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arXiv:1509.06829 · IEEE Transactions on Information Theory

Quantum Error-Correcting Codes for Qudit Amplitude Damping

Markus Grassl, Linghang Kong, Zhaohui Wei, Zhang‐qi Yin, Bei Zeng

Published Jan 8, 2018• 56 citations
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Abstract

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.

DOI
10.1109/tit.2018.2790423

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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.
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 study the error-correction properties of multi-mode Fock-state codes under amplitude-damping (AD) noise, focusing on the asymptotic regime in which the total excitation of the code states grows without limit and the number of photon losses induced by the noise scales linearly with it. In this setting, existing code families, which correct only sublinearly many photon losses, do not protect against amplitude-damping (AD) noise with a constant loss parameter. As our main result, we address this gap by constructing asymptotically good Fock-state codes relying on random classical codes in the discrete simplex. Unlike many standard constructions of random quantum codes, our construction introduces randomness through the underlying classical indexing structure. Randomization also enables another desirable feature: bounded per-mode occupancy, which limits the number of photons in any individual mode and thereby increases the coherence lifetime of the code states. Finally, via a relation between Fock-state codes and permutation-invariant codes, our results also yield asymptotically good families of qudit permutation-invariant codes as well as codes in monolithic nuclear state spaces.
We analyze a generalized quantum error-correction code against photon loss where a logical qubit is encoded into a subspace of a single oscillator mode that is spanned by distinct multicomponent cat states (coherent-state superpositions). We present a systematic code construction that includes the extension of an existing one-photon-loss code to higher numbers of losses. When subject to a photon loss (amplitude damping) channel, the encoded qubits are shown to exhibit a cyclic behavior where the code and error spaces each correspond to certain multiples of losses, half of which can be corrected. As another generalization we also discuss how to protect logical qudits against photon losses, and as an application we consider a one-way quantum communication scheme in which the encoded qubits are periodically recovered while the coherent-state amplitudes are restored as well at regular intervals.
Traditionally, Quantum Information, and Quantum Communication specifically, have been focused on qubit-based architectures. Recent results, however, highlighted that higher dimensional architectures (qudit-based) may present advantages both in terms of communication and computation; a family of channels called Multi-level Amplitude Damping (MAD) channels, which are a possible qudit generalization of the well known Amplitude Damping Channels, is able to model energy decay processes that may happen during signal transmission. In this work, the Quantum Capacity of 4-dimensional MAD's is studied, relying on a technique for computing it even outside of degradable and antidegradable conditions. We also characterized the complete region of antidegradability and degradability in the parameter space for a generic d-dimensional MAD using both analytical and semi-numerical methods.
Quantum error correction (QEC) is crucial for faithful transmission of quantum information and successful realization of quantum computers. Typically, QEC is designed with minimal assumptions about the noise process, i.e. depolarizing noise that happens independently on each individual particle. In real physical systems, errors are unlikely to be arbitrary. Rather we have reasonable knowledge about the system to model its quantum decoherence process. In this talk, we examine QEC that are adapted to the physical noise models. We start from an overview of the concepts and general methods for channel-adapted QEC. We then examine a technique, called code concatenation, for constructing channel-adapted quantum codes. Applying to the amplitude damping channels, we demonstrate the power of the code concatenation method for constructing good codes for channel-adapted QEC.
The increasing interest in using quantum error correcting codes in practical devices has heightened the need for designing quantum error correcting codes that can correct against specialized errors, such as that of amplitude damping errors which model photon loss. Although considerable research has been devoted to quantum error correcting codes for amplitude damping, not so much attention has been paid to having these codes simultaneously lie within the decoherence free subspace of their underlying physical system. One common physical system comprises of quantum harmonic oscillators, and constant-excitation quantum codes can be naturally stabilized within them. The purpose of this paper is to give constant-excitation quantum codes that not only correct amplitude damping errors, but are also immune against permutations of their underlying modes. To construct such quantum codes, we use the nullspace of a specially constructed matrix based on integer partitions.