Entanglement-assisted fault-tolerant encoding and decoding for qudit stabilizer codes

We propose fault-tolerant quantum encoders for Calderbank-Shor-Steane (CSS) and non-CSS codes over qudits by utilizing $g$-partite maximally entangled states. The qudits are grouped into $g$ contiguous blocks such that entangled states are shared across the blocks, and multiqudit gates are applied locally to restrict the error propagation within each block. We also propose a transversal fault-tolerant block decoder to decode the blocks individually and recover the complete encoded quantum information. The optimum value of $g$ to obtain an improvement in the probability of error propagation of the proposed encoders against non-fault-tolerant schemes is analytically determined for application in the encoding circuit. The proposed fault-tolerant scheme with entanglement assistance scales linearly with the number of $g$-partite entangled states and overcomes the limitations of diminishing coding rates for higher levels of concatenation within the concatenated coding scheme. We validate the proposed ideas by constructing fault-tolerant encoding circuits for the non-CSS ${[[5,1,3]]}_{3}$ code and the ${[[7,1,3]]}_{3}$ CSS code through illustrative examples. We reduced the probability of error propagation from 0.248 to 0.212 for the ${[[5,1,3]]}_{3}$ code and from 0.207 to 0.174 for the ${[[7,1,3]]}_{3}$ code. For example, over ${[[100,70]]}_{2}$ quantum code without any fault tolerance, using three-partite entangled pairs, one can get two orders of improvement (1e-2) in the probability of error propagation compared to the non-fault-tolerant encoder by encoding over 190 physical qubits with the same number of stabilizers.