#Quantum error code
GRAND is a universal decoder, enabling the decoding of any block code of moderate length or high code rate, when dealing with low entropy noise. Īt its core, GRAND is focused on guessing the noise that corrupted the transmitted codeword, rather than exhaustively going through all the possible codewords in order to find the one that fits the ML criterion, and is proven to still lead to ML decoding. Practical decoding of RLCs was only recently made possible with the advent of guessing random additive noise decoding (GRAND) algorithms, motivated by both power- and complexity-limited devices in wireless communications and the shifting of the focus to the finite-blocklength regime. Nonetheless, the decoding step remains prohibitively complex, limiting the real-world use of these codes. The approach taken by classical random linear codes (RLCs) is known to be capacity-achieving in the binary symmetric channel,, while solving the codewords’ storage problem, given that only the random generation matrix needs to be stored. Nonetheless, Shannon’s codes at too complex to be used in practice. Shannon’s random code-construction provably reaches capacity when the length of the codewords tends to infinity ( ) and the decoder picks, among all possible codewords, the most likely one, by applying maximum-likelihood (ML) decoding with the received observation and the codewords’ prior distribution.
This mapping should be random, in the sense that each of the codewords should be assigned to one of the possible words, and should be made according to a uniform distribution. The concept of error correction appeared in tandem with the one of channel capacity in the pioneering work of Claude Shannon, who proved that the capacity of a channel could be attained by using a uniform-at-random code that maps any bits of information onto codewords bits long. Quantum states are notoriously sensitive to noise, so quantum error correction must be employed when establishing entanglements.
Moreover, a number of applications, such as quantum-key distribution, distributed quantum computing, or the connection of different quantum sensors, require the existence of quantum networks and quantum repeaters in order to sustain end-to-end quantum connections between several devices. Implementing quantum computers and other devices bearing quantum memories will require quantum error correction of the stored quantum states. Quantum GRAND algorithm that makes use of quantum noise statistics, not only toīuild an adaptive code membership test, but also to efficiently implement Needed to reach the QRLCs' asymptotic performance, and subsequently proposes a The paper starts by assessing the minimum number of gates in the coding circuit Paper shows that decoding versatile quantum error correction is possible,Īllowing for QECCs that are simple to adapt on the fly to changing conditions. By combining QRLCs and a newly proposed quantum GRAND, this (QRLCs), which were known to be possible to construct but whose decoding was These conditions are particularly suitable for quantum systemsĪnd therefore the paper extends these concepts to quantum random linear codes Universal decoder for classical codes, providing there is a simple code Making use of the noise statistics, GRAND is a noise-centric efficient (RLCs) that perform near the capacity of the finite blocklength regime. Proposed strategy for decoding classical codes called GRAND (guessing randomĪdditive noise decoding) opened doors to decoding classical random linear codes This paper shows that it is possible to both construct and decode QECCs thatĬan attain the maximum performance of the finite blocklength regime, for anyĬhosen code length and when the code rate is sufficiently high. Lengths and code rates, limiting their adaptability to changing requirements. Practical quantum error correction codes, such as stabilizerĬodes, are generally structured to suit a specific use, and present rigid code Quantum error correction codes (QECCs) play a central role both in quantumĬommunications and in quantum computation, given how error-prone quantum
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