140 research outputs found
Quantum Theory is a Quasi-stochastic Process Theory
There is a long history of representing a quantum state using a
quasi-probability distribution: a distribution allowing negative values. In
this paper we extend such representations to deal with quantum channels. The
result is a convex, strongly monoidal, functorial embedding of the category of
trace preserving completely positive maps into the category of quasi-stochastic
matrices. This establishes quantum theory as a subcategory of quasi-stochastic
processes. Such an embedding is induced by a choice of minimal informationally
complete POVM's. We show that any two such embeddings are naturally isomorphic.
The embedding preserves the dagger structure of the categories if and only if
the POVM's are symmetric, giving a new use of SIC-POVM's, objects that are of
foundational interest in the QBism community. We also study general convex
embeddings of quantum theory and prove a dichotomy that such an embedding is
either trivial or faithful.Comment: In Proceedings QPL 2017, arXiv:1802.0973
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
An effect-theoretic reconstruction of quantum theory
An often used model for quantum theory is to associate to every physical
system a C*-algebra. From a physical point of view it is unclear why operator
algebras would form a good description of nature. In this paper, we find a set
of physically meaningful assumptions such that any physical theory satisfying
these assumptions must embed into the category of finite-dimensional
C*-algebras. These assumptions were originally introduced in the setting of
effectus theory, a categorical logical framework generalizing classical and
quantum logic. As these assumptions have a physical interpretation, this
motivates the usage of operator algebras as a model for quantum theory.
In contrast to other reconstructions of quantum theory, we do not start with
the framework of generalized probabilistic theories and instead use effect
theories where no convex structure and no tensor product needs to be present.
The lack of this structure in effectus theory has led to a different notion of
pure maps. A map in an effectus is pure when it is a composition of a
compression and a filter. These maps satisfy particular universal properties
and respectively correspond to `forgetting' and `measuring' the validity of an
effect.
We define a pure effect theory (PET) to be an effect theory where the pure
maps form a dagger-category and filters and compressions are adjoint. We show
that any convex finite-dimensional PET must embed into the category of
Euclidean Jordan algebras. Moreover, if the PET also has monoidal structure,
then we show that it must embed into either the category of real or complex
C*-algebras, which completes our reconstruction.Comment: 20+5 pages. V4: Journal version V3: complete rewrite. Changed name of
the paper from the original 'Reconstruction of quantum theory from universal
filters' to reflect a change of presentatio
Three characterisations of the sequential product
It has already been established that the properties required of an abstract
sequential product as introduced by Gudder and Greechie are not enough to
characterise the standard sequential product on
an operator algebra. We give three additional properties, each of which
characterises the standard sequential product on either a von Neumann algebra
or a Euclidean Jordan algebra. These properties are (1) invariance under
application of unital order isomorphisms, (2) symmetry of the sequential
product with respect to a certain inner product, and (3) preservation of
invertibility of the effects. To give these characterisations we first have to
study convex -sequential effect algebras. We show that these objects
correspond to unit intervals of spectral order unit spaces with a homogeneous
positive cone.Comment: 18 pages + 2 page appendi
Sequential Product Spaces are Jordan Algebras
We show that finite-dimensional order unit spaces equipped with a continuous
sequential product as defined by Gudder and Greechie are homogeneous and
self-dual. As a consequence of the Koecher-Vinberg theorem these spaces
therefore correspond to Euclidean Jordan algebras. We remark on the
significance of this result in the context of reconstructions of quantum
theory. In particular, we show that sequential product spaces that have locally
tomographic tensor products, i.e. their vector space tensor products are also
sequential product spaces, must be C* algebras. Finally we remark on a couple
of ways these results can be extended to the infinite-dimensional setting of
JB- and JBW-algebras and how changing the axioms of the sequential product
might lead to a new characterisation of homogeneous cones.Comment: Original paper title was "Sequential Measurement Characterises
Quantum Theory". It has been changed to reflect a change in focus of the
pape
Constructing quantum circuits with global gates
There are various gate sets that can be used to describe a quantum
computation. A particularly popular gate set in the literature on quantum
computing consists of arbitrary single-qubit gates and 2-qubit CNOT gates. A
CNOT gate is however not always the natural multi-qubit interaction that can be
implemented on a given physical quantum computer, necessitating a compilation
step that transforms these CNOT gates to the native gate set. A particularly
interesting case where compilation is necessary is for ion trap quantum
computers, where the natural entangling operation can act on more than 2 qubits
and can even act globally on all qubits at once. This calls for an entirely
different approach to constructing efficient circuits. In this paper we study
the problem of converting a given circuit that uses 2-qubit gates to one that
uses global gates. Our three main contributions are as follows. First, we find
an efficient algorithm for transforming an arbitrary circuit consisting of
Clifford gates and arbitrary phase gates into a circuit consisting of
single-qubit gates and a number of global interactions proportional to the
number of non-Clifford phases present in the original circuit. Second, we find
a general strategy to transform a global gate that targets all qubits into one
that targets only a subset of the qubits. This approach scales linearly with
the number of qubits that are not targeted, in contrast to the exponential
scaling reported in (Maslov & Nam, N. J. Phys. 2018). Third, we improve on the
number of global gates required to synthesise an arbitrary n-qubit Clifford
circuit from the 12n-18 reported in (Maslov & Nam, N. J. Phys. 2018) to 6n-8.Comment: 13 pages. v2: added some more figures and fixed a number of
(mathematical) typo
Pure Maps between Euclidean Jordan Algebras
We propose a definition of purity for positive linear maps between Euclidean
Jordan Algebras (EJA) that generalizes the notion of purity for quantum
systems. We show that this definition of purity is closed under composition and
taking adjoints and thus that the pure maps form a dagger category (which sets
it apart from other possible definitions.) In fact, from the results presented
in this paper, it follows that the category of EJAs with positive contractive
linear maps is a dagger-effectus, a type of structure originally defined to
study von Neumann algebras in an abstract categorical setting. In combination
with previous work this characterizes EJAs as the most general systems allowed
in a generalized probabilistic theory that is simultaneously a dagger-effectus.
Using the dagger structure we get a notion of dagger-positive maps of the form
f = g*g. We give a complete characterization of the pure dagger-positive maps
and show that these correspond precisely to the Jordan algebraic version of the
sequential product that maps (a,b) to sqrt(a) b sqrt(a). The notion of
dagger-positivity therefore characterizes the sequential product.Comment: In Proceedings QPL 2018, arXiv:1901.0947
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