scholarly journals From Generalization of Bacon-Shor Codes to High Performance Quantum LDPC Codes

2021 ◽  
Author(s):  
Jihao Fan ◽  
Jun Li ◽  
Ya Wang ◽  
Yonghui Li ◽  
Min-Hsiu Hsieh ◽  
...  
Keyword(s):  
Minimum Distance ◽  
High Performance ◽  
Ldpc Codes ◽  
Quantum Error Correction ◽  
Concatenated Codes ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Distance Bound ◽  
Quantum Error ◽  
Minimum Distance Bound

Abstract We utilize a concatenation scheme to construct new families of quantum error correction codes that include the Bacon-Shor codes. We show that our scheme can lead to asymptotically good quantum codes while Bacon-Shor codes cannot. Further, the concatenation scheme allows us to derive quantum LDPC codes of distance Ω(N2/3/loglogN) which can improve Hastings’s recent result [arXiv:2102.10030] by a polylogarithmic factor. Moreover, assisted by the Evra-Kaufman- Zémor distance balancing construction, our concatenation scheme can yield quantum LDPC codes with non-vanishing code rates and better minimum distance upper bound than the hypergraph product quantum LDPC codes. Finally, we derive a family of fast encodable and decodable quan- tum concatenated codes with parameters Q = [[N,Ω(√N),Ω(√N)]] and they also belong to the Bacon-Shor codes. We show that Q can be encoded very efficiently by circuits of size O(N) and depth O(√N), and can correct any adversarial error of weight up to half the minimum distance bound in O(√N) time. To the best of our knowledge, they are the most powerful quantum codes for correcting so many adversarial errors in sublinear time by far.

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 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 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.

On the iterative decoding of sparse quantum codes

2008 ◽  
Vol 8 (10) ◽  
pp. 986-1000
Author(s):  
D. Poulin ◽  
Y. Chung
Keyword(s):  
Iterative Decoding ◽  
Belief Propagation ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Heuristic Methods ◽  
Tanner Graph ◽  
Coding Scheme ◽  
Propagation Algorithm ◽  
Belief Propagation Algorithm ◽  
Quantum Error

We address the problem of decoding sparse quantum error correction codes. For Pauli channels, this task can be accomplished by a version of the belief propagation algorithm used for decoding sparse classical codes. Quantum codes pose two new challenges however. Firstly, their Tanner graph unavoidably contain small loops which typically undermines the performance of belief propagation. Secondly, sparse quantum codes are by definition highly degenerate. The standard belief propagation algorithm does not exploit this feature, but rather it is impaired by it. We propose heuristic methods to improve belief propagation decoding, specifically targeted at these two problems. While our results exhibit a clear improvement due to the proposed heuristic methods, they also indicate that the main source of errors in the quantum coding scheme remains in the decoding.

On the iterative decoding of sparse quantum codes

2008 ◽  
Vol 8 (10) ◽  
pp. 986-1000
Author(s):  
D. Poulin ◽  
Y. Chung
Keyword(s):  
Iterative Decoding ◽  
Belief Propagation ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Heuristic Methods ◽  
Tanner Graph ◽  
Coding Scheme ◽  
Propagation Algorithm ◽  
Belief Propagation Algorithm ◽  
Quantum Error

We address the problem of decoding sparse quantum error correction codes. For Pauli channels, this task can be accomplished by a version of the belief propagation algorithm used for decoding sparse classical codes. Quantum codes pose two new challenges however. Firstly, their Tanner graph unavoidably contain small loops which typically undermines the performance of belief propagation. Secondly, sparse quantum codes are by definition highly degenerate. The standard belief propagation algorithm does not exploit this feature, but rather it is impaired by it. We propose heuristic methods to improve belief propagation decoding, specifically targeted at these two problems. While our results exhibit a clear improvement due to the proposed heuristic methods, they also indicate that the main source of errors in the quantum coding scheme remains in the decoding.

scholarly journals Approximating Decoherence Processes for the Design and Simulation of Quantum Error Correction Codes on Classical Computers

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 172623-172643
Author(s):  
Josu Etxezarreta Martinez ◽  
Patricio Fuentes ◽  
Pedro M. Crespo ◽  
J. Garcia-Frias
Keyword(s):  
Error Correction ◽  
Quantum Error Correction ◽  
Error Correction Codes ◽  
Quantum Error Correction Codes ◽  
Quantum Error

scholarly journals Decoding surface code with a distributed neural network–based decoder

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Savvas Varsamopoulos ◽  
Koen Bertels ◽  
Carmen G. Almudever
Keyword(s):  
Neural Network ◽  
Renormalization Group ◽  
Error Correction ◽  
Quantum Error Correction ◽  
Noise Model ◽  
Error Correction Codes ◽  
Exponential Increase ◽  
Main Challenge ◽  
Surface Code ◽  
Quantum Error

Abstract There has been a rise in decoding quantum error correction codes with neural network–based decoders, due to the good decoding performance achieved and adaptability to any noise model. However, the main challenge is scalability to larger code distances due to an exponential increase of the error syndrome space. Note that successfully decoding the surface code under realistic noise assumptions will limit the size of the code to less than 100 qubits with current neural network–based decoders. Such a problem can be tackled by a distributed way of decoding, similar to the renormalization group (RG) decoders. In this paper, we introduce a decoding algorithm that combines the concept of RG decoding and neural network–based decoders. We tested the decoding performance under depolarizing noise with noiseless error syndrome measurements for the rotated surface code and compared against the blossom algorithm and a neural network–based decoder. We show that a similar level of decoding performance can be achieved between all tested decoders while providing a solution to the scalability issues of neural network–based decoders.

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.

Sign in / Sign up

Export Citation Format

Share Document