Quantum Error Correction and Its Application

Daniel A. Lidar, Todd A. Brun

Eromanga Adermann, Haiyue Kang, Martin Sevior +1 more

At the intersection of quantum computing and machine learning, quantum machine learning (QML) is poised to revolutionize artificial intelligence. However, the vulnerability of the current generation of quantum computers to noise and computational error poses a significant barrier to this vision. Whilst quantum error correction (QEC) offers a promising solution for almost any type of hardware noise, its application requires millions of qubits to encode even a simple logical algorithm, rendering it impractical in the near term. In this chapter, we examine strategies for integrating QEC and quantum error detection (QED) into QML under realistic resource constraints. We first quantify the resource demands of fully error-corrected QML and propose a partial QEC approach that reduces overhead while enabling error correction. We then demonstrate the application of a simple QED method, evaluating its impact on QML performance and highlighting challenges we have yet to overcome before we achieve fully fault-tolerant QML.

Yaakov S. Weinstein

Fault-tolerant protocol assumes the application of error correction after every quantum gate. However, correcting errors is costly in terms of time and number of qubits. Here we demonstrate that quantum error correction (QEC) can be applied significantly less often with only a minimal loss of fidelity. This is done by simulating the implementation of 50 encoded, single-qubit, quantum gates within the [[7,1,3]] QEC code in a noisy, nonequiprobable Pauli error environment with QEC being applied at different intervals. We find that applying QEC after every gate is rarely optimal and even applying QEC only once after all 50 gates sacrifices only a slight amount of fidelity with the benefit of 50-fold saving of resources. In addition, we find that in cases where bit-flip errors are dominant, it is best not to apply QEC at all.

Ray LaPierre

Belal Ehsan Baaquie, L. C. Kwek

Alireza Shabani, Daniel A. Lidar

By introducing an operator sum representation for arbitrary linear maps, we develop a generalized theory of quantum error correction (QEC) that applies to any linear map, in particular maps that are not completely positive (CP). This theory of "linear quantum error correction" is applicable in cases where the standard and restrictive assumption of a factorized initial system-bath state does not apply.