Introducing a few intermediate qutrits for efficient decomposition of 3-qubit unitary gates has been proposed recently to obtain an exponential reduction in the depth of the decomposed circuit. An intermediate qutrit implies that a qubit is operated as a qutrit in a particular execution cycle. This method, primarily for the NISQ era, treats a qubit as a qutrit only for the duration when it requires access to the state \(\left| {2}\right\rangle \) during the computation. In this article, we study the challenges of extending this decomposition to the error-corrected regime. We first we show that if a qubit has to be in state \(\left| {2}\right\rangle \) at any point of time, then it must be encoded using a qutrit quantum error correcting code (QECC), thus resulting in a circuit with both qubits and qutrits. Qutrits being noisier than qubits, the former are expected to require higher levels of concatenation to achieve a particular accuracy than that for qubit-only decomposition. We derive analytically a relation between the levels of concatenation required for qubit-only and that for qubit–qutrit decomposition to achieve the same level of accuracy. Finally, we estimate (i) the degree of concatenation for both qubit–qutrit and qubit-only decompositions as a function of the probability of error and (ii) the criterion for which qubit–qutrit decomposition leads to a lower gate count than that for qubit-only decomposition.
