Abstract
We investigate the usage of highly efficient error correcting codes of multilevel systems to protect encoded quantum information from erasure errors and implementation to repetitively correct these errors. Our scheme makes use of quantum polynomial codes to encode quantum information and generalizes teleportation based error correction for multilevel systems to correct photon losses and operation errors in a fault-tolerant manner. We discuss the application of quantum polynomial codes to one-way quantum repeaters. For various types of operation errors, we identify different parameter regions where quantum polynomial codes can achieve a superior performance compared to qubit based quantum parity codes.
Highlights
A quantum erasure channel replaces a qubit with an “erasure state” that is orthogonal to all the basis states of a qubit with a certain probability, thereby erasing the qubit and enabling the receiver know that it has been erased [1]
We find that the present scheme with quantum polynomial codes (QPyC) can achieve a very small cost coefficient, which is about 5 times less than for quantum parity codes (QPC) with teleportation based error correction (TEC) for Ltot = 10, 000 km in the absence of operation errors
We have investigated efficient codes using multilevel systems that can correct up to 50% erasure error rates, which is the bound set by the no-cloning theorem [1]
Summary
A quantum erasure channel replaces a qubit (qudit) with an “erasure state” that is orthogonal to all the basis states of a qubit (qudit) with a certain probability, thereby erasing the qubit (qudit) and enabling the receiver know that it has been erased [1]. The third generation uses quantum error correction to correct both loss and operation errors, and avoids any form of two-way classical communication between repeater stations, thereby rendering ultrafast communication over transcontinental distances [16, 17, 18, 19, 20, 21]. Quantum error correcting codes of higher-dimensional systems provide a promising alternative to qubit based encoding schemes to correct erasure errors. One can encode a secret qudit into 2k + 1 qudits (with prime dimension d ≥ 2k + 1) and distribute one qudit to each of the many parties, so that at least (k + 1) of them should get together to reconstruct the secret This makes the [[2k + 1, 1, k + 1]]d QPyC a good choice for the correction of erasure errors up to a fraction of k/(2k + 1) → 50% for a large k.
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