The Quantum Decoding Problem

• URL: http://arxiv.org/abs/2310.20651v1
• Date: Tue, 31 Oct 2023 17:21:32 GMT
• Title: The Quantum Decoding Problem
• Authors: Andr\'e Chailloux, Jean-Pierre Tillich
• Abstract summary: We consider the quantum decoding problem, where we are given a superposition of noisy versions of a codeword. We show that when the noise rate is small enough, then the quantum decoding problem can be solved in quantum time. We also show that the problem can in principle be solved quantumly for noise rates for which the associated classical decoding problem cannot be solved.
• Score: 0.23310087539224286
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: One of the founding results of lattice based cryptography is a quantum reduction from the Short Integer Solution problem to the Learning with Errors problem introduced by Regev. It has recently been pointed out by Chen, Liu and Zhandry that this reduction can be made more powerful by replacing the learning with errors problem with a quantum equivalent, where the errors are given in quantum superposition. In the context of codes, this can be adapted to a reduction from finding short codewords to a quantum decoding problem for random linear codes. We therefore consider in this paper the quantum decoding problem, where we are given a superposition of noisy versions of a codeword and we want to recover the corresponding codeword. When we measure the superposition, we get back the usual classical decoding problem for which the best known algorithms are in the constant rate and error-rate regime exponential in the codelength. However, we will show here that when the noise rate is small enough, then the quantum decoding problem can be solved in quantum polynomial time. Moreover, we also show that the problem can in principle be solved quantumly (albeit not efficiently) for noise rates for which the associated classical decoding problem cannot be solved at all for information theoretic reasons. We then revisit Regev's reduction in the context of codes. We show that using our algorithms for the quantum decoding problem in Regev's reduction matches the best known quantum algorithms for the short codeword problem. This shows in some sense the tightness of Regev's reduction when considering the quantum decoding problem and also paves the way for new quantum algorithms for the short codeword problem.

Related papers

• Quantum advantage from soft decoders [0.7728149002473877]
  We provide improvements for some instantiations of the Optimal Polynomial Interpolation (OPI) problem. Our results provide natural and convincing decoding problems for which we believe to have a quantum advantage. In order to be able to use a decoder in the setting of Regev's reduction, we provide a novel reduction from a syndrome to a coset sampling problem.
  arXiv  Detail & Related papers  (2024-11-19T15:12:03Z)
• Limitations of the decoding-to-LPN reduction via code smoothing [59.90381090395222]
  The Learning Parity with Noise (LPN) problem underlies several classic cryptographic primitives. This paper attempts to find a reduction from the decoding problem of linear codes, for which several hardness results exist. We characterize the efficiency of the reduction in terms of the parameters of the decoding and problems.
  arXiv  Detail & Related papers  (2024-08-07T12:54:43Z)
• Breadth-first graph traversal union-find decoder [0.0]
  We develop variants of the union-find decoder that simplify its implementation and provide potential decoding speed advantages. We show how these methods can be adapted to decode non-topological quantum low-density-parity-check codes.
  arXiv  Detail & Related papers  (2024-07-22T18:54:45Z)
• Deep Quantum Error Correction [73.54643419792453]
  Quantum error correction codes (QECC) are a key component for realizing the potential of quantum computing. In this work, we efficiently train novel emphend-to-end deep quantum error decoders. The proposed method demonstrates the power of neural decoders for QECC by achieving state-of-the-art accuracy.
  arXiv  Detail & Related papers  (2023-01-27T08:16:26Z)
• Quantum Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
  We study the problem of designing worst-case to average-case reductions for quantum algorithms. We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
  arXiv  Detail & Related papers  (2022-12-06T22:01:49Z)
• Quantum Error Correction via Noise Guessing Decoding [0.0]
  Quantum error correction codes (QECCs) play a central role in both quantum communications and quantum computation. This paper shows that it is possible to both construct and decode QECCs that can attain the maximum performance of the finite blocklength regime.
  arXiv  Detail & Related papers  (2022-08-04T16:18:20Z)
• Entanglement and coherence in Bernstein-Vazirani algorithm [58.720142291102135]
  Bernstein-Vazirani algorithm allows one to determine a bit string encoded into an oracle. We analyze in detail the quantum resources in the Bernstein-Vazirani algorithm. We show that in the absence of entanglement, the performance of the algorithm is directly related to the amount of quantum coherence in the initial state.
  arXiv  Detail & Related papers  (2022-05-26T20:32:36Z)
• Space-efficient binary optimization for variational computing [68.8204255655161]
  We show that it is possible to greatly reduce the number of qubits needed for the Traveling Salesman Problem. We also propose encoding schemes which smoothly interpolate between the qubit-efficient and the circuit depth-efficient models.
  arXiv  Detail & Related papers  (2020-09-15T18:17:27Z)
• Fault-tolerant Coding for Quantum Communication [71.206200318454]
  encode and decode circuits to reliably send messages over many uses of a noisy channel. For every quantum channel $T$ and every $eps>0$ there exists a threshold $p(epsilon,T)$ for the gate error probability below which rates larger than $C-epsilon$ are fault-tolerantly achievable. Our results are relevant in communication over large distances, and also on-chip, where distant parts of a quantum computer might need to communicate under higher levels of noise.
  arXiv  Detail & Related papers  (2020-09-15T15:10:50Z)
• Efficiently computing logical noise in quantum error correcting codes [0.0]
  We show that measurement errors on readout qubits manifest as a renormalization on the effective logical noise. We derive general methods for reducing the computational complexity of the exact effective logical noise by many orders of magnitude.
  arXiv  Detail & Related papers  (2020-03-23T19:40:56Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.