-level…

### Completeness of the ZH-calculus

- Mathematics
- 2021

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…

### Completeness of the ZX-Calculus

- MathematicsLog. Methods Comput. Sci.
- 2020

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.

### ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity

- Computer ScienceElectronic Proceedings in Theoretical Computer Science
- 2019

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.

### The ZX-calculus is complete for stabilizer quantum mechanics

- Physics, Mathematics
- 2014

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…

### Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics

- PhysicsLICS
- 2018

It is shown that the axiomatisation for Clifford+T is not complete in general but can be completed by adding a single (non linear) axiom, providing a simpler axiom atisation of the ZX-calculus for pure quantum mechanics than the one recently introduced by Ng&Wang.

### Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics

- Mathematics, Physics
- 2018

Recently, we gave a complete axiomatisation of the ZX-calculus [1] for the overall pure qubit quantum mechanics [4]. In this paper, we first simplify the rule of addition (AD) and show that some…

### ZX-Rules for 2-Qubit Clifford+T Quantum Circuits

- MathematicsRC
- 2018

These ZX-rules are much simpler than the complete of set Clifford+T circuit equations due to Selinger and Bian, which indicates that Zx-calculus provides a more convenient arena for quantum circuit rewriting than restricting oneself to circuit equations.

### SZX-Calculus: Scalable Graphical Quantum Reasoning

- Computer ScienceMFCS
- 2019

The Scalable ZX-calculus is introduced, a formal and compact graphical language for the design and verification of quantum computations and two examples of applications are provided, for graph states and error correcting codes.