$i\text{SWAP-type}$ geometric gates induced by paths on the Schmidt sphere

We propose $i\text{SWAP-type}$ quantum gates based on geometric phases purely associated with paths on the Schmidt sphere [Phys. Rev. A 62, 022109 (2000)]. These geometric Schmidt gates can entangle qubit pairs to an arbitrary degree; in particular, they can create maximally entangled states from product states by an appropriate choice of base point on the Schmidt sphere. We identify Hamiltonians that generate pure paths on the Schmidt sphere by reverse engineering and demonstrate explicitly that the resulting Hamiltonians can be implemented in systems of transmon qubits. The geometric Schmidt gates are characterized by vanishing dynamical phases and are complementary to geometric single-qubit gates that take place on the Bloch sphere.

Figure 1

Entangling capacity on the Schmidt sphere with $(x,y,z)=(sin\alpha cos\beta ,sin\alpha sin\beta ,cos\alpha )$. Perfect entanglers (PEs) are certain loops based at points with polar angles ${\alpha }_0\in [\frac{\pi }{4},\frac{\pi }{2}]$ with the special perfect entanglers (SPEs) on the equator ${\alpha }_0=\frac{\pi }{2}$. There are no product states that can be transformed into a maximally entangled state by gates for base points with ${\alpha }_0\in [0,\frac{\pi }{2})$. Product gates, such as ${\mathrm{U}}^{\left(i\right)}\left(\mathcal{C}\right)$ in Eq. (5), are located at the north pole. While we show only the upper half for clarity, exactly the same classification of gates can be found on the lower half of the Schmidt sphere.

Figure 2

Orange-slice curve on the Schmidt sphere that implements the special perfect entangler ${\mathrm{U}}^{\left(ii\right)}\left(\mathcal{C}\right)$ in Eq. (10). The curve starts and ends at ${\mathbf{r}}_0=(0,1,0)$ and consists of two path segments generated by sequentially applying Zeeman and $XY$ interactions to the qubit pair. The enclosed solid angle is $-\pi$.