scholarly journals Hypermap-homology quantum codes

2014 ◽  
Vol 12 (01) ◽  
pp. 1430001 ◽  
Author(s):  
Martin Leslie
Keyword(s):  
Quantum Information ◽  
Error Correcting Code ◽  
Chain Complex ◽  
Error Correcting Codes ◽  
Quantum Computers ◽  
Quantum Codes ◽  
Straightforward Generalization ◽  
New Type ◽  
Quantum Error ◽  
Toric Code

We introduce a new type of sparse CSS quantum error correcting code based on the homology of hypermaps. Sparse quantum error correcting codes are of interest in the building of quantum computers due to their ease of implementation and the possibility of developing fast decoders for them. Codes based on the homology of embeddings of graphs, such as Kitaev's toric code, have been discussed widely in the literature and our class of codes generalize these. We use embedded hypergraphs, which are a generalization of graphs that can have edges connected to more than two vertices. We develop theorems and examples of our hypermap-homology codes, especially in the case that we choose a special type of basis in our homology chain complex. In particular the most straightforward generalization of the m × m toric code to hypermap-homology codes gives us a [(3/2)m2, 2, m] code as compared to the toric code which is a [2m2, 2, m] code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.

ON TWO PROBLEMS OF ASYMMETRIC QUANTUM CODES

2014 ◽  
Vol 28 (06) ◽  
pp. 1450017 ◽  
Author(s):  
RUIHU LI ◽  
GEN XU ◽  
LUOBIN GUO
Keyword(s):  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Bch Code ◽  
Quantum Codes ◽  
Quantum Code ◽  
Asymmetric Quantum ◽  
Asymmetric Quantum Codes ◽  
Special Cases ◽  
Quantum Error ◽  
Better Than

In this paper, we discuss two problems on asymmetric quantum error-correcting codes (AQECCs). The first one is on the construction of a [[12, 1, 5/3]]2 asymmetric quantum code, we show an impure [[12, 1, 5/3 ]]2 exists. The second one is on the construction of AQECCs from binary cyclic codes, we construct many families of new asymmetric quantum codes with dz> δ max +1 from binary primitive cyclic codes of length n = 2m-1, where δ max = 2⌈m/2⌉-1 is the maximal designed distance of dual containing narrow sense BCH code of length n = 2m-1. A number of known codes are special cases of the codes given here. Some of these AQECCs have parameters better than the ones available in the literature.

scholarly journals Transforming graph states using single-qubit operations

2018 ◽  
Vol 376 (2123) ◽  
pp. 20170325 ◽  
Author(s):  
Axel Dahlberg ◽  
Stephanie Wehner
Keyword(s):  
Quantum Information ◽  
Error Correcting Codes ◽  
Graph States ◽  
Target State ◽  
Single Qubit ◽  
Minor Problem ◽  
Fixed Parameter ◽  
Source State ◽  
Quantum Error ◽  
Stabilizer States

Stabilizer states form an important class of states in quantum information, and are of central importance in quantum error correction. Here, we provide an algorithm for deciding whether one stabilizer (target) state can be obtained from another stabilizer (source) state by single-qubit Clifford operations (LC), single-qubit Pauli measurements (LPM) and classical communication (CC) between sites holding the individual qubits. What is more, we provide a recipe to obtain the sequence of LC+LPM+CC operations which prepare the desired target state from the source state, and show how these operations can be applied in parallel to reach the target state in constant time. Our algorithm has applications in quantum networks, quantum computing, and can also serve as a design tool—for example, to find transformations between quantum error correcting codes. We provide a software implementation of our algorithm that makes this tool easier to apply. A key insight leading to our algorithm is to show that the problem is equivalent to one in graph theory, which is to decide whether some graph G ′ is a vertex-minor of another graph G . The vertex-minor problem is, in general, -Complete, but can be solved efficiently on graphs which are not too complex. A measure of the complexity of a graph is the rank-width which equals the Schmidt-rank width of a subclass of stabilizer states called graph states, and thus intuitively is a measure of entanglement. Here, we show that the vertex-minor problem can be solved in time O (| G | 3 ), where | G | is the size of the graph G , whenever the rank-width of G and the size of G ′ are bounded. Our algorithm is based on techniques by Courcelle for solving fixed parameter tractable problems, where here the relevant fixed parameter is the rank width. The second half of this paper serves as an accessible but far from exhausting introduction to these concepts, that could be useful for many other problems in quantum information. This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.

