Quantum error correction nested codes derived from jacket transform

2019 ◽  
Vol 33 (31) ◽  
pp. 1950378
Author(s):  
Yuan Li ◽  
Niansheng Chen ◽  
Yiyuan Luo
Keyword(s):  
Error Correction ◽  
Decomposition Methods ◽  
Quantum Error Correction ◽  
Quantum Codes ◽  
Arbitrary Size ◽  
Novel Approach ◽  
Nested Codes ◽  
Quantum Error

A novel approach for construction of quantum codes employing jacket transform is proposed in this paper. The constructed quantum nested codes has less complexities than the previous quantum codes in encoding procedures as well as decoding procedures. Furthermore, the quantum error codes can be constructed with an arbitrary size using the number decomposition, and the corresponding distance of constructed quantum codes can be adjusted by changing decomposition methods.

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 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 From Quantum Codes to Gravity: A Journey of Gravitizing Quantum Mechanics

Universe ◽  
2021 ◽  
Vol 8 (1) ◽  
pp. 1
Author(s):  
Chun-Jun Cao
Keyword(s):  
Quantum Mechanics ◽  
Quantum Gravity ◽  
Error Correction ◽  
Quantum Error Correction ◽  
Quantum States ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Recent Approach ◽  
Tensor Network ◽  
Quantum Error

In this note, I review a recent approach to quantum gravity that “gravitizes” quantum mechanics by emerging geometry and gravity from complex quantum states. Drawing further insights from tensor network toy models in AdS/CFT, I propose that approximate quantum error correction codes, when re-adapted into the aforementioned framework, also have promise in emerging gravity in near-flat geometries.

scholarly journals New perspectives on covariant quantum error correction

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 521
Author(s):  
Sisi Zhou ◽  
Zi-Wen Liu ◽  
Liang Jiang
Keyword(s):  
Error Correction ◽  
Lower Bounds ◽  
Fault Tolerant ◽  
Symmetry Transformation ◽  
Substantial Improvement ◽  
Quantum Error Correction ◽  
Quantum Codes ◽  
Covariant Quantum ◽  
Continuous Symmetries ◽  
Quantum Error

Covariant codes are quantum codes such that a symmetry transformation on the logical system could be realized by a symmetry transformation on the physical system, usually with limited capability of performing quantum error correction (an important case being the Eastin–Knill theorem). The need for understanding the limits of covariant quantum error correction arises in various realms of physics including fault-tolerant quantum computation, condensed matter physics and quantum gravity. Here, we explore covariant quantum error correction with respect to continuous symmetries from the perspectives of quantum metrology and quantum resource theory, establishing solid connections between these formerly disparate fields. We prove new and powerful lower bounds on the infidelity of covariant quantum error correction, which not only extend the scope of previous no-go results but also provide a substantial improvement over existing bounds. Explicit lower bounds are derived for both erasure and depolarizing noises. We also present a type of covariant codes which nearly saturates these lower bounds.

scholarly journals Key ideas in quantum error correction

2012 ◽  
Vol 370 (1975) ◽  
pp. 4541-4565 ◽  
Author(s):  
Robert Raussendorf
Keyword(s):  
Quantum Information ◽  
Error Correction ◽  
Quantum Computation ◽  
Fault Tolerant ◽  
Quantum Error Correction ◽  
Quantum Gates ◽  
Quantum Codes ◽  
Introductory Article ◽  
The Subject ◽  
Quantum Error

In this introductory article on the subject of quantum error correction and fault-tolerant quantum computation, we review three important ingredients that enter known constructions for fault-tolerant quantum computation, namely quantum codes, error discretization and transversal quantum gates. Taken together, they provide a ground on which the theory of quantum error correction can be developed and fault-tolerant quantum information protocols can be built.

scholarly journals Fast Constructions of Quantum Codes Based on Residues Pauli Block Matrices

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Ying Guo ◽  
Guihu Zeng ◽  
MoonHo Lee
Keyword(s):  
Error Correction ◽  
Abelian Groups ◽  
Quantum Error Correction ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Block Matrices ◽  
Quadratic Residues ◽  
Quantum Error Correction Codes ◽  
Block Transforms ◽  
Quantum Error

We demonstrate how to fast construct quantum error-correction codes based on quadratic residues Pauli block transforms. The present quantum codes have an advantage of being fast designed from Abelian groups on the basis of Pauli block matrices that can be yielded from quadratic residues with much efficiency.

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.

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