Quantum error-correcting code for ternary logic

Abstract

Ternary quantum systems are being studied because these provide more computational state space per unit of information, known as qutrit. A qutrit has three basis states, thus a qubit may be considered as a special case of a qutrit where the coefficient of one of the basis states is zero. Hence both $(2 \times 2)$-dimensional as well as $(3 \times 3)$-dimensional Pauli errors can occur on qutrits. In this paper, we (i) explore the possible $(2 \times 2)$-dimensional as well as $(3 \times 3)$-dimensional Pauli errors in qutrits and show that any pairwise bit swap error can be expressed as a linear combination of shift errors and phase errors, (ii) propose a new type of error called quantum superposition error and show its equivalence to arbitrary rotation, (iii) formulate a nine qutrit code which can correct a single error in a qutrit, and (iv) provide its stabilizer and circuit realization.