scholarly journals Approximate quantum error correction, random codes, and quantum channel capacity

2007 ◽  
Vol 75 (6) ◽  
Author(s):  
Rochus Klesse
Keyword(s):  
Error Correction ◽  
Channel Capacity ◽  
Quantum Channel ◽  
Quantum Error Correction ◽  
Random Codes ◽  
Quantum Error

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.

scholarly journals Digital System Design for Quantum Error Correction Codes

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Othman O. Khalifa ◽  
Nur Amirah bt Sharif ◽  
Rashid A Saeed ◽  
S. Abdel-Khalek ◽  
Abdulaziz N. Alharbi ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
System Design ◽  
Error Correction ◽  
Quantum Channel ◽  
Quantum Error Correction ◽  
Digital System ◽  
Error Correction Codes ◽  
Error Correction Code ◽  
Digital System Design ◽  
Quantum Error

Quantum computing is a computer development technology that uses quantum mechanics to perform the operations of data and information. It is an advanced technology, yet the quantum channel is used to transmit the quantum information which is sensitive to the environment interaction. Quantum error correction is a hybrid between quantum mechanics and the classical theory of error-correcting codes that are concerned with the fundamental problem of communication, and/or information storage, in the presence of noise. The interruption made by the interaction makes transmission error during the quantum channel qubit. Hence, a quantum error correction code is needed to protect the qubit from errors that can be caused by decoherence and other quantum noise. In this paper, the digital system design of the quantum error correction code is discussed. Three designs used qubit codes, and nine-qubit codes were explained. The systems were designed and configured for encoding and decoding nine-qubit error correction codes. For comparison, a modified circuit is also designed by adding Hadamard gates.

A Random Coding Based Proof for the Quantum Coding Theorem

2008 ◽  
Vol 15 (01) ◽  
pp. 21-45 ◽  
Author(s):  
Rochus Klesse
Keyword(s):  
Information Transmission ◽  
Error Correction ◽  
Channel Coding ◽  
Quantum Channel ◽  
Quantum Error Correction ◽  
Random Coding ◽  
Quantum Coding ◽  
Quantum Error ◽  
Original Treatment

We present a proof for the quantum channel coding theorem which relies on the fact that a randomly chosen code space typically is highly suitable for quantum error correction. In this sense, the proof is close to Shannon's original treatment of information transmission via a noisy classical channel.

scholarly journals 2-designs and redundant syndrome extraction for quantum error correction

2021 ◽  
Vol 20 (3) ◽  
Author(s):  
Vickram N. Premakumar ◽  
Hele Sha ◽  
Daniel Crow ◽  
Eric Bach ◽  
Robert Joynt
Keyword(s):  
Error Correction ◽  
Quantum Error Correction ◽  
Quantum Error

scholarly journals Exponential suppression of bit or phase errors with cyclic error correction

Nature ◽  
2021 ◽  
Vol 595 (7867) ◽  
pp. 383-387
Author(s):  
◽  
Zijun Chen ◽  
Kevin J. Satzinger ◽  
Juan Atalaya ◽  
Alexander N. Korotkov ◽  
...  
Keyword(s):  
Error Correction ◽  
Error Detection ◽  
Fault Tolerant ◽  
Error Rates ◽  
Quantum Error Correction ◽  
Superconducting Qubits ◽  
Two Dimensional ◽  
Logical Error ◽  
Quantum Error ◽  
Logical Qubit

AbstractRealizing the potential of quantum computing requires sufficiently low logical error rates1. Many applications call for error rates as low as 10−15 (refs. 2–9), but state-of-the-art quantum platforms typically have physical error rates near 10−3 (refs. 10–14). Quantum error correction15–17 promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device18,19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.

scholarly journals Impact of correlations and heavy tails on quantum error correction

2021 ◽  
Vol 103 (5) ◽  
Author(s):  
B. D. Clader ◽  
Colin J. Trout ◽  
Jeff P. Barnes ◽  
Kevin Schultz ◽  
Gregory Quiroz ◽  
...  
Keyword(s):  
Error Correction ◽  
Heavy Tails ◽  
Quantum Error Correction ◽  
Quantum Error
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