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

@article{Duncan2020GraphtheoreticSO, title={Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus}, author={Ross Duncan and Aleks Kissinger and Simon Perdrix and John van de Wetering}, journal={Quantum}, year={2020}, volume={4}, pages={279} }

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…

## 104 Citations

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- 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.

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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.

### ZX-calculus for the working quantum computer scientist

- Computer Science
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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.

### Vanishing 2-Qubit Gates with Non-Simplification ZX-Rules

- Computer Science
- 2022

This work uses a pair of congruences based on the graph-theoretic notions of local complementation and pivoting to generate local variants of a simpliﬁed ZX-diagram and outperforms state-of-the-art optimization techniques for low-qubit ( < 10) circuits.

### 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…

### 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…

### 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…

### Reducing the number of non-Clifford gates in quantum circuits

- Computer SciencePhysical Review A
- 2020

We present a method for reducing the number of non-Clifford quantum gates, in particularly T-gates, in a circuit, an important task for efficiently implementing fault-tolerant quantum computations.…

### 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.

### Reducing T-count with the ZX-calculus

- Computer Science
- 2019

A new method for reducing the number of T-gates in a quantum circuit based on the ZX-calculus is presented, which matches or beats previous approaches to T-count reduction on the majority of benchmark circuits in the ancilla-free case, in some cases yielding up to 50% improvement.

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