Decomposing the generalized Toffoli gate with qutrits

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 $2N-3$ 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.

Figure 1

Decomposition of two-qutrit gate $U_{i\to j}$ (a) and its inverse $U_{i\to j}^{†}$ (b) using ${\mathtt{iSWAP}}^{\mathtt{02}}$ gate, which is native for superconducting qutrit-based platforms, is shown. In (c) the transformation of ${\mathtt{iSWAP}}^{\mathtt{02}}$ into ${\mathtt{iSWAP}}^{\mathtt{20}}$ using local operations is depicted.

Figure 2

Three types of operations with the tree that correspond to ${\mathtt{C}}^{N-1}\mathtt{Z}$ gate decomposition and evolution of the qutrits' states during these operations. Before the decomposition, qutrits are initialized with the state of zero population on the ancillary level, therefore corresponding truth tables are presented for these qubit states of qutrits only. In (a) the circuit structure for elementary folding operation is presented. It changes the tree structure by collapsing all siblings $\mathbf{s}|\mathbf{1},...,\mathbf{s}\left|n\right(\mathbf{s})$ to their parent $\mathbf{s}$. In (b) the circuit structure for the basic operation that applies a phase factor $-1$ to the input state if all qutrit of the level-one subtree are in the state $|1\rangle$ is shown. It does not change single-level tree structure. In (c) the circuit structure for elementary unfolding operation that serves as uncomputation of elementary folding operation is illustrated.

Figure 3

Example of the ${\mathtt{C}}^5\mathtt{Z}$ gate decomposition for the existing qutrit-based superconducting quantum processor. In (a) coupling map of the Aspen-9 architecture [58] is shown. Each incoming arrow indicates a transmon that transitions to the second excited state $|2\rangle$ in the corresponding two-qutrit operation. Subgraph $\tilde{E}$ corresponds to the qutrits that are chosen for decomposition of ${\mathtt{C}}^5\mathtt{Z}$ gate. In (b) the resulting decomposition circuit for the ${\mathtt{C}}^5\mathtt{Z}$ gate is illustrated.