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arXiv:quant-ph/9603022 · Physical Review A

Quantum error correction in the presence of spontaneous emission

Martin Bodo Plenio, Vlatko Vedral, P. L. Knight

Published Jan 1, 1997• 85 citations
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Abstract

We present a quantum error correcting code that is invariant under the conditional time evolution between spontaneous emissions and which can correct for one general error. The code presented here generalizes previous error correction codes in that not all errors lead to different error syndromes. This idea may lead to shorter codes than previously expected.

DOI
10.1103/physreva.55.67

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We present a quantum error correcting code that is invariant under the conditional time evolution between spontaneous emissions and which can correct for one general error. The code presented here generalizes previous error correction codes in that not all errors lead to different error syndromes. This idea may lead to shorter codes than previously expected.
In this paper we show how the fault-tolerant quantum error-correction scheme recently proposed by DiVincenzo and Shor [Phys. Rev. Lett. 77, 3260 (1996)] can be made more efficient. Our scheme, unlike the earlier one, can deal with a single error between the quantum gate operations that are required for the implementation of the quantum error correction without repeated generation of error syndromes. For errors during gate operations we have to produce fewer error syndromes as we generate them conditionally on the previous one. We show how to generalize the method to arbitrary error-correcting codes.
We construct a new family of permutationally in-variant codes that correcttttPauli errors for anyt⩾1t\geqslant 1t⩾1. We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. This is an extended abstract of the preprint [1].
A method for quantum computation in the presence of spontaneous emission is proposed. The method combines strong and fast (dynamical decoupling) pulses and a quantum error correcting code that encodes n logical qubits into only n+1n+1n+1 physical qubits. Universal, fault-tolerant, quantum computation is shown to be possible in this scheme using Hamiltonians relevant to a range of promising proposals for the physical implementation of quantum computers.
There are well known necessary and sufficient conditions for a quantum code to correct a set of errors. We study weaker conditions under which a quantum code may correct errors with probabilities that may be less than one. We work with stabilizer codes and as an application study how the nine qubit code, the seven qubit code, and the five qubit code perform when there are errors on more than one qubit. As a second application, we discuss the concept of syndrome quality and use it to suggest a way that quantum error correction can be practically improved.
We construct a new family of permutationally invariant codes that correct t Pauli errors for any t≥1 . We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases, which also admit transversal gates. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. Furthermore, our new code family includes a new ((4,2,2)) optimal single-deletion-correcting code. As a separate result, we generalize the conditions for permutationally invariant codes to correct t Pauli errors from the previously known results for t=1 to any number of errors. For small t , these conditions can be used to construct new examples of codes by computer.
Quantum error-correcting codes require that the environment cannot distinguish relevant error processes acting on different codewords. In practice, residual Zeeman, Stark, or anharmonic interactions shift the underlying states, making errors distinguishable and weakening the conditions for perfect recovery. We study this effect with an analytically solvable four-level model and a canonical spontaneous-emission noise model for atoms and molecules. Distinguishability raises the effective Kraus rank of the noise channel, lowers the recovered fidelity, and sets an operational checking time that shortens as the frequency separation between the emitted photons increases. We find that rapid syndrome checking suppresses the buildup of distinguishability and quantify how fast one needs to check to restore near-perfect recovery. Error distinguishability is thus a practical limitation of monolithic error correction, and our analysis identifies fast syndrome extraction as a route to mitigating it.