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arXiv:quant-ph/9605031 · Physical Review Letters

Fault-Tolerant Error Correction with Efficient Quantum Codes

David Peter Divincenzo, Peter W. Shor

Published Oct 7, 1996• 337 citations
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Abstract

We exhibit a simple, systematic procedure for detecting and correcting errors using any of the recently reported quantum error-correcting codes. The procedure is shown explicitly for a code in which one qubit is mapped into five. The quantum networks obtained are fault tolerant, that is, they can function successfully even if errors occur during the error correction. Our construction is derived using a recently introduced group-theoretic framework for unifying all known quantum codes.

DOI
10.1103/physrevlett.77.3260

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We exhibit a simple, systematic procedure for detecting and correcting errors using any of the recently reported quantum error-correcting codes. The procedure is shown explicitly for a code in which one qubit is mapped into five. The quantum networks obtained are fault tolerant, that is, they can function successfully even if errors occur during the error correction. Our construction is derived using a recently introduced group-theoretic framework for unifying all known quantum codes.
A major obstacle towards realizing a practical quantum computer is the noise that arises due to system-environment interactions. While it is very well known that quantum error correction (QEC) provides a way to protect against errors that arise due to the noise affecting the system, a perfect quantum code requires atleast five physical qubits to observe a noticeable improvement over the no-QEC scenario. However, in cases where the noise structure in the system is already known, it might be more useful to consider quantum codes that are adapted to specific noise models. It is already known in the literature that such codes are resource efficient and perform on par with the standard codes. In this spirit, we address the following questions concerning such adaptive quantum codes in this thesis. (a) Construction: Given a noise model, we propose a simple and fast numerical optimization algorithm to search for good quantum codes. (b) Application: As a simple application of noise-adapted codes, we propose an adaptive QEC protocol that allows transmission of quantum information from one site to the other over a 1-d spin chain with high fidelity. (c) Fault-tolerance: Finally, we address the question of whether such noise-adapted QEC protocols can be made fault-tolerant starting with a [[4,1]] code and obtain a rigorous lower bound on threshold.
Noise rates in quantum computing experiments have dropped dramatically, but reliable qubits remain precious. Fault-tolerance schemes with minimal qubit overhead are therefore essential. We introduce fault-tolerant error-correction procedures that use only two extra qubits. The procedures are based on adding "flags" to catch the faults that can lead to correlated errors on the data. They work for various distance-three codes. In particular, our scheme allows one to test the ⟦5,1,3⟧ code, the smallest error-correcting code, using only seven qubits total. Our techniques also apply to the ⟦7,1,3⟧ and ⟦15,7,3⟧ Hamming codes, thus allowing us to protect seven encoded qubits on a device with only 17 physical qubits.