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arXiv:quant-ph/9704002 · Physical Review A

Approximate quantum error correction can lead to better codes

Debbie Leung, Michael A. Nielsen, Isaac L. Chuang, Y. Yamamoto

Published Oct 1, 1997• 214 citations
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Abstract

We present relaxed criteria for quantum error correction that are useful when the specific dominant quantum noise process is known. As an example, we provide a four-bit code that corrects for a single amplitude damping error. This code violates the usual Hamming bound calculated for a Pauli description of the error process and has no simple explanation in terms of the usual Pauli basis GF(4) codes.

DOI
10.1103/physreva.56.2567

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We present relaxed criteria for quantum error correction that are useful when the specific dominant quantum noise process is known. As an example, we provide a four-bit code that corrects for a single amplitude damping error. This code violates the usual Hamming bound calculated for a Pauli description of the error process and has no simple explanation in terms of the usual Pauli basis GF(4) codes.
Traditional quantum error correction involves the redundant encoding of kkk quantum bits using nnn quantum bits to allow the detection and correction of any ttt bit error. The smallest general t=1t=1t=1 code requires n=5n=5n=5 for k=1k=1k=1. However, the dominant error process in a physical system is often well known, thus inviting the following question: Given a specific error model, can more efficient codes be devised? We demonstrate alternative codes that correct just amplitude damping errors that allow, for example, a t=1t=1t=1, k=1k=1k=1 code using effectively n=4.6n=4.6n=4.6. Our scheme is based on using bosonic states of photons in a finite number of optical modes. We present necessary and sufficient conditions for the codes and describe construction algorithms, physical implementation, and performance bounds.
Quantum error correction (QEC) plays a critical role in preventing information loss in quantum systems and provides a framework for reliable quantum computation. Identifying quantum codes with nice code parameters for physically motivated noise models remains an interesting challenge. While past work has primarily focused on qubit codes, here we identify a [4,1] qudit error correcting code tailored to protect against amplitude-damping noise. We show that this four-qudit code satisfies the error correction conditions for all single-qudit and a few two-qudit damping errors up to the leading order in the damping parameter $\ensuremath{\gamma}$. We devise a protocol to extract syndromes that unambiguously identify this set of errors, leading to a noise-adapted recovery scheme that achieves a fidelity loss of $O({\ensuremath{\gamma}}^{2})$. For the d=2d=2d=2 case, our QEC scheme is identical to the known example of the four-qubit code and the associated syndrome-based recovery. We also assess the performance of this code using the Petz recovery map and note some interesting deviations from the qubit case. Finally, we generalize this construction to a family of [2M+2,M][2M+2,M][2M+2,M] qudit codes that can approximately correct all the single-qudit and a few two-qudit amplitude-damping errors.
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 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.
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.