Quantum error-correcting code for ternary logic

TL;DR

This paper introduces a novel quantum error-correcting code for qutrits, addressing both Pauli and superposition errors, and provides a stabilizer framework and circuit implementation.

Contribution

It proposes a new nine-qutrit code capable of correcting single errors, including a novel quantum superposition error, with explicit stabilizer and circuit design.

Findings

The code corrects arbitrary single-qutrit errors.

Introduces a new quantum superposition error model.

Provides stabilizer formalism and circuit realization for the code.

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.