The security of code-based cryptosystems such as the McEliece cryptosystem relies primarily on the difficulty of decoding random linear codes. The best decoding algorithms are all improvements of an old algorithm due to Prange: they are known under the name of information set decoding techniques. It is also important to assess the security of such cryptosystems against a quantum computer. This research thread started in Overbeck and Sendrier's 2009 survey on code-based cryptography, and the best algorithm to date has been Bernstein's quantising of the simplest information set decoding algorithm, namely Prange's algorithm. It consists in applying Grover's quantum search to obtain a quadratic speed-up of Prange's algorithm. In this paper, we quantise other information set decoding algorithms by using quantum walk techniques which were devised for the subset-sum problem by Bernstein, Jeffery, Lange and Meurer. This results in improving the worst-case complexity of $2^{0.06035n}$ of Bernstein's algorithm to $2^{0.05869n}$ with the best algorithm presented here (where $n$ is the codelength).The security of code-based cryptosystems such as the McEliece cryptosystem relies primarily on the difficulty of decoding random linear codes. The best decoding algorithms are all improvements of an old algorithm due to Prange: they are known under the name of information set decoding techniques. It is also important to assess the security of such cryptosystems against a quantum computer. This research thread started in Overbeck and Sendrier's 2009 survey on code-based cryptography, and the best algorithm to date has been Bernstein's quantising of the simplest information set decoding algorithm, namely Prange's algorithm. It consists in applying Grover's quantum search to obtain a quadratic speed-up of Prange's algorithm. In this paper, we quantise other information set decoding algorithms by using quantum walk techniques which were devised for the subset-sum problem by Bernstein, Jeffery, Lange and Meurer. This results in improving the worst-case complexity of $2^{0.06035n}$ of Bernstein's algorithm to $2^{0.05869n}$ with the best algorithm presented here (where $n$ is the codelength).

arXiv:1703.00263Mar 1, 201711 citations

Advantage of Quantum Neural Networks as Quantum Information Decoders

Weishun Zhong, Oles Shtanko, Ramis Movassagh

A promising strategy to protect quantum information from noise-induced errors is to encode it into the low-energy states of a topological quantum memory device. However, readout errors from such memory under realistic settings is less understood. We study the problem of decoding quantum information encoded in the groundspaces of topological stabilizer Hamiltonians in the presence of generic perturbations, such as quenched disorder. We first prove that the standard stabilizer-based error correction and decoding schemes work adequately well in such perturbed quantum codes by showing that the decoding error diminishes exponentially in the distance of the underlying unperturbed code. We then prove that Quantum Neural Network (QNN) decoders provide an almost quadratic improvement on the readout error. Thus, we demonstrate provable advantage of using QNNs for decoding realistic quantum error-correcting codes, and our result enables the exploration of a wider range of non-stabilizer codes in the near-term laboratory settings.A promising strategy to protect quantum information from noise-induced errors is to encode it into the low-energy states of a topological quantum memory device. However, readout errors from such memory under realistic settings is less understood. We study the problem of decoding quantum information encoded in the groundspaces of topological stabilizer Hamiltonians in the presence of generic perturbations, such as quenched disorder. We first prove that the standard stabilizer-based error correction and decoding schemes work adequately well in such perturbed quantum codes by showing that the decoding error diminishes exponentially in the distance of the underlying unperturbed code. We then prove that Quantum Neural Network (QNN) decoders provide an almost quadratic improvement on the readout error. Thus, we demonstrate provable advantage of using QNNs for decoding realistic quantum error-correcting codes, and our result enables the exploration of a wider range of non-stabilizer codes in the near-term laboratory settings.

arXiv:2401.06300Jan 11, 20240 citations

2^{0.06035n}

of Bernstein's algorithm to

2^{0.05869n}

with the best algorithm presented here (where

n

is the codelength).