scholarly journals Extraspecial two-Groups, generalized Yang-Baxter equations and braiding quantum gates

2010 ◽  
Vol 10 (7&8) ◽  
pp. 685-702
Author(s):  
E.C. Rowell ◽  
Y. Zhang ◽  
Y.-S. Wu ◽  
M.-L. Ge
Keyword(s):  
Quantum Computing ◽  
Braid Group ◽  
Quantum Error Correction ◽  
Unitary Representations ◽  
Quantum Gates ◽  
Group Representations ◽  
Unitary Evolution ◽  
Ghz States ◽  
Topological Quantum ◽  
Quantum Error

In this paper we describe connections among extraspecial 2-groups, unitary representations of the braid group and multi-qubit braiding quantum gates. We first construct new representations of extraspecial 2-groups. Extending the latter by the symmetric group, we construct new unitary braid representations, which are solutions to generalized Yang-Baxter equations and use them to realize new braiding quantum gates. These gates generate the GHZ (Greenberger-Horne-Zeilinger) states, for an arbitrary (particularly an \emph{odd}) number of qubits, from the product basis. We also discuss the Yang-Baxterization of the new braid group representations, which describes unitary evolution of the GHZ states. Our study suggests that through their connection with braiding gates, extraspecial 2-groups and the GHZ states may play an important role in quantum error correction and topological quantum computing.

Braiding and Majorana fermions

2017 ◽  
Vol 26 (09) ◽  
pp. 1743001 ◽  
Author(s):  
Louis H. Kauffman
Keyword(s):  
Quantum Computing ◽  
Clifford Algebra ◽  
General Result ◽  
Braid Group ◽  
Clifford Algebras ◽  
Majorana Fermions ◽  
Group Representations ◽  
Topological Quantum

In this paper, we study unitary braid group representations associated with Majorana fermions. Majorana fermions are represented by Majorana operators, elements of a Clifford algebra. The paper proves a general result about braid group representations associated with Clifford algebras and compares this result with the Ivanov braiding associated with Majorana operators and with other braiding representations associated with Majorana fermions such as the Fibonacci model for universal topological quantum computing.

scholarly journals GENERALIZED YANG–BAXTER EQUATIONS AND BRAIDING QUANTUM GATES

2012 ◽  
Vol 21 (09) ◽  
pp. 1250087 ◽  
Author(s):  
REBECCA S. CHEN
Keyword(s):  
Braid Group ◽  
Knot Theory ◽  
Information Science ◽  
Quantum Gates ◽  
Quantum Computers ◽  
Baxter Equation ◽  
Group Representations ◽  
Ongoing Research ◽  
Topological Quantum ◽  
Mathematics And Physics

Solutions to the Yang–Baxter equation — an important equation in mathematics and physics — and their afforded braid group representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, unitary representations of the braid group are desired because they generate braiding quantum gates. These are actively studied in the ongoing research into topological quantum computing. A generalized Yang–Baxter equation was proposed a few years ago by Eric Rowell et al. By finding solutions to the generalized Yang–Baxter equation, we obtain new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.

Quantum Error Correction in the Presence of Initial System-Environment Correlations

2020 ◽  
Vol 27 (02) ◽  
pp. 2050009
Author(s):  
A. Türkmen ◽  
A. Verçin
Keyword(s):  
Error Correction ◽  
Reference System ◽  
Quantum Error Correction ◽  
Communication Process ◽  
Initial System ◽  
Unitary Evolution ◽  
System Environment ◽  
Necessary And Sufficient ◽  
Quantum Error ◽  
Unitary Action

Quantum error correction is studied in a framework consisting of an open quantum system and its environment, jointly subjected to a unitary action, and an interaction-free reference system. It has been shown that coherent information between the initially correlated open system and reference system is conserved in the transmission stage of any quantum communication process, provided that the tripartite input is any pure Markov state and the overall evolution preserves its form. This conservation constitutes the necessary and sufficient condition for accomplishment of perfect error correction by a recovery channel even in the presence of initial system-environment correlations. Explicit expressions of the recovery operators and examples of the joint unitary evolution preserving the form of inputs are given for all classes of pure Markov states.

scholarly journals Non-Pauli topological stabilizer codes from twisted quantum doubles

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 398
Author(s):  
Julio Carlos Magdalena de la Fuente ◽  
Nicolas Tarantino ◽  
Jens Eisert
Keyword(s):  
Quantum Information ◽  
Error Correction ◽  
Lattice Models ◽  
Full Rank ◽  
Error Correcting Codes ◽  
Quantum Error Correction ◽  
Topological Order ◽  
Topological Quantum ◽  
Surface Code ◽  
Quantum Error

It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group cohomological data sitting at the heart of these lattice models, we transmute the terms of these Hamiltonians into full-rank, pairwise commuting operators, defining commuting stabilizers. The resulting codes are defined by non-Pauli commuting stabilizers, with local systems that can either be qubits or higher dimensional quantum systems. Thus, this work establishes a new connection between condensed matter physics and quantum information theory, and constructs tools to systematically devise new topological quantum error correcting codes beyond toric or surface code models.

scholarly journals The surface code with a twist

Quantum ◽  
2017 ◽  
Vol 1 ◽  
pp. 2 ◽  
Author(s):  
Theodore J. Yoder ◽  
Isaac H. Kim
Keyword(s):  
Quantum Error Correction ◽  
New Variety ◽  
Near Surface ◽  
Topological Quantum ◽  
Clifford Group ◽  
Logical Qubits ◽  
Surface Code ◽  
Quantum Error ◽  
Qubit Efficiency ◽  
Logical Qubit

The surface code is one of the most successful approaches to topological quantum error-correction. It boasts the smallest known syndrome extraction circuits and correspondingly largest thresholds. Defect-based logical encodings of a new variety called twists have made it possible to implement the full Clifford group without state distillation. Here we investigate a patch-based encoding involving a modified twist. In our modified formulation, the resulting codes, called triangle codes for the shape of their planar layout, have only weight-four checks and relatively simple syndrome extraction circuits that maintain a high, near surface-code-level threshold. They also use 25% fewer physical qubits per logical qubit than the surface code. Moreover, benefiting from the twist, we can implement all Clifford gates by lattice surgery without the need for state distillation. By a surgical transformation to the surface code, we also develop a scheme of doing all Clifford gates on surface code patches in an atypical planar layout, though with less qubit efficiency than the triangle code. Finally, we remark that logical qubits encoded in triangle codes are naturally amenable to logical tomography, and the smallest triangle code can demonstrate high-pseudothreshold fault-tolerance to depolarizing noise using just 13 physical qubits.

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