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NATO science for peace and security series. D, Information and communication security

Quantum jump codes and related combinatorial designs

Masakazu Jimbo, Keisuke Shiromoto

Published Jan 1, 2011• 10 citations
Publisher

Abstract

Quantum jump codes were introduced by Alber et al. (2001). Quantum jump codes have a close connection with combinatorial designs called t-SEED (t-spontaneaus emission error design). In this paper, we give a brief survey of quantum jump codes together with some new results. Firstly, fundamental properties of a t-error correcting quantum jump code are described. Secondly, a few examples of jump codes are given and an upper bound on the dimension of a jump code with a fixed length and given error correcting ability is derived. A relation between a t-SEED and a jump code is discussed and various constructions of t-SEEDs are given.

DOI
10.3233/978-1-60750-663-8-285

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Quantum jump codes were introduced by Alber et al. (2001). Quantum jump codes have a close connection with combinatorial designs called t-SEED (t-spontaneaus emission error design). In this paper, we give a brief survey of quantum jump codes together with some new results. Firstly, fundamental properties of a t-error correcting quantum jump code are described. Secondly, a few examples of jump codes are given and an upper bound on the dimension of a jump code with a fixed length and given error correcting ability is derived. A relation between a t-SEED and a jump code is discussed and various constructions of t-SEEDs are given.
A new class of embedded error correcting quantum codes is discussed. These detected-jump correcting quantum codes are capable of stabilizing distinguishable qubits against spontaneous decay. Due to their low redundancy and due to the small number of control measurements and recovery operations required these new error-correcting quantum codes offer interesting perspectives for quantum computation. In this contribution main ideas underlying these embedded error correcting quantum codes and their links to fundamental concepts of combinatorial design theory are discussed. In addition, possibilities are explored for implementing universal quantum gates within these code spaces.
The recently introduced detected-jump-correcting quantum codes are capable of stabilizing qubit systems against spontaneous decay processes arising from couplings to statistically independent reservoirs. These embedded quantum codes exploit classical information about which qubit has emitted spontaneously and correspond to an active error-correcting code embedded in a passive error-correcting code. The construction of a family of one-detected-jump-error-correcting quantum codes is shown and the optimal redundancy, encoding, and recovery as well as general properties of detected-jump-error-correcting quantum codes are discussed. By the use of design theory, multiple-jump-error-correcting quantum codes can be constructed. The performance of one-jump-error-correcting quantum codes under nonideal conditions is studied numerically by simulating a quantum memory and Grover's algorithm.
We explore the design of quantum error-correcting codes for cases where the decoherence events of qubits are correlated. In particular, we consider the case where only spatially contiguous qubits decohere, which is analogous to the case of burst errors in classical coding theory. We present several different efficient schemes for constructing families of such codes. For example, one can find one-dimensional quantum codes of length n=13 and 15 that correct burst errors of width b<3; as a comparison, a random-error correcting quantum code that corrects t=3 errors must have length n/spl ges/17. In general, we show that it is possible to build quantum burst-correcting codes that have near optimal dimension. For example, we show that for any constant b, there exist b-burst-correcting quantum codes with length n, and dimension k=n-log n-O(1); as a comparison, the Hamming bound for the case with t (constant) random errors yields k/spl les/n-tlog n+O(1).
Suppressing the effect of decoherence plays a vital role in the theory of quantum information processing.Despite many skepticisms about its feasibility, the existence of error correction schemes in the quantum do- main was proved in the last century, giving the birth to the field of quantum error correction.Since then, the new field has seen remarkably rapid progress in various directions including experimental realizations of quantum error-correcting codes.While quantum information science draws on many branches of physics, computer science, and mathematics, combinatorics is rapidly becoming an indispensable mathematical tool to the study of quantum error correction.In this note, we review some of the latest developments in quantum error correction where combinatorial design theory and coding theory play the central role.
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.