102 research outputs found
Communication protocols and quantum error-correcting codes from the perspective of topological quantum field theory
Topological quantum field theories (TQFTs) provide a general,
minimal-assumption language for describing quantum-state preparation and
measurement. They therefore provide a general language in which to express
multi-agent communication protocols, e.g. local operations, classical
communication (LOCC) protocols. Here we construct LOCC protocols using TQFT,
and show that LOCC protocols induce quantum error-correcting codes (QECCs) on
the agent-environment boundary. Such QECCs can be regarded as implementing, or
inducing the emergence of, spacetimes on such boundaries. We investigate this
connection between inter-agent communication and spacetime using BF and
Chern-Simons theories, and then using topological M-theory.Comment: 52 page
An algorithm for Tambara-Yamagami quantum invariants of 3-manifolds, parameterized by the first Betti number
Quantum topology provides various frameworks for defining and computing
invariants of manifolds. One such framework of substantial interest in both
mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum
construction, which uses the data of a spherical fusion category to define
topological invariants of triangulated 3-manifolds via tensor network
contractions. In this work we consider a restricted class of state sum
invariants of 3-manifolds derived from Tambara-Yamagami categories. These
categories are particularly simple, being entirely specified by three pieces of
data: a finite abelian group, a bicharacter of that group, and a sign .
Despite being one of the simplest sources of state sum invariants, the
computational complexities of Tambara-Yamagami invariants are yet to be fully
understood.
We make substantial progress on this problem. Our main result is the
existence of a general fixed parameter tractable algorithm for all such
topological invariants, where the parameter is the first Betti number of the
3-manifold with coefficients. We also explain that
these invariants are sometimes #P-hard to compute (and we expect that this is
almost always the case).
Contrary to other domains of computational topology, such as graphs on
surfaces, very few hard problems in 3-manifold topology are known to admit FPT
algorithms with a topological parameter. However, such algorithms are of
particular interest as their complexity depends only polynomially on the
combinatorial representation of the input, regardless of size or combinatorial
width. Additionally, in the case of Betti numbers, the parameter itself is
easily computable in polynomial time.Comment: 24 pages, including 3 appendice
An algorithm for Tambara-Yamagami quantum invariants of 3-manifolds, parameterized by the first Betti number
24 pages, including 3 appendicesQuantum topology provides various frameworks for defining and computing invariants of manifolds. One such framework of substantial interest in both mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum construction, which uses the data of a spherical fusion category to define topological invariants of triangulated 3-manifolds via tensor network contractions. In this work we consider a restricted class of state sum invariants of 3-manifolds derived from Tambara-Yamagami categories. These categories are particularly simple, being entirely specified by three pieces of data: a finite abelian group, a bicharacter of that group, and a sign . Despite being one of the simplest sources of state sum invariants, the computational complexities of Tambara-Yamagami invariants are yet to be fully understood. We make substantial progress on this problem. Our main result is the existence of a general fixed parameter tractable algorithm for all such topological invariants, where the parameter is the first Betti number of the 3-manifold with coefficients. We also explain that these invariants are sometimes #P-hard to compute (and we expect that this is almost always the case). Contrary to other domains of computational topology, such as graphs on surfaces, very few hard problems in 3-manifold topology are known to admit FPT algorithms with a topological parameter. However, such algorithms are of particular interest as their complexity depends only polynomially on the combinatorial representation of the input, regardless of size or combinatorial width. Additionally, in the case of Betti numbers, the parameter itself is easily computable in polynomial time
The exact evaluation of hexagonal spin-networks and topological quantum neural networks
The physical scalar product between spin-networks has been shown to be a
fundamental tool in the theory of topological quantum neural networks (TQNN),
which are quantum neural networks previously introduced by the authors in the
context of quantum machine learning. However, the effective evaluation of the
scalar product remains a bottleneck for the applicability of the theory. We
introduce an algorithm for the evaluation of the physical scalar product
defined by Noui and Perez between spin-network with hexagonal shape. By means
of recoupling theory and the properties of the Haar integration we obtain an
efficient algorithm, and provide several proofs regarding the main steps. We
investigate the behavior of the TQNN evaluations on certain classes of
spin-networks with the classical and quantum recoupling. All results can be
independently reproduced through the "idea.deploy"
framework~\href{https://github.com/lullimat/idea.deploy}{\nolinkurl{https://github.com/lullimat/idea.deploy}}Comment: 15 pages (2 columns, 12+3), 16 figures. Comments are welcome
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
