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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Single Error Correcting Quantum Codes for the Amplitude Damping Channel Based on Classical Codes over GF(7)

Keisuke Kodaira, Mihoko Wada, Tomoharu Shibuya

Published Jan 1, 2014• 0 citations
Publisher

Abstract

The amplitude damping (AD) quantum channel is one of the models describing evolution of quantum states. The construction of quantum error correcting codes for the AD channel based on classical codes has been presented, and Shor et al. proposed a class of classical codes over F3 which are efficiently applicable to this construction. In this study, we expand Shor's construction to that over F7, and succeeded to construct an AD code that has better parameters than AD codes constructed by Shor et al.

DOI
10.1587/transfun.e97.a.2247

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The amplitude damping (AD) quantum channel is one of the models describing evolution of quantum states. The construction of quantum error correcting codes for the AD channel based on classical codes has been presented, and Shor et al. proposed a class of classical codes over F3 which are efficiently applicable to this construction. In this study, we expand Shor's construction to that over F7, and succeeded to construct an AD code that has better parameters than AD codes constructed by Shor et al.
We construct families of high performance quantum amplitude damping codes. All of our codes are nonadditive and most modestly outperform the best possible additive codes in terms of encoded dimension. One family is built from nonlinear error-correcting codes for classical asymmetric channels, with which we systematically construct quantum amplitude damping codes with parameters better than any prior construction known for any block lengthn≥ 8 exceptn=2r-1. We generalize this construction to employ classical codes overGF(3) with which we numerically obtain better performing codes up to length 14. Because the resulting codes are of the codeword stabilized (CWS) type, conceptually simple (though potentially computationally expensive) encoding and decoding circuits are available.
Traditional quantum error-correcting codes are designed for the depolarizing channel modeled by generalized Pauli errors occurring with equal probability. Amplitude damping channels, in general, model the decay process of a multilevel atom or energy dissipation of a bosonic system at zero temperature. We discuss quantum error-correcting codes adapted to amplitude damping channels for higher dimensional systems (qudits). For multi-level atoms, we consider a natural kind of decay process, and for bosonic systems, we consider the qudit amplitude damping channel obtained by truncating the Fock basis of the bosonic modes to a certain maximum occupation number. We construct families of single-error-correcting quantum codes that can be used for both cases. Our codes have larger code dimensions than the previously known single-error-correcting codes of the same lengths.
We introduce a family of high-rate amplitude-damping (AD) Shor codes, designed to effectively correct AD errors while maintaining immunity to collective coherent (CC) errors. The proposed [[(w+1)(w+K),K]][[(w+1)(w+K),K]][[(w+1)(w+K),K]] AD codes can approximately correct up to www AD errors, with flexible parameters (w,K)(w,K)(w,K), and we provide a rigorous proof that these codes satisfy the approximate quantum error-correction conditions. These codes leverage structured stabilizer measurements, enabling efficient detection of AD errors using only local operations and ancillary qubits. We further construct a family of CC-AD Shor codes by concatenating these AD codes with the dual-rail code.
We construct new families of multi-error-correcting quantum codes for the amplitude damping channel. Our key observation is that, with proper encoding, two uses of the amplitude damping channel simulate a quantum erasure channel. This allows us to use concatenated codes with quantum erasure-correcting codes as outer codes for correcting multiple amplitude damping errors. Our new codes are degenerate stabilizer codes and have parameters which are better than the amplitude damping codes obtained by any previously known construction.
Traditional quantum error-correcting codes are designed for the depolarizing channel modeled by generalized Pauli errors occurring with equal probability. Amplitude damping channels model, in general, the decay process of a multilevel atom or energy dissipation of a bosonic system with Markovian bath at zero temperature. We discuss quantum error-correcting codes adapted to amplitude damping channels for higher dimensional systems (qudits). For multi-level atoms, we consider a natural kind of decay process, and for bosonic systems, we consider the qudit amplitude damping channel obtained by truncating the Fock basis of the bosonic modes (e.g., the number of photons) to a certain maximum occupation number. We construct families of single-error-correcting quantum codes that can be used for both cases. Our codes have larger code dimensions than the previously known single-error-correcting codes of the same lengths. In addition, we present families of multi-error correcting codes for these two channels, as well as generalizations of our construction technique to error-correcting codes for the qutrit V and Λ channels.
We describe the smallest quantum error correcting (QEC) code to correct for amplitude-damping (AD) noise, namely, a three-qubit code that corrects all the single-qubit damping errors. We generalize this construction to a family of codes that correct AD noise up to any fixed order of the damping strength. We underpin the fundamental connection between the structure of our codes and the noise structure, via a relaxed form of the Knill-Laflamme conditions, different from existing formulations of approximate QEC conditions. Although the recovery procedure for this code is nondeterministic, our codes are optimal with respect to overheads and outperform existing codes to tackle AD noise in terms of entanglement fidelity. This formulation of probabilistic QEC further leads us to a family of quantum codes tailored to AD noise and also gives rise to a noise-adapted quantum Hamming bound for AD noise. Finally, we construct a set of universal logical gates for the three-qubit code, thus providing a potential pathway to fault tolerance via this class of codes.