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Designing Ternary Quantum Error Correcting Codes from Binary Codes
Ritajit Majumdar and Susmita Sur-Kolay
Higher dimensional quantum error correcting codes (QECC) are expected to be carried over directly from the corresponding binary QECC. However, the 9-qutrit QECC in [25] as a direct ternary carryover of Shor code using the generalized π dimensional π and π operators failed to correct an error in a single step, leading to a significant increase in gate count and depth of the QECC circuit. In this article, we show that generalized π and π operators alone are not sufficient to allow the design of ternary QECCs as direct extensions of their binary counterparts. We propose operators π1, π2 and π1, π2, which span the 3 Γ 3 operator space, and show that π1, π2 as well as π1 are necessary to retain the stabilizer structure of the binary QECC in its ternary version. We devise a 9-qutrit QECC using these three operators and retrieve the stabilizer structure of Shor code, yielding a reduction of 51.9% in circuit cost and 23.07% in depth over the one in [25]. We also show a similar extension of Steane and Laflammeβs code to the ternary regime. These results provide a necessary requirement for easy design of ternary QECCs from existing binary ones.