Optimal quantum circuits for general two-qubit gates

In order to demonstrate nontrivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 controlled-NOT (CNOT) gates and 15 elementary one-qubit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requries at most 2 CNOT and 12 one-qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, $y$-rotation, $z$-rotation, and phase gates.

Figure 1

A circuit for implementing the magic gate $\mathcal{M}$.

Figure 2

A circuit for implementing a general transform in $\mathbf{SO}\left(4\right)$, where $A,B∊\mathbf{SU}\left(2\right)$, $S_1=R_z\left(\pi ∕2\right)$, and $R_1=R_y\left(\pi ∕2\right)$.

Figure 3

A circuit for implementing a transform in $\mathbf{O}\left(4\right)$ determinant equal to $-1$, where $A,B∊\mathbf{SU}\left(2\right)$, $S_1=R_z\left(\pi ∕2\right)$, and $R_1=R_y\left(\pi ∕2\right)$.

Figure 4

A circuit for implementing $N\left(\alpha ,\beta ,\gamma \right)$; first version.

Figure 5

A circuit for implementing $N\left(\alpha ,\beta ,\gamma \right)$; second version. Here $S_1=R_z\left(\pi ∕2\right)$, $S_2=R_z\left(2\gamma -\pi ∕2\right)$, $T_1=R_y\left(\pi ∕2-2\alpha \right)$, and $T_2=R_y\left(2\beta -\pi ∕2\right)$.

Figure 6

A circuit for implementing $N\left(\alpha ,\beta ,\gamma \right)$; third version. A global $e^{i(\pi ∕4)}$ phase is missing here.

Figure 7

A circuit for implementing a transform in $\mathbf{U}\left(4\right)$.

Figure 8

A circuit consisting of two CNOT gates in terms of CZ gates.