Entanglement-assisted quantum codes achieving the quantum singleton bound but violating the quantum hamming bound

2014 ◽  
Vol 14 (13&14) ◽  
pp. 1107-1116
Author(s):  
Ruihu Li ◽  
Luobin Guo ◽  
Zongben Xu
Keyword(s):  
Error Correction ◽  
Infinite Family ◽  
Error Correcting Codes ◽  
Quantum Codes ◽  
Singleton Bound ◽  
Minimum Distances ◽  
Quantum Error

We give an infinite family of degenerate entanglement-assisted quantum error-correcting codes (EAQECCs) which violate the EA-quantum Hamming bound for non-degenerate EAQECCs and achieve the EA-quantum Singleton bound, thereby proving that the EA-quantum Hamming bound does not asymptotically hold for degenerate EAQECCs. Unlike the previously known quantum error-correcting codes that violate the quantum Hamming bound by exploiting maximally entangled pairs of qubits, our codes do not require local unitary operations on the entangled auxiliary qubits during encoding. The degenerate EAQECCs we present are constructed from classical error-correcting codes with poor minimum distances, which implies that, unlike the majority of known EAQECCs with large minimum distances, our EAQECCs take more advantage of degeneracy and rely less on the error correction capabilities of classical codes.

Constructing new q-ary quantum MDS codes with distances bigger than q/2 from generator matrices

2018 ◽  
Vol 18 (3&4) ◽  
pp. 223-230
Author(s):  
Xianmang He
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Minimum Distance ◽  
Quantum Information Theory ◽  
Error Correcting Codes ◽  
Mds Codes ◽  
Quantum Error ◽  
Generator Matrices ◽  
Quantum Mds Codes

The construction of quantum error-correcting codes has been an active field of quantum information theory since the publication of \cite{Shor1995Scheme,Steane1998Enlargement,Laflamme1996Perfect}. It is becoming more and more difficult to construct some new quantum MDS codes with large minimum distance. In this paper, based on the approach developed in the paper \cite{NewHeMDS2016}, we construct several new classes of quantum MDS codes. The quantum MDS codes exhibited here have not been constructed before and the distance parameters are bigger than q/2.

scholarly journals ON OPTIMAL QUANTUM CODES

2004 ◽  
Vol 02 (01) ◽  
pp. 55-64 ◽  
Author(s):  
MARKUS GRASSL ◽  
THOMAS BETH ◽  
MARTIN RÖTTELER
Keyword(s):  
Minimum Distance ◽  
Prime Power ◽  
Quantum Systems ◽  
Error Correcting Codes ◽  
Mds Codes ◽  
Quantum Codes ◽  
Maximum Distance Separable ◽  
Shortened Codes ◽  
Quantum Error ◽  
Quantum Mds Codes

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters 〚n, n - 2d + 2, d〛q exist for all 3≤n≤q and 1≤d≤n/2+1. We also present quantum MDS codes with parameters 〚q2, q2-2d+2, d〛q for 1≤d≤q which additionally give rise to shortened codes 〚q2-s, q2-2d+2-s, d〛q for some s.

ON THE AUTOMORPHISM GROUPS OF A FAMILY OF BINARY QUANTUM ERROR-CORRECTING CODES

2006 ◽  
Vol 04 (06) ◽  
pp. 1013-1022
Author(s):  
TAILIN LIU ◽  
FENGTONG WEN ◽  
QIAOYAN WEN
Keyword(s):  
Semidirect Product ◽  
Automorphism Groups ◽  
Error Correcting Code ◽  
Error Correcting Codes ◽  
Index Formula ◽  
Product Group ◽  
Simplex Code ◽  
Quantum Error ◽  
The Relationship ◽  
Free Element

Based on the classical binary simplex code [Formula: see text] and any fixed-point-free element f of [Formula: see text], Calderbank et al. constructed a binary quantum error-correcting code [Formula: see text]. They proved that [Formula: see text] has a normal subgroup H, which is a semidirect product group of the centralizer Z(f) of f in GLm(2) with [Formula: see text], and the index [Formula: see text] is the number of elements of Ff = {f, 1 - f, 1/f, 1 - 1/f, 1/(1 - f), f/(1 - f)} that are conjugate to f. In this paper, a theorem to describe the relationship between the quotient group [Formula: see text] and the set Ff is presented, and a way to find the elements of Ff that are conjugate to f is proposed. Then we prove that [Formula: see text] is isomorphic to S3 and H is a semidirect product group of [Formula: see text] with [Formula: see text] in the linear case. Finally, we generalize a result due to Calderbank et al.

