Approximate Ternary Quantum Error Correcting Code with Low Circuit Cost

Quantum error correcting code (QECC) with encoding and decoding circuits having low gate count, is an important criterion for realizing quantum computing systems. CSS QECCs are known to have simple circuits. Shaw et al. showed that it is not possible to have a 6-qubit CSS type QECC without sharing entanglement between the encoder and the decoder. In this paper, we propose a 6-qutrit approximate QECC (AQECC) of CSS structure which can simultaneously correct phase errors in upto six qutrits, and one bit error in only four of the six qutrits, without sharing prior entanglement. Our AQECC corrects a single error with probability 0.75 for symmetric error model, and probability 0.9988 for asymmetric error model. It also maintains CSS structure without sharing prior entanglement. Furthermore, the quantum cost of a circuit for this AQECC is 55.72% less than that for the 9-qutrit exact QECC. Overall, low qutrit count, low cost circuit realization with low depth and the ability to correct multiple phase errors make our proposed AQECC a suitable candidate for real life quantum channels.