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arXiv:0810.2524 · IEEE Transactions on Information Theory

Channel-Optimized Quantum Error Correction

Soraya Taghavi, Robert L. Kosut, Daniel A. Lidar

Published Mar 1, 2010• 30 citations
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Abstract

We develop a theory for finding quantum error correction (QEC) procedures which are optimized for given noise channels. Our theory accounts for uncertainties in the noise channel, against which our QEC procedures are robust. We demonstrate, via numerical examples, that our optimized QEC procedures always achieve a higher channel fidelity than the standard error correction method, which is agnostic about the specifics of the channel. This demonstrates the importance of channel characterization before QEC procedures are applied. Our main novel finding is that in the setting of a known noise channel the recovery ancillas are redundant for optimized quantum error correction. We show this using a general rank minimization heuristic and supporting numerical calculations. Therefore, one can further improve the fidelity by utilizing all the available ancillas in the encoding block.

DOI
10.1109/tit.2009.2039162

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We develop a theory for finding quantum error correction (QEC) procedures which are optimized for given noise channels. Our theory accounts for uncertainties in the noise channel, against which our QEC procedures are robust. We demonstrate, via numerical examples, that our optimized QEC procedures always achieve a higher channel fidelity than the standard error correction method, which is agnostic about the specifics of the channel. This demonstrates the importance of channel characterization before QEC procedures are applied. Our main novel finding is that in the setting of a known noise channel the recovery ancillas are redundant for optimized quantum error correction. We show this using a general rank minimization heuristic and supporting numerical calculations. Therefore, one can further improve the fidelity by utilizing all the available ancillas in the encoding block.
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 a semidefinite program optimization approach to quantum error correction that yields codes and recovery procedures that are robust against significant variations in the noise channel. Our approach allows us to optimize the encoding, recovery, or both, and is amenable to approximations that significantly improve computational cost while retaining fidelity. We illustrate our theory numerically for optimized 5-qubit codes, using the standard [5,1,3] code as a benchmark. Our optimized encoding and recovery yields fidelities that are uniformly higher by 1-2 orders of magnitude against random unitary weight-2 errors compared to the [5,1,3] code with standard recovery.
Using convex optimization, we propose entanglement-assisted quantum error-correction procedures that are optimized for given noise channels. We demonstrate through numerical examples that such an optimized error-correction method achieves higher channel fidelities than existing methods. This improved performance, which leads to perfect error correction for a larger class of error channels, is interpreted in at least some cases by quantum teleportation, but for general channels this interpretation does not hold.
The purpose of quantum error correction (QEC) is to preserve a quantum state despite the presence of decoherence. As we see throughout this book, the desired robustness exacts a cost in resources, most commonly the inclusion of redundant qubits. The following are reasonable engineering queries: How much will it cost to provide the desired robustness? For a fixed cost, what is the best performance I can achieve? In this chapter, we present some numerical tools to illuminate these kinds of questions. To understand the cost/performance trade-off quantitatively, we need a clear measure of performance and a model for permissible operations. With this in mind, we will revisit the concepts of fidelity and quantum operations. As it turns out, this quantitative approach can yield very well structured convex optimization problems. Using powerful numerical tools, we determine optimal encodings and recovery operations. Even if the optimal results are not directly implemented, the optimization tools can provide insight into practical solutions by providing the ultimate performance limits. Limitation of the independent arbitrary errors model As pointed out in Chapter 2, our rich history of classical error correction has provided a significant foundation for QEC methods. As such, the initial QEC breakthroughs involved importing classical coding concepts into a framework that made robust quantum codes (such as CSS codes or the stabilizer code formalism). We learned that we could create general purpose codes that made minimal assumptions about the structure of the decoherence process.
This paper presents a novel approach to quantum error correction (QEC) and adaptive quantum channel mapping (QECM) designed to enhance the robustness of quantum information transmission through noisy channels. The core idea is to leverage the redundancy inherent in QEC codes and combine it with a machine learning-based adaptive mapping algorithm. Unlike traditional methods that rely on pre-characterized channel models, this approach dynamically optimizes the QEC encoding and channel mapping strategies based on real-time channel feedback. The system utilizes an error correction matrix derived from the chosen QEC code to construct a quantum channel model. A neural network, specifically a variational quantum neural network (VQNN), is then trained to learn the optimal quantum channel mapping policy. This dynamic optimization addresses the limitations of fixed channel models and promises improved performance in practical quantum communication scenarios. The effectiveness of the QECM framework is demonstrated through theoretical analysis and a conceptual model, paving the way for robust long-distance quantum communication. The system's adaptability allows it to cope with unknown or time-varying channel characteristics, a critical factor for successful quantum communication implementations. The ultimate goal is to achieve efficient and reliable quantum information transfer, regardless of the underlying channel noise.
We demonstrate that there exists a universal, near-optimal recovery map---the transpose channel---for approximate quantum error-correcting codes, where optimality is defined using the worst-case fidelity. Using the transpose channel, we provide an alternative interpretation of the standard quantum error correction (QEC) conditions, and generalize them to a set of conditions for approximate QEC (AQEC) codes. This forms the basis of a simple algorithm for finding AQEC codes. Our analytical approach is a departure from earlier work relying on exhaustive numerical search for the optimal recovery map, with optimality defined based on entanglement fidelity. For the practically useful case of codes encoding a single qubit of information, our algorithm is particularly easy to implement.