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.

New optimal asymmetric quantum codes constructed from constacyclic codes

2017 ◽  
Vol 31 (05) ◽  
pp. 1750030 ◽  
Author(s):  
Gen Xu ◽  
Ruihu Li ◽  
Luobin Guo ◽  
Liangdong Lü
Keyword(s):  
Prime Power ◽  
Error Correcting Codes ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Code Length ◽  
Theor Phys ◽  
Singleton Bound ◽  
Asymmetric Quantum ◽  
Asymmetric Quantum Codes ◽  
Quantum Error

In this paper, we propose the construction of asymmetric quantum codes from two families of constacyclic codes over finite field [Formula: see text] of code length [Formula: see text], where for the first family, [Formula: see text] is an odd prime power with the form [Formula: see text] ([Formula: see text] is integer) or [Formula: see text] ([Formula: see text] is integer) and [Formula: see text]; for the second family, [Formula: see text] is an odd prime power with the form [Formula: see text] or [Formula: see text] ([Formula: see text] is integer) and [Formula: see text]. As a result, families of new asymmetric quantum codes [Formula: see text] with [Formula: see text] distance larger than [Formula: see text] are obtained, which are not covered by the asymmetric quantum error-correcting codes (AQECCs) in Refs. 32 and 33 [J.-Z. Chen, J.-P. Li and J. Lin, Int. J. Theor. Phys. 53, 72 (2014); L. Wang and S. Zhu, Int. J. Quantum Inf. 12, 1450017 (2014)] that [Formula: see text]. Also, all the newly obtained asymmetric quantum codes are optimal according to the singleton bound for asymmetric quantum codes.

scholarly journals GRAPHICAL QUANTUM ERROR-CORRECTING CODES BASED ON ENTANGLEMENT OF SUBGRAPHS

2012 ◽  
Vol 26 (04) ◽  
pp. 1250024
Author(s):  
YUAN LI ◽  
MANTAO XU ◽  
QIANG SUN
Keyword(s):  
Error Correction ◽  
Error Correcting Codes ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Quantum Code ◽  
Generator Matrix ◽  
Code Construction ◽  
Nested Codes ◽  
Colorable Graph ◽  
Quantum Error

Graphical approach provides a more intuitive and simple way to construct error correction codes. How to obtain generator matrix is the key problem of constructing graphical quantum codes. In this paper, we further generalize the graphical quantum code construction method by entangling its disconnected subgraphs, so that the corresponding generator matrix of quantum nondegenerate codes can be easily obtained. By making use of the method of subgraphs entangling, we also point out its application in adjacency matrix constructions of larger colorable graph and graphical quantum nested codes.

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 Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes

2021 ◽  
Author(s):  
Dongsheng Wang ◽  
Yunjiang Wang ◽  
Ningping Cao ◽  
Bei Zeng ◽  
Raymond Lafflamme
Keyword(s):  
Error Correction ◽  
Quantum Computation ◽  
Fault Tolerant ◽  
Valence Bond ◽  
Quantum Error Correction ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Exact Quantum ◽  
Valence Bond Solid ◽  
Quantum Error

Abstract In this work, we develop the theory of quasi-exact fault-tolerant quantum (QEQ) computation, which uses qubits encoded into quasi-exact quantum error-correction codes (``quasi codes''). By definition, a quasi code is a parametric approximate code that can become exact by tuning its parameters. The model of QEQ computation lies in between the two well-known ones: the usual noisy quantum computation without error correction and the usual fault-tolerant quantum computation, but closer to the later. Many notions of exact quantum codes need to be adjusted for the quasi setting. Here we develop quasi error-correction theory using quantum instrument, the notions of quasi universality, quasi code distances, and quasi thresholds, etc. We find a wide class of quasi codes which are called valence-bond-solid codes, and we use them as concrete examples to demonstrate QEQ computation.

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.

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.

scholarly journals Flag fault-tolerant error correction with arbitrary distance codes

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 53 ◽  
Author(s):  
Christopher Chamberland ◽  
Michael E. Beverland
Keyword(s):  
Error Correction ◽  
Fault Tolerant ◽  
Quantum Error Correction ◽  
Quantum Codes ◽  
Parity Check ◽  
Code Length ◽  
Stabilizer Codes ◽  
Low Density Parity Check ◽  
Quantum Error ◽  
First Time

In this paper we introduce a general fault-tolerant quantum error correction protocol using flag circuits for measuring stabilizers of arbitrary distance codes. In addition to extending flag error correction beyond distance-three codes for the first time, our protocol also applies to a broader class of distance-three codes than was previously known. Flag circuits use extra ancilla qubits to signal when errors resulting fromvfaults in the circuit have weight greater thanv. The flag error correction protocol is applicable to stabilizer codes of arbitrary distance which satisfy a set of conditions and uses fewer qubits than other schemes such as Shor, Steane and Knill error correction. We give examples of infinite code families which satisfy these conditions and analyze the behaviour of distance-three and -five examples numerically. Requiring fewer resources than Shor error correction, flag error correction could potentially be used in low-overhead fault-tolerant error correction protocols using low density parity check quantum codes of large code length.

scholarly journals A Simple Comparative Analysis of Exact and Approximate Quantum Error Correction

2014 ◽  
Vol 21 (03) ◽  
pp. 1450002 ◽  
Author(s):  
Carlo Cafaro ◽  
Peter van Loock
Keyword(s):  
Comparative Analysis ◽  
Error Correction ◽  
Quantum Error Correction ◽  
Quantum Codes ◽  
Similarities And Differences ◽  
Quantum Error ◽  
Noise Models ◽  
Recovery Schemes

We present a comparative analysis of exact and approximate quantum error correction by means of simple unabridged analytical computations. For the sake of clarity, using primitive quantum codes, we study the exact and approximate error correction of the two simplest unital (Pauli errors) and nonunital (non-Pauli errors) noise models, respectively. The similarities and differences between the two scenarios are stressed. In addition, the performances of quantum codes quantified by means of the entanglement fidelity for different recovery schemes are taken into consideration in the approximate case. Finally, the role of self-complementarity in approximate quantum error correction is briefly addressed.

scholarly journals Entropy of a Quantum Error Correction Code

2008 ◽  
Vol 15 (04) ◽  
pp. 329-343 ◽  
Author(s):  
David W. Kribs ◽  
Aron Pasieka ◽  
Karol Życzkowski
Keyword(s):  
Error Correction ◽  
Detailed Analysis ◽  
Quantum Channel ◽  
Error Correcting Codes ◽  
Quantum Error Correction ◽  
Maximal Entropy ◽  
Initial State ◽  
Error Correction Code ◽  
Quantum Error ◽  
Decoherence Free Subspaces

We define and investigate the notion of entropy for quantum error correcting codes. The entropy of a code for a given quantum channel has a number of equivalent realisations, such as through the coefficients associated with the Knill-Laflamme conditions and the entropy exchange computed with respect to any initial state supported on the code. In general the entropy of a code can be viewed as a measure of how close it is to the minimal entropy case, which is given by unitarily correctable codes (including decoherence-free subspaces), or the maximal entropy case, which from dynamical Choi matrix considerations corresponds to non-degenerate codes. We consider several examples, including a detailed analysis of the case of binary unitary channels, and we discuss an extension of the entropy to operator quantum error correcting subsystem codes.

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