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arXiv:quant-ph/9610043 · Physical Review A

Bosonic quantum codes for amplitude damping

Isaac L. Chuang, Debbie Leung, Y. Yamamoto

Published Aug 1, 1997• 214 citations
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Abstract

Traditional quantum error correction involves the redundant encoding of $k$ quantum bits using $n$ quantum bits to allow the detection and correction of any $t$ bit error. The smallest general $t=1$ code requires $n=5$ for $k=1$. 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 $t=1$, $k=1$ code using effectively $n=4.6$. 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.

DOI
10.1103/physreva.56.1114

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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.
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.
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.
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.
We consider quantum error-correction codes for multimode bosonic systems, such as optical fields, that are affected by amplitude damping. We demonstrate that the most accessible method of transforming optical systems with the help of passive linear networks has limited usefulness in preparing and manipulating such codes. These limitations stem directly from the recoverability condition for one-photon loss. We introduce a three-photon code protecting against the first order of amplitude damping, i.e., a single photon loss, and discuss its preparation using linear optics with single-photon sources and conditional detection. Quantum state and process tomography in the code subspace can be implemented using passive linear optics and photon counting. An experimental proof-of-principle demonstration of elements of the proposed quantum error correction scheme for a one-photon erasure lies well within present technological capabilities.
We reexamine a non-Gaussian quantum error-correction code designed to protect optical coherent-state qubits against errors due to an amplitude-damping channel. We improve on a previous result [R. Wickert, N. K. Bernardes, and P. van Loock, Phys. Rev. A 81, 062344 (2010)] by providing a tighter upper bound on the performance attained when considering realistic assumptions, which constrain the operation of the gates employed in the scheme. The quantitative characterization is performed through measures of fidelity and concurrence, the latter obtained by employing the code as an entanglement distillation protocol. We find that, when running the code in fully deterministic error-correction mode, direct transmission can only be beaten for certain combinations of channel and input state parameters. In contrast, in error-detection mode, the usage of higher repetition encodings remains beneficial throughout, however, at the expense of diminishing success probabilities.