Related papers: On some projective unitary qutrit gates
We construct the local Hamiltonian description of the Chern-Simons theory with discrete non-Abelian gauge group on a lattice. We show that the theory is fully determined by the phase factors associated with gauge transformations and…
Below we construct non-cyclic and torsion-free abelian quotients for subgroups of braid groups generated by cube powers of half-twists. In the case of 3 and 4 strands we compute the abelianization of these groups. Also, we get…
Seven commuting elements of the Clifford algebra $Cl_{7,7}$ define seven binary eigenvalues that distinguish the $2^7=128$ states of 32 fermions, and determine their parity, electric charge and interactions. Three commuting elements of the…
Chern-Simons field theory based on a compact non-abelian gauge group is studied as a theory of knots and links in three dimensions. A method to obtain the invariants for links made from braids of upto four strands is developed. This…
In this paper, we show the equivalence of the set of unitaries computable by the circuits over the Clifford and T library and the set of unitaries over the ring $\mathbb{Z}[\frac{1}{\sqrt{2}},i]$, in the single-qubit case. We report an…
In a topological quantum computer, universality is achieved by braiding and quantum information is natively protected from small local errors. We address the problem of compiling single-qubit quantum operations into braid representations…
Homotopy braid group description including cyclotron motion of charged interacting 2D particles at strong magnetic field presence is developed in order to explain, in algebraic topology terms, Laughlin correlations in fractional quantum…
It is shown that the generators of Clifford algebras behave as creation and annihilation operators for fermions and bosons. They can create extended objects, such as strings and branes, and can induce curved metric of our spacetime. At a…
It is shown that ternary qubit trees with the same number of nodes can be transformed by the naturally defined sequence of Clifford gates into each other or into standard representation as 1D chain corresponding to Jordan-Wigner transform.
We use our Clifford algebra technique, that is nilpotents and projectors which are binomials of the Clifford algebra objects $\gamma^a$ with the property $\{\gamma^a,\gamma^b\}_+ = 2 \eta^{ab}$, for representing quantum gates and quantum…
The fundamental group $\pi_1(L)$ of a knot or link $L$ may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states.…
Topological quantum computing promises intrinsic fault tolerance by encoding quantum information in non-Abelian anyons, where quantum gates are implemented via braiding. While braiding operations are robust against local perturbations, a…
We propose two definitions of configuration Lie groupoids and in both the cases we prove a Fadell-Neuwirth type fibration theorem for a class of Lie groupoids. We show that this is the best possible extension, in the sense that, for the…
To build large-scale quantum computers while minimizing resource requirements, one may want to use high-rate quantum error-correcting codes that can efficiently encode information. However, realizing an addressable gate$\unicode{x2014}$a…
The group of automorphisms of Euclidean (embedded in $\mathbb{R}^n$) dense lattices such as the root lattices $D_4$ and $E_8$, the Barnes-Wall lattice $BW_{16}$, the unimodular lattice $D_{12}^+$ and the Leech lattice $\Lambda_{24}$ may be…
Suppose $4|n$, $n\geq 8$, $F=F_n=\mathbb{Q}(\zeta_n+\bar{\zeta}_n)$, and there is one prime $\mathfrak{p}=\mathfrak{p}_n$ above $2$ in $F_n$. We study amalgam presentations for $\operatorname{PU_{2}}(\mathbb{Z}[\zeta_n, 1/2])$ and…
Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1D are explored. Many of our field theories are highly-interacting without free quadratic analogs.…
We provide a comprehensive systematic method for the numerical computation of elementary braid operations in topological quantum computation (TQC). This {procedure} is systematically applicable to all anyon models, including $SU(2)_k$.…
We classify extended Abelian Chern-Simons theories with gauge group $U(1)^n$ as extended $(2+1)$-dimensional topological quantum field theories. For an even integral nondegenerate lattice $(\Lambda,K)$, let $(G_K,q_K)$ denote its…
We study an efficient algorithm to hash any single qubit gate (or unitary matrix) into a braid of Fibonacci anyons represented by a product of icosahedral group elements. By representing the group elements by braid segments of different…