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arXiv:0710.1052 · IEEE Transactions on Information Theory

Channel-Adapted Quantum Error Correction for the Amplitude Damping Channel

Andrew S. Fletcher, Peter W. Shor, Moe Z. Win

Published Dec 1, 2008• 119 citations
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Abstract

Error correction procedures are considered which are designed specifically for the amplitude damping channel. Amplitude damping errors are analyzed in the stabilizer formalism. This analysis allows a generalization of the[4,1]ldquoapproximaterdquo amplitude damping code. This generalization is presented as a class of[2(M+1),M] codes; quantum circuits for encoding and recovery operations are presented. A[7,3]amplitude damping code based on the classical Hamming code is presented. All of these are stabilizer codes whose encoding and recovery operations can be completely described with Clifford group operations. Finally, optimization options are described in which recovery operations may be further adapted according to the damping probabilitygamma.

DOI
10.1109/tit.2008.2006458

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Error correction procedures are considered which are designed specifically for the amplitude damping channel. Amplitude damping errors are analyzed in the stabilizer formalism. This analysis allows a generalization of the[4,1]ldquoapproximaterdquo amplitude damping code. This generalization is presented as a class of[2(M+1),M] codes; quantum circuits for encoding and recovery operations are presented. A[7,3]amplitude damping code based on the classical Hamming code is presented. All of these are stabilizer codes whose encoding and recovery operations can be completely described with Clifford group operations. Finally, optimization options are described in which recovery operations may be further adapted according to the damping probabilitygamma.
Quantum error correction (QEC) is an essential concept for any quantum information processing device. Typically, QEC is designed with minimal assumptions about the noise process; this generic assumption exacts a high cost in efficiency and performance. In physical systems, errors are not likely to be arbitrary; rather we will have reasonable models for the structure of quantum decoherence. We may choose quantum error correcting codes and recovery operations that specifically target the most likely errors. We present a convex optimization method to determine the optimal (in terms of average entanglement fidelity) recovery operation for a given channel, encoding, and information source. This is solvable via a semidefinite program (SDP). We present computational algorithms to generate near-optimal recovery operations structured to begin with a projective syndrome measurement. These structured operations are more computationally scalable than the SDP required for computing the optimal; we can thus numerically analyze longer codes. Using Lagrange duality, we bound the performance of the structured recovery operations and show that they are nearly optimal in many relevant cases. We present two classes of channel-adapted quantum error correcting codes specifically designed for the amplitude damping channel. These have significantly higher rates with shorter block lengths than corresponding generic quantum error correcting codes. Both classes are stabilizer codes, and have good fidelity performance with stabilizer recovery operations. The encoding, syndrome measurement, and syndrome recovery operations can all be implemented with Clifford group operations.
We present analytic estimates of the performances of various approximate quantum error-correction schemes for the generalized amplitude-damping (GAD) qubit channel. Specifically, we consider both stabilizer and nonadditive quantum codes. The performance of such error-correcting schemes is quantified by means of the entanglement fidelity as a function of the damping probability and the nonzero environmental temperature. The recovery scheme employed throughout our work applies, in principle, to arbitrary quantum codes and is the analog of the perfect Knill-Laflamme recovery scheme adapted to the approximate quantum error-correction framework for the GAD error model. We also analytically recover and/or clarify some previously known numerical results in the limiting case of vanishing temperature of the environment, the well-known traditional amplitude-damping channel. In addition, our study suggests that degenerate stabilizer codes and self-complementary nonadditive codes are especially suitable for the error correction of the GAD noise model. Finally, comparing the properly normalized entanglement fidelities of the best performant stabilizer and nonadditive codes characterized by the same length, we show that nonadditive codes outperform stabilizer codes not only in terms of encoded dimension but also in terms of entanglement fidelity.
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 construct two simple error correction schemes adapted to amplitude damping noise for Bacon-Shor codes and investigate their prospects for fault-tolerant implementation. Both consist solely of Clifford gates and require far fewer qubits, relative to the standard method, to achieve exact correction to a desired order in the damping rate. The first, employing one-bit teleportation and single-qubit measurements, needs only one-fourth as many physical qubits, while the second, using just stabilizer measurements and Pauli corrections, needs only half. The improvements stem from the fact that damping events need only be detected, not corrected, and that effective phase errors arising due to undamped qubits occur at a lower rate than damping errors. For error correction that is itself subject to damping noise, we show that existing fault-tolerance methods can be employed for the latter scheme, while the former can be made to avoid potential catastrophic errors and can easily cope with damping faults in ancilla qubits.
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.