A novel realization of the Virasoro algebra in number state space (original) (raw)

Spin Networks, Quantum Topology and Quantum Computation

Lecture Notes in Computer Science, 2007

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. The simplest case of these models is the Fibonacci model, itself universal for quantum computation. We here formulate these braid group representations in a form suitable for computation and algebraic work.

Algebras and universal quantum computations with higher dimensional systems

2002

Here is discussed application of the Weyl pair to construction of universal set of quantum gates for high-dimensional quantum system. An application of Lie algebras (Hamiltonians) for construction of universal gates is revisited first. It is shown next, how for quantum computation with qubits can be used two-dimensional analog of this Cayley-Weyl matrix algebras, i.e. Clifford algebras, and discussed well known applications to product operator formalism in NMR, Jordan-Wigner construction in fermionic quantum computations. It is introduced universal set of quantum gates for higher dimensional system (``qudit''), as some generalization of these models. Finally it is briefly mentioned possible application of such algebraic methods to design of quantum processors (programmable gates arrays) and discussed generalization to quantum computation with continuous variables.

Topological Methods In Quantum Computations

Basic concepts of quantum information theory, principles of quantum calculations and the possibility of creation on this basis unique on calculation power and functioning principle device, named quantum computer, are concerned. The main blocks of quantum logic, schemes of quantum calculations implementation, as well as some known today effective quantum algorithms, called to realize ad- vantages of quantum calculations upon classical, are presented here. Among them special place is taken by Shor’s algorithm of num- ber factorization and Grover’s algorithm of unsorted database search. Phenomena of decoherence, its influence on quantum computer stability and methods of quantum errors correction are described. Topological quantum computation conception is stated. It hasn’t more computational power than the conventional quantum computation has, but is noiseless by its nature. Anyon statistics necessity for qubits is shown and representative anyon model of topological quantum information processing is presented. More recent by Steven Duplij: "Innovative Quantum Computing" (IOP Publishing, Bristol-London) 2023, 178 pages) https://iopscience.iop.org/book/mono/978-0-7503-5281-9 https://www.amazon.com/gp/product/0750352795

Two paradigms for topological quantum computation

Contemporary Mathematics, 2009

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 group representations. While at least parts of these paradigms are well-known to experts, we provide supporting evidence for them in terms of recent results. We give a fairly comprehensive list of known examples and formulate two conjectures that would further support the paradigms.

Spin networks, quantum automata and link invariants

Journal of Physics: Conference Series, 2006

The spin network simulator model represents a bridge between (generalized) circuit schemes for standard quantum computation and approaches based on notions from Topological Quantum Field Theories (TQFT). More precisely, when working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum automata which in turn represent discrete versions of topological quantum computation models. Such a quantum combinatorial scheme, which essentially encodes SU(2) Racah--Wigner algebra and its braided counterpart, is particularly suitable to address problems in topology and group theory and we discuss here a finite states--quantum automaton able to accept the language of braid group in view of applications to the problem of estimating link polynomials in Chern--Simons field theory.

Space of Quantum Theory Representations of Natural Numbers, Integers, and Rational Numbers

This paper extends earlier work on quantum theory representations of natural numbers N, integers I, and rational numbers Ra to describe a space of these representations and transformations on the space. The space is parameterized by 4-tuple points in a parameter set. Each point, (k,m,h,g), labels a specific representation of X = N, I, Ra as a Fock space F^{X}_{k,m,h} of states of finite length strings of qukits q and a string state basis B^{X}_{k,m,h,g}. The pair (m,h) locates the q string in a square integer lattice I \times I, k is the q base, and the function g fixes the gauge or basis states for each q. Maps on the parameter set induce transformations on on the representation space. There are two shifts, a base change operator W_{k',k}, and a basis or gauge transformation function U_{k}. The invariance of the axioms and theorems for N, I, and Ra under any transformation is discussed along with the dependence of the properties of W_{k',k} on the prime factors of k' an...

Quantum computational structures

Mathematica Slovaca, 2004

Quantum computation has suggested new forms of quantum logic, called quantum computational logics ([CDCGL01]). The basic semantic idea is the following: the meaning of a sentence is identified with a quregister, representing a possible pure state of a compound physical system, whose associated Hilbert space is an n-fold tensor product ⊗ n C 2 . The generalization to density operators, which might be useful to analyse entanglement-phenomena, is due to Gudder [Gu03]. In this paper we study structural properties of density operators systems, where some basic quantum logical gates are defined. We introduce the notions of standard reversible and standard irreversible quantum computational structure. We prove that the second structure is isomorphic to an algebra based on a particular set of complex numbers.

A conjecture on the use of quantum algebras in the treatment of discrete systems

Physics Letters A, 2010

The interactions between atomic spin-states, and between them and an external radiation field, can be described in terms of quantum algebras by a trade-off of bosonic and fermionic degrees of freedom and q-deformed schemes. In this Letter we discuss the use of this concept concerning the calculation of a spin observable, like the spin squeezing.

Spin networks and anyonic topological computing II

Proceedings of SPIE, 2007

Character varieties and algebraic surfaces for the topology of quantum computing

MPDI Symmetry , 2022

It is shown that the representation theory of some finitely presented groups thanks to their SL 2 (C) character variety is related to algebraic surfaces. We make use of the Enriques-Kodaira classification of algebraic surfaces and the related topological tools to make such surfaces explicit. We study the connection of SL 2 (C) character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link H, whose character variety is a Del Pezzo surface f H (the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups as well as that of the fundamental group for the singular fibersẼ 6 andD 4 contain f H. A surface birationally equivalent to a K 3 surface is another compound of their character varieties.

A novel realization of the Virasoro algebra in number state space (original) (raw)

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