100 research outputs found
Simulating quantum computation by contracting tensor networks
The treewidth of a graph is a useful combinatorial measure of how close the
graph is to a tree. We prove that a quantum circuit with gates whose
underlying graph has treewidth can be simulated deterministically in
time, which, in particular, is polynomial in if
. Among many implications, we show efficient simulations for
log-depth circuits whose gates apply to nearby qubits only, a natural
constraint satisfied by most physical implementations. We also show that
one-way quantum computation of Raussendorf and Briegel (Physical Review
Letters, 86:5188--5191, 2001), a universal quantum computation scheme with
promising physical implementations, can be efficiently simulated by a
randomized algorithm if its quantum resource is derived from a small-treewidth
graph.Comment: 7 figure
Belief propagation in monoidal categories
We discuss a categorical version of the celebrated belief propagation
algorithm. This provides a way to prove that some algorithms which are known or
suspected to be analogous, are actually identical when formulated generically.
It also highlights the computational point of view in monoidal categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
Duality of Graphical Models and Tensor Networks
In this article we show the duality between tensor networks and undirected
graphical models with discrete variables. We study tensor networks on
hypergraphs, which we call tensor hypernetworks. We show that the tensor
hypernetwork on a hypergraph exactly corresponds to the graphical model given
by the dual hypergraph. We translate various notions under duality. For
example, marginalization in a graphical model is dual to contraction in the
tensor network. Algorithms also translate under duality. We show that belief
propagation corresponds to a known algorithm for tensor network contraction.
This article is a reminder that the research areas of graphical models and
tensor networks can benefit from interaction
Constant-degree graph expansions that preserve the treewidth
Many hard algorithmic problems dealing with graphs, circuits, formulas and
constraints admit polynomial-time upper bounds if the underlying graph has
small treewidth. The same problems often encourage reducing the maximal degree
of vertices to simplify theoretical arguments or address practical concerns.
Such degree reduction can be performed through a sequence of splittings of
vertices, resulting in an _expansion_ of the original graph. We observe that
the treewidth of a graph may increase dramatically if the splittings are not
performed carefully. In this context we address the following natural question:
is it possible to reduce the maximum degree to a constant without substantially
increasing the treewidth?
Our work answers the above question affirmatively. We prove that any simple
undirected graph G=(V, E) admits an expansion G'=(V', E') with the maximum
degree <= 3 and treewidth(G') <= treewidth(G)+1. Furthermore, such an expansion
will have no more than 2|E|+|V| vertices and 3|E| edges; it can be computed
efficiently from a tree-decomposition of G. We also construct a family of
examples for which the increase by 1 in treewidth cannot be avoided.Comment: 12 pages, 6 figures, the main result used by quant-ph/051107
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
On traces of tensor representations of diagrams
Let be a set, of {\em types}, and let \iota,o:T\to\oZ_+. A {\em
-diagram} is a locally ordered directed graph equipped with a function
such that each vertex of has indegree
and outdegree . (A directed graph is {\em locally ordered} if at
each vertex , linear orders of the edges entering and of the edges
leaving are specified.)
Let be a finite-dimensional \oF-linear space, where \oF is an
algebraically closed field of characteristic 0. A function on assigning
to each a tensor is called a {\em tensor representation} of . The {\em trace} (or {\em
partition function}) of is the \oF-valued function on the
collection of -diagrams obtained by `decorating' each vertex of a
-diagram with the tensor , and contracting tensors along
each edge of , while respecting the order of the edges entering and
leaving . In this way we obtain a {\em tensor network}.
We characterize which functions on -diagrams are traces, and show that
each trace comes from a unique `strongly nondegenerate' tensor representation.
The theorem applies to virtual knot diagrams, chord diagrams, and group
representations
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