A recursive construction of noiseless subsystem for qudits

The noiseless subsystem is a method of using the inherent permutation symmetry of the noise to protect a subsystem against errors. Its construction becomes a formidable task with the growing number of qudits. In this work, we describe a recursive way of constructing noiseless subsystem for qudits, that is robust against collective noise of the form $W^{\otimes n}$, where $n$ is the number of qudits and $W$ is the Kraus operator acting on a single site. This kind of error appears when the wavelength of an environmental disturbance is much larger than the size of the quantum system, which makes it natural to assume all the qubits in the register suffer from the same error operator. The presented recursive scheme is a direct generalization of the recursive scheme described in \emph{Phys. Rev. A, {\bf 84}, (2011) 044301} for qubits. We show that the quantum error correction rate, i.e., the ratio of correctable qudits and the number of transmitted qudits, approaches $1/d$ as $n$ goes to infinity in this recursive scheme.