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… 

Figures from this paper

Hybrid Quantum-Classical Circuit Simplification with the ZX-Calculus

TLDR
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.

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

TLDR
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

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.

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

TLDR
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

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

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

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

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

TLDR
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

TLDR
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.
...

References

SHOWING 1-10 OF 52 REFERENCES

Reducing the number of non-Clifford gates in quantum circuits

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.

Reducing T-count with the ZX-calculus

TLDR
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.

Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics

TLDR
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.

Automated optimization of large quantum circuits with continuous parameters

TLDR
An automated methods for optimizing quantum circuits of the size and type expected in quantum computations that outperform classical computers are developed and implemented and a collection of fast algorithms capable of optimizing large-scale quantum circuits are reported.

Optimal synthesis of linear reversible circuits

TLDR
Simulation results show that even for relatively small n the authors' algorithm is faster and yields smaller circuits than the standard method, and can be interpreted as a matrix decomposition algorithm, yielding an asymptotically efficient decomposition of a binary matrix into a product of elementary matrices.

Making the stabilizer ZX-calculus complete for scalars

TLDR
This work replaces those scalar-free rewrite rules with correctly scaled ones and shows that, by adding one new diagram element and a new rewrite rule, the calculus can be made complete for pure qubit stabilizer quantum mechanics with scalars.

CNOT circuit extraction for topologically-constrained quantum memories

TLDR
A new technique for quantum circuit mapping, based on Gaussian elimination constrained to certain optimal spanning trees called Steiner trees, is given, which significantly out-performs general-purpose routines on CNOT circuits.

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

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.

Classical simulation of quantum computation, the Gottesman-Knill theorem, and slightly beyond

We study classical simulation of quantum computation, taking the Gottesman-Knilltheorem as a starting point. We show how each Clifford circuit can be reduced to anequivalent, manifestly simulatable
...