Designing Ternary Quantum Error Correcting Codes from Binary Codes

Article Type

Research Article

Publication Title

Journal of Multiple-Valued Logic and Soft Computing

Abstract

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 d dimensional X and Z 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 X and Z operators alone are not sufficient to allow the design of ternary QECCs as direct extensions of their binary counterparts. We propose operators X1, X2 and Z1, Z2, which span the 3 × 3 operator space, and show that Z1, Z2 as well as X1 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 Laflamme's code to the ternary regime. These results provide a necessary requirement for easy design of ternary QECCs from existing binary ones.

First Page

179

Last Page

201

Publication Date

1-1-2023

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