Near-Optimal Error Correcting Code For Ternary Quantum Systems
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Abstract
Multi-valued quantum systems can store more information than binary ones for a given number of quantum states. For reliable operation of multi-valued quantum systems, error correction is mandated. In this paper, we propose a 6-qutrit quantum error correcting code by extending the ternary Hamming code to the quantum domain. We prove that 5 qutrits are necessary to correct a single error, which makes our proposed code near-optimal in the number of qutrits. We also provide the stabilizer formulation and the circuit realization for this code. This code outperforms the previous 9-qutrit code in (i) the number of qutrits required for encoding, (ii) our code can correct any arbitrary (3*3) error, (ii) our code can readily correct bit errors in a single step as opposed to the two-step correction used previously, (iii) phase error correction does not require correcting individual subspaces, and (iv) the quantum cost of our proposed circuit is 68 as opposed to 244 for the 9-qutrit code.