26 research outputs found
A Diagrammatic Axiomatisation of Fermionic Quantum Circuits
We introduce the fermionic ZW calculus, a string-diagrammatic language for fermionic quantum computing (FQC). After defining a fermionic circuit model, we present the basic components of the calculus, together with their interpretation, and show how the main physical gates of interest in FQC can be represented in the language. We then list our axioms, and derive some additional equations. We prove that the axioms provide a complete equational axiomatisation of the monoidal category whose objects are quantum systems of finitely many local fermionic modes, with operations that preserve or reverse the parity (number of particles mod 2) of states, and the tensor product, corresponding to the composition of two systems, as monoidal product. We achieve this through a procedure that rewrites any diagram in a normal form. We conclude by showing, as an example, how the statistics of a fermionic Mach-Zehnder interferometer can be calculated in the diagrammatic language
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
Light-matter interaction in the ZXW calculus
In this paper, we develop a graphical calculus to rewrite photonic circuits
involving light-matter interactions and non-linear optical effects. We
introduce the infinite ZW calculus, a graphical language for linear operators
on the bosonic Fock space which captures both linear and non-linear photonic
circuits. This calculus is obtained by combining the QPath calculus, a
diagrammatic language for linear optics, and the recently developed qudit ZXW
calculus, a complete axiomatisation of linear maps between qudits. It comes
with a 'lifting' theorem allowing to prove equalities between infinite
operators by rewriting in the ZXW calculus. We give a method for representing
bosonic and fermionic Hamiltonians in the infinite ZW calculus. This allows us
to derive their exponentials by diagrammatic reasoning. Examples include phase
shifts and beam splitters, as well as non-linear Kerr media and Jaynes-Cummings
light-matter interaction.Comment: 27 pages, lots of figures, a previous version accepted to QPL 202
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
Towards a Minimal Stabilizer ZX-calculus
The stabilizer ZX-calculus is a rigorous graphical language for reasoning
about quantum mechanics. The language is sound and complete: one can transform
a stabilizer ZX-diagram into another one using the graphical rewrite rules if
and only if these two diagrams represent the same quantum evolution or quantum
state. We previously showed that the stabilizer ZX-calculus can be simplified
by reducing the number of rewrite rules, without losing the property of
completeness [Backens, Perdrix & Wang, EPTCS 236:1--20, 2017]. Here, we show
that most of the remaining rules of the language are indeed necessary. We do
however leave as an open question the necessity of two rules. These include,
surprisingly, the bialgebra rule, which is an axiomatisation of
complementarity, the cornerstone of the ZX-calculus. Furthermore, we show that
a weaker ambient category -- a braided autonomous category instead of the usual
compact closed category -- is sufficient to recover the meta rule 'only
connectivity matters', even without assuming any symmetries of the generators.Comment: 29 pages, minor updates for v
Diagrammatic Analysis for Parameterized Quantum Circuits
Diagrammatic representations of quantum algorithms and circuits offer novel
approaches to their design and analysis. In this work, we describe extensions
of the ZX-calculus especially suitable for parameterized quantum circuits, in
particular for computing observable expectation values as functions of or for
fixed parameters, which are important algorithmic quantities in a variety of
applications ranging from combinatorial optimization to quantum chemistry. We
provide several new ZX-diagram rewrite rules and generalizations for this
setting. In particular, we give formal rules for dealing with linear
combinations of ZX-diagrams, where the relative complex-valued scale factors of
each diagram must be kept track of, in contrast to most previously studied
single-diagram realizations where these coefficients can be effectively
ignored. This allows us to directly import a number useful relations from the
operator analysis to ZX-calculus setting, including causal cone and quantum
gate commutation rules. We demonstrate that the diagrammatic approach offers
useful insights into algorithm structure and performance by considering several
ans\"atze from the literature including realizations of hardware-efficient
ans\"atze and QAOA. We find that by employing a diagrammatic representation,
calculations across different ans\"atze can become more intuitive and
potentially easier approach systematically than by alternative means. Finally,
we outline how diagrammatic approaches may aid in the design and study of new
and more effective quantum circuit ans\"atze
Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus
The ZX-calculus is a universal graphical language for qubit quantum
computation, meaning that every linear map between qubits can be expressed in
the ZX-calculus. Furthermore, it is a complete graphical rewrite system: any
equation involving linear maps that is derivable in the Hilbert space formalism
for quantum theory can also be derived in the calculus by rewriting. It has
widespread usage within quantum industry and academia for a variety of tasks
such as quantum circuit optimisation, error-correction, and education.
The ZW-calculus is an alternative universal graphical language that is also
complete for qubit quantum computing. In fact, its completeness was used to
prove that the ZX-calculus is universally complete. This calculus has advanced
how quantum circuits are compiled into photonic hardware architectures in the
industry.
Recently, by combining these two calculi, a new calculus has emerged for
qubit quantum computation, the ZXW-calculus. Using this calculus,
graphical-differentiation, -integration, and -exponentiation were made
possible, thus enabling the development of novel techniques in the domains of
quantum machine learning and quantum chemistry.
Here, we generalise the ZXW-calculus to arbitrary finite dimensions, that is,
to qudits. Moreover, we prove that this graphical rewrite system is complete
for any finite dimension. This is the first completeness result for any
universal graphical language beyond qubits.Comment: 47 pages, lots of figure