The use of a few intermediate qutrits for efficient decomposition of 3-qubit unitary gates has been proposed, 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
∣2⟩ during the computation. In this article, we study the challenges of including fault-tolerance in such a decomposition. We first show that any qubit that requires access to the state
∣2⟩ at any point in the circuit, must be encoded using a qutrit quantum error correcting code (QECC), thus resulting in a circuit with both qubits and qutrits at the outset. Since qutrits are noisier than qubits, the former is expected to require higher levels of concatenation to achieve a particular accuracy than that for qubit-only decomposition. Next, we derive analytically (i) the number of levels of concatenation required for 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 qubit-only decomposition. We present numerical results for these two types of decomposition and obtain the situation where qubit-qutrit decomposition excels for the example circuit of the quantum adder by considering different values for quantum hardware-noise and non-transversal implementation of the 2-controlled ternary CNOT gate.
The Toffoli gate, which has practical applications in the implementation of practical quantum algorithms and quantum error correction, is usually constructed from concatenated two-qubit or qubit-qutrit gates. Here we show that the Toffoli gate for three qubits can be implemented through a single
XY interaction plus two single-qubit rotations. This method greatly simplifies the experimental implementation of the Toffoli gate and promises a much higher fidelity compared to those based on elementary gate decomposition. We present an example for the implementation of the required
XY interaction in cavity QED.
The problem of finding efficient decompositions of multiqubit gates is of importance for quantum computing, especially, in application to existing noisy intermediate-scale quantum devices, whose resources are substantially limited. Here we propose a decomposition scheme for a generalized
N-qubit Toffoli gate with the use of $2N\ensuremath{-}3$ two-qutrit gates for arbitrary connectivity. The fixed number of the required additional levels (the choice of qutrits is optimal) and the use of the iSWAP gate as a native operation make our approach directly applicable for ongoing experiments with superconducting quantum processors. Specifically, we present a blueprint of the realization of the proposed scheme for the Aspen-9 processor supporting quantum operations with qutrits.