General qubit errors cannot be corrected

Shigeaki Yoshizumi, Masanari Asano, Satoshi Iriyama +1 more

Ryszard Kukulski, Łukasz Pawela, Zbigniew Puchała

Probabilistic quantum error correction is an error-correcting procedure which uses postselection to determine if the encoded information was successfully restored. In this work, we analyze the probabilistic version of the error-correcting procedure for general noise. We generalize the Knill-Laflamme conditions for probabilistically correctable errors. We show that for some noise channels, the initial information has to be encoded into a mixed state to maximize the probability of successful error correction. Finally, the probabilistic error-correcting procedure offers an advantage over the deterministic procedure. Reducing the probability of successful error correction allows for correcting errors generated by a broader class of noise channels. Significantly, if the errors are caused by a unitary interaction with an auxiliary qubit system, we can probabilistically restore a qubit state by using only one additional physical qubit.

Samuel L. Braunstein

We show how to perform error correction of single qubit dephasing by encoding a single qubit into a minimum of three. This may be performed in a manner closely analogous to classical error correction schemes. Further, the resulting quantum error correction schemes are trivially generalized to the minimal encoding of arbitrarily many qubits so as to allow for multiqubit dephasing correction under the sole condition that the environment acts independently on each qubit.

T Wagner, Hermann Kampermann, Dagmar Bruß

Abstract Graph states have been used to construct quantum error correction codes for independent errors. Hypergraph states generalize graph states, and symmetric hypergraph states have been shown to allow for the correction of correlated errors. In this paper, it is shown that symmetric hypergraph states are not useful for the correction of independent errors, at least for up to 30 qubits. Furthermore, error correction for error models with protected qubits is explored. A class of known graph codes for this scenario is generalized to hypergraph codes.

C. H. Wu