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Chinese Physics B

Entanglement fidelity of channel adaptive quantum codes

Yun Zhan, Xiaoyu Chen

Published Jan 1, 2013• 4 citations
Publisher

Abstract

We study the performances of quantum channel adaptive [4,1] code transmitting in a joint amplitude damping and dephasing channel, the [6,2] code transmitting in an amplitude damping channel by combining the encoding, noise process, and decoding as one effective channel. We explicitly obtain the entanglement fidelities. The recovery operators of the [6,2] code are given. The performance is nearly optimal compared with that of the optimal method of semidefinite programming.

DOI
10.1088/1674-1056/22/1/010308

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We study the performances of quantum channel adaptive [4,1] code transmitting in a joint amplitude damping and dephasing channel, the [6,2] code transmitting in an amplitude damping channel by combining the encoding, noise process, and decoding as one effective channel. We explicitly obtain the entanglement fidelities. The recovery operators of the [6,2] code are given. The performance is nearly optimal compared with that of the optimal method of semidefinite programming.
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.
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.
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.
We present a class of numerical algorithms which adapt a quantum error correction scheme to a channel model. Given an encoding and a channel model, it was previously shown that the quantum operation that maximizes the average entanglement fidelity may be calculated by a semidefinite program (SDP), which is a convex optimization. While optimal, this recovery operation is computationally difficult for long codes. Furthermore, the optimal recovery operation has no structure beyond the completely positive trace-preserving constraint. We derive methods to generate structured channel-adapted error recovery operations. Specifically, each recovery operation begins with a projective error syndrome measurement. The algorithms to compute the structured recovery operations are more scalable than the SDP and yield recovery operations with an intuitive physical form. Using Lagrange duality, we derive performance bounds to certify near-optimality.
This paper presents a novel approach to quantum information coding and decoding leveraging the unique properties of quantum entanglement. The core idea is to develop a self-adaptive system capable of dynamically adjusting encoding and decoding parameters based on the observed noise levels and channel losses within the quantum channel. This adaptive strategy aims to significantly improve the efficiency and reliability of quantum information transmission. The proposed framework utilizes a quantum decoherence model to accurately represent channel losses and employs an optimization algorithm rooted in quantum control theory to optimize the generation and measurement strategies of entangled states in real-time. The algorithm's objective is to minimize information loss. The key innovation lies in the system's ability to respond in real-time to fluctuating channel conditions, moving beyond the limitations of static encoding/decoding schemes commonly employed in existing quantum communication protocols. Mathematical models and formulations are presented to illustrate the design and optimization process, focusing on key performance metrics such as channel capacity and fidelity. The system's potential impact on future quantum communication networks is discussed.
Entanglement fidelity quantifies how well a quantum channel preserves the correlations between a transmitted system and an inaccessible reference system. We derive closed-form expressions for the entanglement fidelity associated with several standard quantum noise models, including the random Pauli-X, dephasing, depolarizing, Werner-Holevo, generalized Pauli (Weyl), and amplitude-damping channels. For each model, we express the entanglement fidelity in terms of a general input density operator ρρρ, using Schumacher's Kraus-operator approach, which provides a channel-agnostic recipe applicable to any completely positive trace-preserving (CPTP) map with a finite Kraus representation. We then specialize to a communication scenario in which the source emits a two-letter parametric alphabet, thereby making explicit the dependence of entanglement preservation on both channel and source parameters. The resulting expressions enable direct comparisons of channel performance and rankings for representative families of input states, including common qubit states.