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We provide a procedure to construct entanglement-assisted Calderbank-Shor-Steane (CSS) codes over qudits from the parity check matrices of two classical codes over F q , where q = p k , p is prime, and k is a positive integer. The construction procedure involves the proposed Euclidean Gram-Schmidt orthogonalization algorithm, followed by a procedure to extend the quantum operators to obtain stabilizers of the code. Using this construction, we provide a construction of entanglement-assisted tensor product codes from classical tensor product codes over F q . We further show that a nonzero rate entanglement-assisted tensor product code can be obtained from a classical tensor product code whose component codes yield zero rate entanglement-assisted CSS codes. We view this result as the coding analog of superadditivity.INDEX TERMS Entanglement-assisted (EA) codes, entanglement-assisted Calderbank-Shor-Steane (CSS) code, extension procedure, quantum error correction, tensor product codes (TPCs). Engineering uantum Transactions on IEEENadkarni and Garani: Coding Analog of Superadditivity Using Entanglement-Assisted Quantum Tensor Product Codes

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Quantum coding schemes over qudits using pre-shared entanglement between the encoder and decoder can provide better error correction capability than without it. In this paper, we develop procedures for constructing encoding operators for entanglement-unassisted and entanglement-assisted qudit stabilizer codes over F p k , with p prime and k ≥ 1 from first principles, generalizing prior works on qubit based codes and codes that work over F p . We also provide quantum encoding architectures based on the proposed encoding procedures using one and two qudit gates, useful towards realizing coded quantum computing and communication systems using qudits.INDEX TERMS Quantum error correction, Entanglement-assisted codes, Encoding architecture, Qudit stabilizer codes, Non-binary codes. 1 The encoding circuit complexity linearly increases with codelength. To realize the practical implementation of large length codes, we need to construct all the stabilizers or their extensions from a carefully chosen group of quantum operators, making it a computationally challenging problem.2 Preliminary version on the encoding procedure for entanglementunassisted stabilizer codes over qudits was presented at IEEE Globecom 2018, Abu Dhabi [14].

We introduce a stochastic resonance based decoding paradigm for quantum codes using an error correction circuit made of a combination of noisy and noiseless logic gates. The quantum error correction circuit is based on iterative syndrome decoding of quantum low-density parity check codes, and uses the positive effect of errors in gates to correct errors due to decoherence. We analyze how the proposed stochastic algorithm can escape from short cycle trapping sets present in the dual containing Calderbank, Shor and Steane (CSS) codes. Simulation results show improved performance of the stochastic algorithm over the deterministic decoder.

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