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arXiv:quant-ph/0405012 · Physical Review A

Effect of ancilla’s structure on quantum error correction using the seven-qubit Calderbank-Shor-Steane code

Pedro J. Salas, Angel L. Sanz

Published May 17, 2004• 6 citations

Abstract

In this work we discuss the ability of different types of ancillas to control the decoherence of a qubit interacting with an environment. The error is introduced into the numerical simulation via a depolarizing isotropic channel. The ranges of values considered are ${10}^{\ensuremath{-}4}\ensuremath{\leqslant}\ensuremath{\epsilon}\ensuremath{\leqslant}{10}^{\ensuremath{-}2}$ for memory errors and $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}\ensuremath{\leqslant}\ensuremath{\gamma}∕7\ensuremath{\leqslant}{10}^{\ensuremath{-}2}$ for gate errors. After the correction we calculate the fidelity as a quality criterion for the qubit recovered. We observe that a recovery method with a three-qubit ancilla provides reasonably good results bearing in mind its economy. If we want to go further, we have to use fault tolerant ancillas with a high degree of parallelism, even if this condition implies introducing additional ancilla verification qubits.

DOI: 10.1103/physreva.69.052322

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In this work we discuss the ability of different types of ancillas to control the decoherence of a qubit interacting with an environment. The error is introduced into the numerical simulation via a depolarizing isotropic channel. The ranges of values considered are

{10}^{\ensuremath{-}4}\ensuremath{\leqslant}\ensuremath{\epsilon}\ensuremath{\leqslant}{10}^{\ensuremath{-}2}

for memory errors and

3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}\ensuremath{\leqslant}\ensuremath{\gamma}∕7\ensuremath{\leqslant}{10}^{\ensuremath{-}2}

for gate errors. After the correction we calculate the fidelity as a quality criterion for the qubit recovered. We observe that a recovery method with a three-qubit ancilla provides reasonably good results bearing in mind its economy. If we want to go further, we have to use fault tolerant ancillas with a high degree of parallelism, even if this condition implies introducing additional ancilla verification qubits.

Decoherence is the main problem to be solved before quantum computers can be built. To control decoherence, it is possible to use error correction methods, but these methods are themselves noisy quantum computation processes. In this work, we study the ability of Steane's and Shor's fault-tolerant recovering methods, as well as a modification of Steane's ancilla network, to correct errors in qubits. We test a way to measure correctly ancilla's fidelity for these methods, and state the possibility of carrying out an effective error correction through a noisy quantum channel, even using noisy error correction methods.

The error correcting capabilities of the Calderbank-Shor-Steane [[7,1,3]] quantum code, together with a fault-tolerant syndrome extraction by means of several ancilla states, have been numerically studied. A simple probability expression to characterize the code ability for correcting an encoded qubit has been considered. This probability, as a correction quality criterion, permits the error correction capabilities among different recovery schemes to be compared. The memory error threshold is calculated by means of the best method of those considered.

The error correcting capabilities of the Calderbank-Shor-Steane [[7,1,3]] quantum code, together with a fault-tolerant syndrome extraction by means of several ancilla states, have been numerically studied. A simple probability expression to characterize the code ability for correcting an encoded qubit has been considered. This probability, as a correction quality criterion, permits the error correction capabilities among different recovery schemes to be compared. The memory error threshold is calculated by means of the best method of those considered.

The error correcting capabilities of the Calderbank-Shor-Steane [[7,1,3]] quantum code, together with a fault-tolerant syndrome extraction by means of several ancilla states, have been numerically studied. A simple probability expression to characterize the code ability for correcting an encoded qubit has been considered. This probability, as a correction quality criterion, permits the error correction capabilities among different recovery schemes to be compared. The memory error threshold is calculated by means of the best method of those considered.

Fault tolerant quantum error correction is a procedure which satisfies the feature that if one of the gates in the procedure has failed then the failure causes at most one error in the output qubits of the encoded block. Quantum computer is based on the idea of two quantum state systems (Qubits). However, the majority of systems are constructed from higher than two- level subspace. Bad control and environmental interactions in these systems lead to leakage fault. Leakage errors are errors that couple the states inside a code subspace to the states outside a code subspace. One example for leakage fault is loss errors. Since the fault tolerant procedure may be unable to recognize the leakage fault because it was designed to deal with Pauli errors. In that case a single leakage fault might disrupt the fault tolerant technique. In this thesis we investigate the effect of ancilla losses on fault-tolerant quantum error correction in the [[7,1,3]] Steane code. We proved that both Shor and Steane methods are still fault tolerant if loss errors occur.

Ancilla postselection is a common means of achieving fault tolerance in quantum error correction. However, it can lead to additional data errors due to movement or wait operations. Alternatives to postselection may achieve lower overall failure rates due to avoiding such errors. We present numerical simulation results comparing the logical error rates for the fault-tolerant [[7,1,3]] Steane code using techniques of ancilla verification vs the newer method of ancilla decoding, as described in Phys. Rev. Lett. 98, 020501 (2007). We simulate QEC procedures in which the possibility of ancilla verification failures requires the creation and storage of additional ancillae and/or additional waiting of the data until a new ancilla can be created. We find that the decoding method, which avoids verification failures, is advantageous in terms of overall error rate in various cases, even when measurement operations are no slower than others. We analyze the effect of different classes of physical error on the relative performance of these two methods.