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 Quantum error correction for the toric code using deep reinforcement learning

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 183 ◽  
Author(s):  
Philip Andreasson ◽  
Joel Johansson ◽  
Simon Liljestrand ◽  
Mats Granath
Keyword(s):  
Reinforcement Learning ◽  
Error Correction ◽  
Minimum Weight ◽  
Error Correcting Codes ◽  
Error Rates ◽  
Quantum Error Correction ◽  
Translational Invariance ◽  
Quantum Error ◽  
Q Function ◽  
Toric Code

We implement a quantum error correction algorithm for bit-flip errors on the topological toric code using deep reinforcement learning. An action-value Q-function encodes the discounted value of moving a defect to a neighboring site on the square grid (the action) depending on the full set of defects on the torus (the syndrome or state). The Q-function is represented by a deep convolutional neural network. Using the translational invariance on the torus allows for viewing each defect from a central perspective which significantly simplifies the state space representation independently of the number of defect pairs. The training is done using experience replay, where data from the algorithm being played out is stored and used for mini-batch upgrade of the Q-network. We find performance which is close to, and for small error rates asymptotically equivalent to, that achieved by the Minimum Weight Perfect Matching algorithm for code distances up to d=7. Our results show that it is possible for a self-trained agent without supervision or support algorithms to find a decoding scheme that performs on par with hand-made algorithms, opening up for future machine engineered decoders for more general error models and error correcting codes.

On quantum codes obtained from cyclic codes over A2

2015 ◽  
Vol 13 (03) ◽  
pp. 1550031 ◽  
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Senol Eren
Keyword(s):  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Gray Map ◽  
Quantum Codes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Arbitrary Length ◽  
Necessary And Sufficient ◽  
Quantum Error

In this paper, quantum codes from cyclic codes over A2 = F2 + uF2 + vF2 + uvF2, u2 = u, v2 = v, uv = vu, for arbitrary length n have been constructed. It is shown that if C is self orthogonal over A2, then so is Ψ(C), where Ψ is a Gray map. A necessary and sufficient condition for cyclic codes over A2 that contains its dual has also been given. Finally, the parameters of quantum error correcting codes are obtained from cyclic codes over A2.

scholarly journals A complete hierarchy for the pure state marginal problem in quantum mechanics

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Xiao-Dong Yu ◽  
Timo Simnacher ◽  
Nikolai Wyderka ◽  
H. Chau Nguyen ◽  
Otfried Gühne
Keyword(s):  
Quantum Mechanics ◽  
Quantum State ◽  
Error Correcting Codes ◽  
Quantum Codes ◽  
Semidefinite Programs ◽  
Marginal Problem ◽  
Separability Problem ◽  
Pure Quantum State ◽  
Maximally Entangled States ◽  
Quantum Error

AbstractClarifying the relation between the whole and its parts is crucial for many problems in science. In quantum mechanics, this question manifests itself in the quantum marginal problem, which asks whether there is a global pure quantum state for some given marginals. This problem arises in many contexts, ranging from quantum chemistry to entanglement theory and quantum error correcting codes. In this paper, we prove a correspondence of the marginal problem to the separability problem. Based on this, we describe a sequence of semidefinite programs which can decide whether some given marginals are compatible with some pure global quantum state. As an application, we prove that the existence of multiparticle absolutely maximally entangled states for a given dimension is equivalent to the separability of an explicitly given two-party quantum state. Finally, we show that the existence of quantum codes with given parameters can also be interpreted as a marginal problem, hence, our complete hierarchy can also be used.

Sign in / Sign up

Export Citation Format

Share Document