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arXiv:quant-ph/9804014 · Physical Review A

Optimal quantum codes for preventing collective amplitude damping

L.-M. Duan, Guang‐Can Guo

Published Nov 1, 1998• 62 citations
arXivHTMLPDF

Abstract

Collective decoherence is possible if the distance between quantum bits is smaller than the effective wavelength of the noise field. Collectivity in the decoherence helps us to devise more efficient quantum codes. We present a class of optimal quantum codes for preventing collective amplitude damping to a reservoir at zero temperature. It is shown that two quantum bits (qubits) are enough to protect one bit quantum information, and approximately $L+\frac{1}{2}{\mathrm{ln}}_{2}(\ensuremath{\pi}L/2)$ qubits are enough to protect L-qubit information when L is large. For preventing collective amplitude damping, these codes are much more efficient than the previously discovered quantum error correcting or avoiding codes.

DOI
10.1103/physreva.58.3491

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Collective decoherence is possible if the distance between quantum bits is smaller than the effective wavelength of the noise field. Collectivity in the decoherence helps us to devise more efficient quantum codes. We present a class of optimal quantum codes for preventing collective amplitude damping to a reservoir at zero temperature. It is shown that two quantum bits (qubits) are enough to protect one bit quantum information, and approximately L+12ln2(\ensuremathπL/2)L+\frac{1}{2}{\mathrm{ln}}_{2}(\ensuremath{\pi}L/2)L+21​ln2​(\ensuremathπL/2) qubits are enough to protect L-qubit information when L is large. For preventing collective amplitude damping, these codes are much more efficient than the previously discovered quantum error correcting or avoiding codes.
A scheme is presented for protecting one-qubit quantum information against decoherence due to a general environment and local exchange interactions. The scheme operates essentially by distributing information over two pairs of qubits and through error-prevention procedures. In the scheme, quantum information is encoded through a decoherence-free subspace for collective phase errors and exchange errors affecting the qubits in pairs; leakage out of the encoding space due to amplitude damping is reduced by quantum Zeno effect. In addition, how to construct decoherence-free states for n-qubit information against phase and exchange errors is discussed.
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 correcting (QEC) plays a crucial role in protecting quantum information against decoherence and enabling scalable, reliable quantum computing. One of the most realistic and ubiquitous sources of noise affecting quantum hardware today is generalized amplitude-damping (GAD) noise. Conventional, deterministic QEC codes struggle to correct for GAD noise because of their inherent structure, leading to fidelity losses that scale linearly with the damping strength. In this work, we introduce the framework of probabilistic approximate quantum error correction (PAQEC), that combines the flexibility of approximate QEC with the potential of post-selected recovery, enabling high-fidelity, resource-efficient error correction. We construct a five-qubit permutation-invariant code that, under probabilistic recovery, achieves a fidelity loss quadratic in the damping strength, thus outperforming existing QEC codes. Formulating PAQEC as an optimization problem, we present a numerical technique based on Charnes-Cooper and semidefinite programming to identify the optimal recovery map for any PAQEC code. Our results establish PAQEC as a powerful tool for developing resource-efficient, high-fidelity quantum codes tailored to realistic noise, with promising implications for near-term quantum devices and future fault-tolerant architectures.
Decoherence is a fundamental obstacle to the implementation of large-scale and low-noise quantum information-processing devices. We suggest an approach for suppressing errors by employing preprocessing and postprocessing unitary operations, which precede and follow the action of a decoherence channel. In contrast to quantum error correction and measurement-based methods, the suggested approach relies on specifically designed unitary operators for a particular state without the need in ancillary qubits or postselection procedures. We consider the case of decoherence channels acting on a single qubit belonging to a many-qubit state. Preprocessing and postprocessing operators can be either individual, which is acting on the qubit effected by the decoherence channel only, or collective, which is acting on the whole multiqubit state. We give a classification of possible strategies for the protection scheme, analyze them, and derive expressions for the optimal unitary operators providing the maximal value of the fidelity regarding initial and final states. Specifically, we demonstrate the equivalence of the schemes where one of the unitary operations is individual while the other is collective. We then consider the realization of our approach for the basic decoherence models, which include single-qubit depolarizing, dephasing, and amplitude damping channels. We also demonstrate that the decoherence robustness of multiqubit states for these decoherence models is determined by the entropy of the reduced state of the qubit undergoing the decoherence channel.
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 show a way to correct collective error through a three-qubit quantum code. The encoding and decoding (i.e., error recovery) operations for such a code, which protects one qubit of quantum information perfectly (with infinite ``distance'') against collective decoherence, are presented. The code is actually a manifestation of a noiseless subsystem first reported by Knill, Laflamme, and Viola [Phys. Rev. Lett. 84, 2525 (2000)], and provides a good illustration of the relationship between noiseless subsystems and error-correcting codes already noted by these authors. We also show in detail how this code can be used to preserve an arbitrary joint state (including entangled states) of two qubits, by pairing each of them with two other qubits, even in the case in which the two resulting clusters interact in nonequivalent ways with a common environment.