TQFTs and quantum computing

• URL: http://arxiv.org/abs/2210.03556v1
• Date: Fri, 7 Oct 2022 13:41:43 GMT
• Title: TQFTs and quantum computing
• Authors: Mahmud Azam, Steven Rayan
• Abstract summary: Quantum computing is captured in the formalism of the monoidal subcategory of $textbfVect_mathbb C$. We formalize this connection by equipping cobordisms with machinery for producing linear maps by parallel transport. We realize quantum circuits as images of cobordisms under monoidal double functors.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum computing is captured in the formalism of the monoidal subcategory of $\textbf{Vect}_{\mathbb C}$ generated by $\mathbb C^2$ -- in particular, quantum circuits are diagrams in $\textbf{Vect}_{\mathbb C}$ -- while topological quantum field theories, in the sense of Atiyah, are diagrams in $\textbf{Vect}_{\mathbb C}$ indexed by cobordisms. We initiate a program that formalizes this connection. In doing so, we equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a double category. Finite-dimensional complex vector spaces and linear maps between them are given a suitable double categorical structure which we call $\mathbb F\textbf{Vect}_{\mathbb C}$. We realize quantum circuits as images of cobordisms under monoidal double functors from these modified cobordisms to $\mathbb F\textbf{Vect}_{\mathbb C}$, which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain double category.

Related papers

• Architectures and random properties of symplectic quantum circuits [0.0]
  Parametrized and random unitary $n$-qubit circuits play a central role in quantum information. We present a universal set of generators $mathcalG$ for the symplectic algebra $imathfraksp(d/2)$. We find that the operators in $mathcalG$ cannot generate arbitrary local symplectic unitaries. We then review the Schur-Weyl duality between the symplectic group and the Brauer algebra, and use tools from Weingarten calculus to prove that Pauli measurements can converge
  arXiv  Detail & Related papers  (2024-05-16T17:15:39Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• Quantum charges of harmonic oscillators [55.2480439325792]
  We show that the energy eigenfunctions $psi_n$ with $nge 1$ are complex coordinates on orbifolds $mathbbR2/mathbbZ_n$. We also discuss "antioscillators" with opposite quantum charges and the same positive energy.
  arXiv  Detail & Related papers  (2024-04-02T09:16:18Z)
• Quantum Current and Holographic Categorical Symmetry [62.07387569558919]
  A quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. The condition for quantum currents to be superconducting is also specified, which corresponds to condensation of anyons in one higher dimension.
  arXiv  Detail & Related papers  (2023-05-22T11:00:25Z)
• Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model [77.34726150561087]
  We provide a systematic treatment of boundaries based on subgroups $Ksubseteq G$ with the Kitaev quantum double $D(G)$ model in the bulk. The boundary sites are representations of a $*$-subalgebra $Xisubseteq D(G)$ and we explicate its structure as a strong $*$-quasi-Hopf algebra. As an application of our treatment, we study patches with boundaries based on $K=G$ horizontally and $K=e$ vertically and show how these could be used in a quantum computer
  arXiv  Detail & Related papers  (2022-08-12T15:05:07Z)
• Quantum teleportation in the commuting operator framework [63.69764116066747]
  We present unbiased teleportation schemes for relative commutants $N'cap M$ of a large class of finite-index inclusions $Nsubseteq M$ of tracial von Neumann algebras. We show that any tight teleportation scheme for $N$ necessarily arises from an orthonormal unitary Pimsner-Popa basis of $M_n(mathbbC)$ over $N'$.
  arXiv  Detail & Related papers  (2022-08-02T00:20:46Z)
• Topological phases of unitary dynamics: Classification in Clifford category [0.0]
  A quantum cellular automaton (QCA) or a causal unitary is by definition an automorphism of local operator algebra. A Clifford QCA is one that maps any Pauli operator to a finite tensor product of Pauli operators.
  arXiv  Detail & Related papers  (2022-05-18T18:00:38Z)
• Markovian Repeated Interaction Quantum Systems [0.0]
  We study a class of dynamical semigroups $(mathbbLn)_ninmathbbN$ that emerge, by a Feynman--Kac type formalism, from a random quantum dynamical system. As a physical application, we consider the case where the $mathcalL_omega$'s are the reduced dynamical maps describing the repeated interactions of a system with thermal probes.
  arXiv  Detail & Related papers  (2022-02-10T20:52:40Z)
• Annihilating Entanglement Between Cones [77.34726150561087]
  We show that Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation.
  arXiv  Detail & Related papers  (2021-10-22T15:02:39Z)
• From Torus Bundles to Particle-Hole Equivariantization [15.857538570676667]
  We consider an infinite family of 3-manifolds, that is, torus bundles over the circle. We show that the modular data are realized by the $mathbbZ$-equivariantization of certain pointed premodular categories. It is our hope that this extensive class of examples will shed light on how to improve the program to recover the full data of a premodular category.
  arXiv  Detail & Related papers  (2021-06-03T16:06:26Z)
• Inequivalent quantizations from gradings and ${\mathbb Z}_2\times {\mathbb Z}_2$ parabosons [0.0]
  It accommodates four kinds of particles: ordinary bosons and three types of parabosons which mutually anticommute when belonging to different type. It is shown how to detect $mathbb Ztimes mathbb Z$-graded parabosons in the multi-particle sector of a quantum model.
  arXiv  Detail & Related papers  (2021-04-19T23:56:33Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.