368 research outputs found
PyZX: Large Scale Automated Diagrammatic Reasoning
The ZX-calculus is a graphical language for reasoning about ZX-diagrams, a
type of tensor networks that can represent arbitrary linear maps between
qubits. Using the ZX-calculus, we can intuitively reason about quantum theory,
and optimise and validate quantum circuits. In this paper we introduce PyZX, an
open source library for automated reasoning with large ZX-diagrams. We give a
brief introduction to the ZX-calculus, then show how PyZX implements methods
for circuit optimisation, equality validation, and visualisation and how it can
be used in tandem with other software. We end with a set of challenges that
when solved would enhance the utility of automated diagrammatic reasoning.Comment: In Proceedings QPL 2019, arXiv:2004.1475
Universal MBQC with generalised parity-phase interactions and Pauli measurements
We introduce a new family of models for measurement-based quantum computation
which are deterministic and approximately universal. The resource states which
play the role of graph states are prepared via 2-qubit gates of the form
. When , these are equivalent, up
to local Clifford unitaries, to graph states. However, when , their
behaviour diverges in two important ways. First, multiple applications of the
entangling gate to a single pair of qubits produces non-trivial entanglement,
and hence multiple parallel edges between nodes play an important role in these
generalised graph states. Second, such a state can be used to realise
deterministic, approximately universal computation using only Pauli and
measurements and feed-forward. Even though, for , the relevant resource
states are no longer stabiliser states, they admit a straightforward, graphical
representation using the ZX-calculus. Using this representation, we are able to
provide a simple, graphical proof of universality. We furthermore show that for
every this family is capable of producing all Clifford gates and all
diagonal gates in the -th level of the Clifford hierarchy.Comment: 19 pages, accepted for publication in Quantum (quantum-journal.org).
A previous version of this article had the title: "Universal MBQC with
M{\o}lmer-S{\o}rensen interactions and two measurement bases
The ZX-calculus is complete for the single-qubit Clifford+T group
The ZX-calculus is a graphical calculus for reasoning about pure state qubit
quantum mechanics. It is complete for pure qubit stabilizer quantum mechanics,
meaning any equality involving only stabilizer operations that can be derived
using matrices can also be derived pictorially. Stabilizer operations include
the unitary Clifford group, as well as preparation of qubits in the state |0>,
and measurements in the computational basis. For general pure state qubit
quantum mechanics, the ZX-calculus is incomplete: there exist equalities
involving non-stabilizer unitary operations on single qubits which cannot be
derived from the current rule set for the ZX-calculus. Here, we show that the
ZX-calculus for single qubits remains complete upon adding the operator T to
the single-qubit stabilizer operations. This is particularly interesting as the
resulting single-qubit Clifford+T group is approximately universal, i.e. any
unitary single-qubit operator can be approximated to arbitrary accuracy using
only Clifford operators and T.Comment: In Proceedings QPL 2014, arXiv:1412.810
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, meaning any pure state, unitary operation and post-selected pure
projective measurement can be expressed in the ZX-calculus. The calculus is
also sound, i.e. any equality that can be derived graphically can also be
derived using matrix mechanics. Here, we show that the ZX-calculus is complete
for pure qubit stabilizer quantum mechanics, meaning any equality that can be
derived using matrices can also be derived pictorially. The proof relies on
bringing diagrams into a normal form based on graph states and local Clifford
operations.Comment: 26 page
Completeness of the ZX-Calculus
The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum
mechanics and quantum information theory. It comes equipped with an equational
presentation. We focus here on a very important property of the language:
completeness, which roughly ensures the equational theory captures all of
quantum mechanics. We first improve on the known-to-be-complete presentation
for the so-called Clifford fragment of the language - a restriction that is not
universal - by adding some axioms. Thanks to a system of back-and-forth
translation between the ZX-Calculus and a third-party complete graphical
language, we prove that the provided axiomatisation is complete for the first
approximately universal fragment of the language, namely Clifford+T.
We then prove that the expressive power of this presentation, though aimed at
achieving completeness for the aforementioned restriction, extends beyond
Clifford+T, to a class of diagrams that we call linear with Clifford+T
constants. We use another version of the third-party language - and an adapted
system of back-and-forth translation - to complete the language for the
ZX-Calculus as a whole, that is, with no restriction. We briefly discuss the
added axioms, and finally, we provide a complete axiomatisation for an altered
version of the language which involves an additional generator, making the
presentation simpler
A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond
We consider a ZX-calculus augmented with triangle nodes which is well-suited
to reason on the so-called Toffoli-Hadamard fragment of quantum mechanics. We
precisely show the form of the matrices it represents, and we provide an
axiomatisation which makes the language complete for the Toffoli-Hadamard
quantum mechanics. We extend the language with arbitrary angles and show that
any true equation involving linear diagrams which constant angles are multiple
of Pi are derivable. We show that a single axiom is then necessary and
sufficient to make the language equivalent to the ZX-calculus which is known to
be complete for Clifford+T quantum mechanics. As a by-product, it leads to a
new and simple complete axiomatisation for Clifford+T quantum mechanics.Comment: In Proceedings QPL 2018, arXiv:1901.09476. Contains Appendi
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