arXiv research

Topological quantum computers use hyperbolic knots for computations.

We present two paradigms relating algebraic, topological and quantum computational statistics for the topological model for quantum computation. In particular we suggest correspondences between the computational power of topological quantum computers, computational complexity of link invariants and images of braid grou…

Researchers prove quantum invariants remain hard even when restricted.

Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned topological models ha…

Simple construction for universal quantum gates.

A surprising image of the stock market arises if the price time series of all Dow Jones Industrial Average stock components are represented in one chart at once. The chart evolves into a braid representation of the stock market by taking into account only the crossing of stocks and fixing a convention defining overcros…

We show that the topological modular functor from Witten-Chern-Simons theory is universal for quantum computation in the sense a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor's state space. A computational model based on Chern-Simons theory at a fifth ro…

Program connects quantum computing and topological field theories.

The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liqu…

Algorithm calculates quantum invariants of 3-manifolds with polynomial time complexity.

This paper explores the interactions between knot theory and quantum computing. On one side, knot theory has been used to create models of quantum computing, and on the other, it is a source of computational problems. Knot theory is often used to introduce topological idea to people without a formal mathematical backgr…

Graph potentials link to topological QFTs, with computational methods.

Link homology theories connect to 4-manifold invariants and TQFTs.

We introduce a recoupling theory for virtual braided trees. This recoupling theory can be utilized to incorporate swap gates into anyonic models of quantum computation.

Tropical geometry aids in computing topological quantum field theories.

Efficient algorithms for WRT invariants of torus bundles using algebraic structures.

Kauffman and Lomonaco explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In the work of G. Alagic, M. Jarret, and S. Jordan it is shown that entanglement is a necessary condit…

Study characterizes memory capacity of quantum reservoirs using transmon qubits.

Quantum model for knotted graphs from knot theory.

Researchers compute large quantum invariants for 3-manifolds.

Quantum physics model uses knot theory for fragile topology.

Study topological quantum mechanics on orbifolds with geometric interpretation.

Quantum computing improves graph neural network aggregation.

The SL_3 colored Jones polynomial of the trefoil knot is a $q$-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this $q$-holonomic sequence as a case study. On the one hand, our results are new and useful to q…

Researchers create projective representations of Hecke groups using TQFT.

Using mirror symmetry as described by Hori and Vafa, we compute the quantum equivariant cohomology ring of toric manifolds. This ring arises naturally in topological gauged sigma-models and is related to the Hamiltonian Gromov-Witten invariants of the target manifold.

This work shows how to efficiently simulate parts of quantum landscapes using classical computers.

Develops trace class operators and inverse Laplacian theory for infinite dimensions.

Constructs 3D topological field theories from a specific quantum group, linking to physics invariants.

New proof and formula linking fusion trees to quantum knot invariants.

Fix a finite group $G$. We analyze the computational complexity of the problem of counting homomorphisms $π_1(X) \to G$, where $X$ is a topological space treated as computational input. We are especially interested in requiring $G$ to be a fixed, finite, nonabelian, simple group. We then consider two cases: when the in…

We define a topological quantum membrane theory on a seven dimensional manifold of $G_2$ holonomy. We describe in detail the path integral evaluation for membrane geometries given by circle bundles over Riemann surfaces. We show that when the target space is $CY_3\times S^1$ quantum amplitudes of non-local observables …

We present an explicit expression for the topological invariants associated to $SU(2)$ monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding topological quantum field theory, and it turns out that they can be expressed in …

New non-semisimple Ising anyons enable robust universal quantum computation.

In this article, we discuss a (2+1)-dimensional topological quantum field theory, for short TQFT, with a Verlinde basis. As a conclusion of this general theory, we have a Dehn surgery formula. We show that Turaev-Viro-Ocneanu TQFT has a Verlinde basis. Several applications of this theorem are exposed. Based on Izumi's …

Fast algorithm for braid group Hecke representation, applied to knot invariants.

I review some recent results on four-manifold invariants which have been obtained in the context of topological quantum field theory. I focus on three different aspects: (a) the computation of correlation functions, which give explicit results for the Donaldson invariants of non-simply connected manifolds, and for gene…

New stable homotopy refinement of quantum annular Khovanov homology.

Study SKK groups of manifolds to classify non-unitary TQFTs.

Topological theory for qLDPC codes enables non-Clifford gates and magic state injection.

Let $F$ be a finite type surface and $ζ$ a complex root of unity. The Kauffman bracket skein algebra $K_ζ(F)$ is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Fie…

This is a survey talk on one of the best known quantum knot invariants, the colored Jones polynomial of a knot, and its relation to the algebraic/geometric topology and hyperbolic geometry of the knot complement. We review several aspects of the colored Jones polynomial, emphasizing modularity, stability and effective …

Decorated TQFTs compute invariants with additional structures.

Novel framework explains generalization in deep neural networks.

Quantum algorithm approximates Khovanov homology ranks.

We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We approach this using the theory of modular operads by introducing a new operad governing associative algebras with involution. This operad is Koszul an…

New framework for quantum invariants of 3-manifolds using homology.

Develops Hermitian TQFTs from quantum groups, defining new topological phases.