Generalized Toffoli gates using qudit catalysis

We present quantum networks for a $n$-qubit controlled gate $C^{n-1}\left(U\right)$ which use a higher-dimensional (qudit) ancilla as a catalyzer. In its simplest form the network has only $n$ two-particle gates (qubit-qudit)—this is the minimum number of two-body interactions needed to couple all $n+1$ subsystems ($n$ qubits plus one ancilla). This class of controlled gates includes the generalized Toffoli gate $C^{n-1}\left(X\right)$ on $n$ qubits, which plays an important role in several quantum algorithms and error correction. A particular example implementing this model is given by the dispersive limit of a generalized Jaynes-Cummings Hamiltonian of an effective spin $s$ interacting with a cavity mode.

Figure 1

(Color online) An $n$-qubit controlled-$U$ gate $C^{n-1}\left(U\right)$; if $U=X$ this corresponds to a generalized Toffoli gate. Qubits $i_1,\dots ,i_n$ are denoted by thin black lines; the ancilla (red, thick line) is a qudit with a $n$-dimensional Hilbert space. The controlled-$U$ gate in the middle between a qubit and the qudit ancilla is performed only if the ancilla is in the $|1⟩$ state, i.e., $C\left(U\right)|i_n⟩|j⟩=\left[U^{{\delta }_{j1}}|i_n⟩\right]|j⟩$ (symbolized by 1 below the gate).

Figure 2

(Color online) A measurement-based $C^{n-1}\left(U\right)$ gate. The $n-1$ gates $C\left(X_n\right)$ used to disentangle the ancilla from the qubits are replaced by a measurement of the ancilla in the Fourier basis (gate $F$ on the qudit) and feed forward. The gates $P^a=\text{diag}\left(1,{\omega }^a\right)$ are conditionally applied depending the value $a$ of the measured ancilla; the same gate $P^a$ is applied to all control qubits, resulting in a homogeneous design.

Figure 3

(Color online) A quantum network equivalent to Fig. 2, using the identity $X_n=F^{-1}{Z}_{n}F$.