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arXiv:1901.00984

Quantum Insertion-Deletion Channels

Leahy, Janet, Dave Touchette, Penghui Yao

Published Jan 4, 2019• 9 citations
arXivHTMLPDF

Abstract

We introduce a model of quantum insertion-deletion (insdel) channels. Insdel channels are meant to represent, for example, synchronization errors arising in data transmission. In the classical setting, they represent a strict generalization of the better-understood corruption error channels, and until recently, had mostly resisted effort toward a similar understanding as their corruption counterparts. They have received considerable attention in recent years. Very recently, Haeupler and Shahrasbi developed a framework, using what they call synchronisation strings, that allows one to turn insdel-type errors into corruption-type errors. These can then be handled by the use of standard error-correcting codes. We show that their framework can be extended to the quantum setting, providing a way to turn quantum insdel errors into quantum corruption errors, which can be handled with standard quantum error-correcting codes.

DOI
10.48550/arxiv.1901.00984

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We introduce a model of quantum insertion-deletion (insdel) channels. Insdel channels are meant to represent, for example, synchronization errors arising in data transmission. In the classical setting, they represent a strict generalization of the better-understood corruption error channels, and until recently, had mostly resisted effort toward a similar understanding as their corruption counterparts. They have received considerable attention in recent years. Very recently, Haeupler and Shahrasbi developed a framework, using what they call synchronisation strings, that allows one to turn insdel-type errors into corruption-type errors. These can then be handled by the use of standard error-correcting codes. We show that their framework can be extended to the quantum setting, providing a way to turn quantum insdel errors into quantum corruption errors, which can be handled with standard quantum error-correcting codes.
Quantum synchronisation errors are a class of errors that change the number of qubits in a quantum system. The classical error correction of synchronisation errors has been well-studied, including an insertion-deletion equivalence more than a half-century ago, but little progress has been made towards the quantum counterpart since the birth of quantum error correction. We address the longstanding problem of a quantum insertion-deletion equivalence on permutation-invariant codes, detailing the conditions under which such codes are ttt-insertion error-correctable. We extend these conditions to quantum insertion-deletion errors, formulating a more restrictive set of conditions under which permutation-invariant codes are (t,s)(t,s)(t,s)-insdel error-correctable. For completeness, we extend our approach to qudit errors on qudit permutation-invariant codes. Our work resolves many of the outstanding questions regarding the quantum error correction of synchronisation errors.
In recent years, significant progress has been made in the research on errors in quantum deletion/insertion channels along with the development of error-correcting codes. However, the current channel model may not be applicable in specific scenarios, such as when using coherent-state qubits as funda-mental qubits. Here, we materialize the conventional error model of synchronization errors to accommodate coherent-state qubits and their superposition states. Additionally, we quantitatively assess the discrepancies between the conventional error model and our more realistic approach, underscoring the limitations of the former in practical applications.
Insertion-deletion (insdel for short) codes have received extensive attention due to their ability to correct synchronization errors. It is usually a very challenging problem to determine the insdel distances of linear codes. In this paper, a good lower bound on the insdel distance of a linear code with a certain algebraic structure is provided and it indeed gives an affirmative answer to an open problem proposed by Hao Chen (IEEE Transactions on Information Theory, 68(8): 5126–5132, 2022). Applying this lower bound to binary first-order Reed-Muller codes and binary simplex codes, we obtain that they possess linear subcodes that can correct arbitrary insdel errors while maintaining Hamming distances robustness. The application of this bound on Reed-Muller codes completely solves an open problem left by Lara Dolecek and Venkat Anantharam (IEEE Transactions on Information Theory, 53(4): 1430–1443, 2007). In order to enhance the code rate, we further perform puncturing on these subcodes and determine the Hamming distances of the punctured subcodes while ensuring that their insdel distances remain unchanged.
It is conjectured that a quantum code CCC corrects ttt deletion errors if and only if CCC corrects a total of ttt insertion and deletion errors. Although this conjecture has been proved only under certain conditions, it has not been proved in general. In this paper, we prove this conjecture for general quantum codes. Specifically, we show that a quantum state after an insertion error is represented as a product state of the original multi-qudit system and the inserted quantum state. Then, we prove this conjecture by combining the previous results and this property. In addition, we define generalized insertion errors by removing the quantum channel requirement from insertion errors and present their properties.
Quantum deletions, which are harder to correct than erasure errors, occur in many realistic settings. It is therefore pertinent to develop quantum coding schemes for quantum deletion channels. To date, not much is known about which explicit quantum error correction codes can combat quantum deletions. We note that any permutation-invariant quantum code that has a distance oft+1t+1t+1can correcttttquantum deletions for any positive integertttin both the qubit and the qudit setting. Leveraging on coding properties of permutation-invariant quantum codes under erasure errors, we derive corresponding coding bounds for permutation-invariant quantum codes under quantum deletions. We focus our attention on a specific family ofNNN-qubit permutation-invariant quantum codes, which we call shifted gnu codes. The main result of this work is that their encoding and decoding algorithms can be performed inO(N)O(N)O(N)andO(N2)O(N^{2})O(N2).
This manuscript is an extended abstract version of the paper entitled ``Quantum Deletion Codes derived from Classical Deletion Codes.'' The paper contributes to the fundamental theory for quantum deletion error-correcting codes. The paper proposes a code construction condition for a partition of classical deletion error-correcting codes to derive quantum deletion error-correcting codes. The construction methods in this paper give examples of quantum codes that can correct single-quantum deletion errors and have a code rate arbitrarily close to 1, while the previously known quantum deletion code rates are close to 0 for long length. This manuscript omits the proofs of the statements in the paper.