The free energy principle induces neuromorphic development
We show how any finite physical system with morphological, i.e. three-dimensional embedding or shape, degrees of freedom and locally limited free energy will, under the constraints of the free energy principle, evolve over time towards a neuromorphic morphology that supports hierarchical computations in which each ‘level’ of the hierarchy enacts a coarse-graining of its inputs, and dually, a fine-graining of its outputs. Such hierarchies occur throughout biology, from the architectures of intracellular signal transduction pathways to the large-scale organization of perception and action cycles in the mammalian brain. The close formal connections between cone-cocone diagrams (CCCD) as models of quantum reference frames on the one hand, and between CCCDs and topological quantum field theories on the other, allow the representation of such computations in the fully-general quantum-computational framework of topological quantum neural networks
Tensor types and their use in physics
The content of this paper can be roughly organized into a three-level
hierarchy of generality. At the first, most general level, we introduce a new
language which allows us to express various categorical structures in a
systematic and explicit manner in terms of so-called 2-schemes. Although
2-schemes can formalize categorical structures such as symmetric monoidal
categories, they are not limited to this, and can be used to define structures
with no categorical analogue. Most categorical structures come with an
effective graphical calculus such as string diagrams for symmetric monoidal
categories, and the same is true more generally for interesting 2-schemes. In
this work, we focus on one particular non-categorical 2-scheme, whose instances
we refer to as tensor types. At the second level of the hierarchy, we work out
different flavors of this 2-scheme in detail. The effective graphical calculus
of tensor types is that of tensor networks or Penrose diagrams, that is, string
diagrams without a flow of time. As such, tensor types are similar to compact
closed categories, though there are various small but potentially important
differences. Also, the two definitions use completely different mechanisms
despite both being examples of 2-schemes. At the third level of the hierarchy,
we provide a long list of different families of concrete tensor types, in a way
which makes them accessible to concrete computations, motivated by their
potential use in physics. Different tensor types describe different types of
physical models, such as classical or quantum physics, deterministic or
statistical physics, many-body or single-body physics, or matter with or
without symmetries or fermions
New Directions in Geometric and Applied Knot Theory
The aim of this book is to present recent results in both theoretical and applied knot theory—which are at the same time stimulating for leading researchers in the field as well as accessible to non-experts. The book comprises recent research results while covering a wide range of different sub-disciplines, such as the young field of geometric knot theory, combinatorial knot theory, as well as applications in microbiology and theoretical physics
Excursions at the Interface of Topological Phases of Matter and Quantum Error Correction
Topological quantum error-correcting codes are a family of stabilizer codes that are built using a lattice of qubits covering some manifold. The stabilizers of the code are local with respect to the underlying lattice, and logical information is encoded in the non-local degrees of freedom. The locality of stabilizers in these codes makes them especially suitable for experiments. From the condensed matter perspective, their code space corresponds to the ground state subspace of a local Hamiltonian belonging to a non-trivial topological phase of matter. The stabilizers of the code correspond to the Hamiltonian terms, and errors can be thought of as excitations above the ground state subspace. Conversely, one can use fixed point Hamiltonian of a topological phase of matter to define a topological quantum error-correcting code.This close connection has motivated numerous studies which utilize insights from one view- point to address questions in the other. This thesis further explores the possibilities in this di- rection. In the first two chapters, we present novel schemes to implement logical gates, which are motivated by viewing topological quantum error-correcting codes as topological phases of
matter. In the third chapter, we show how the quantum error correction perspective could be used to realize robust topological entanglement phases in monitored random quantum circuits. And in the last chapter, we explore the possibility of extending this connection beyond topological quan- tum error-correcting codes. In particular, we introduce an order parameter for detecting k-local non-trivial states, which can be thought of as a generalization of topological states that includes codewords of any quantum error-correcting code
This Week's Finds in Mathematical Physics (1-50)
These are the first 50 issues of This Week's Finds of Mathematical Physics,
from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity,
topological quantum field theory, knot theory, and applications of
-categories to these subjects. However, there are also digressions into Lie
algebras, elliptic curves, linear logic and other subjects. They were typeset
in 2020 by Tim Hosgood. If you see typos or other problems please report them.
(I already know the cover page looks weird).Comment: 242 page
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