Electric nonadiabatic geometric entangling gates on spin qubits

Producing and maintaining entanglement reside at the heart of the optimal construction of quantum operations and are fundamental issues in the realization of universal quantum computation. We here introduce a setup of spin qubits that allows the geometric implementation of entangling gates between the register qubits with any arbitrary entangling power. We show this by demonstrating a circuit through a spin chain, which performs universal nonadiabatic holonomic two-qubit entanglers. The proposed gates are all electric and geometric, which would help to realize fast and robust entangling gates on spin qubits. This family of entangling gates contains gates that are as efficient as the cnot gate in quantum algorithms. We examine the robustness of the circuit to some extent.

Figure 1

Two register spin qubits labeled by $S^{\left(1\right)}$ and $S^{\left(2\right)}$ are coupled through an intermediate ancilla spin qubit labeled as $S^{\left(a\right)}$. The ancilla qubit allows us to evolve two register qubits by means of nonadiabatic quantum holonomy and generate geometric entanglement between the two register spin qubits.

Figure 2

Schematic circuit diagram of the geometric two-qubit entangling gate $U\left(C_0\right)$.

Figure 3

Tetrahedral ($O{A}_1{A}_2{A}_3$) representation of nonlocal two-qubit operations, known as the Weyl chamber. The points $L$, $M$, $N$, $P$, and $Q$, respectively, are the midpoints of the line segments $A_{1}O,A_1{A}_2,A_1{A}_3,A_{3}O$, and $A_{2}O$ with $A_1=(\pi ,0,0)$, $A_2=(\frac{\pi }{2},\frac{\pi }{2},0)$, and $A_3=(\frac{\pi }{2},\frac{\pi }{2},\frac{\pi }{2})$. Every point in the Weyl chamber corresponds to a local equivalence class of nonlocal two-qubit operations. The polyhedron $LMNPQ{A}_2$ corresponds to perfect entanglers in the Weyl chamber. Line $L{A}_2$ identifies special perfect entanglers. The two-qubit gate $U\left(C_0\right)$ belongs to the edge $A_{2}O$ illustrated in blue. Thus, the $U\left(C_0\right)$ is a perfect entangler when it lies along the line segment $A_{2}Q$, which here corresponds to $\pi /8\le \theta \le \pi /4$. In particular, for $\theta =\pi /4$ the gate $U\left(C_0\right)$ is represented by the point $A_2$ in the Weyl chamber, which is the dcnot local equivalence class of special perfect entanglers with local invariants $G1=0$, $G2=-1$ and therefore a maximum entangling power of $2/9$.

Figure 4

Entangling power $e_p\left[U\left(C_0\right)\right]$ as a function of the control parameter $\theta$. For perfect entanglers $\frac{1}{6}\le e_p\le \frac{2}{9}$, which corresponds to $\pi /8\le \theta \le \pi /4$. Maximum entangling power of $2/9$ is achieved by special perfect entanglers at $\theta =\frac{\pi }{4}$.

Figure 5

Schematic illustration of the principal bundle $\mathrm{\Gamma }$ with total space $\mathcal{S}(8,4)$, base space $\mathcal{G}(8,4)$, and the fibers given by the unitary group $\text{U}\left(4\right)$. Each fiber represented by a blue vertical line in the total space is mapped via the natural projection $\pi$ to a point in the base space. For an initial choice of frame $\mathcal{B}\left(0\right)$ for ${\mathcal{H}}_0$, the time-dependent Schrödinger equation lifts the closed path $C_0$ based at ${\mathcal{H}}_0$ in the Grassmann manifold $\mathcal{G}(8,4)$ into the unique horizontal curve ${\mathcal{C}}_0:[0,\tau ]\ni t\to \mathcal{B}\left(t\right)$ in the Stiefel manifold $\mathcal{S}(8,4)$. In general, the horizontal lift is not closed and its two end points are connected by a holonomy element $U\left(\tau \right)$ of the principal bundle $\mathrm{\Gamma }$ associated with the closed path $C_0$, with respect to the connection form $\mathcal{A}$. The holonomy $U\left(\tau \right)$ is the two-qubit entangling gate $U\left(C_0\right)$ when the initial frame $\mathcal{B}\left(0\right)$ is given by the register two-qubit computational frame.

Figure 6

Fidelity $F$ of the special perfect entangler $U(C_0;\theta =\frac{\pi }{4})$, which is carried out only with the XY interaction Hamiltonian term $H_{\text{XY}}$ in Eq. (1), against dimensionless parameters $d_i=\frac{2\mathrm{\Omega }\omega }{\hslash D_i^z}=\frac{\sqrt{|J_1{|}^2+{\left|J_2\right|}^2}}{D_i^z}$, $i=1,2$, corresponding to the Dzyalozhinsky-Moriya spin-orbit interaction. Here, we have used a square pulse function with amplitude $\mathrm{\Omega }$ for the scaling function $\mathrm{\Omega }\left(t\right)$.

Figure 7

Fidelity $F$ of the special perfect entangler $U(C_0;\theta =\frac{\pi }{4})$ against parameter noise. A square pulse function with amplitude $\mathrm{\Omega }$ has been considered for the scaling function $\mathrm{\Omega }\left(t\right)$. For parameter noises, we assume the pulse intensity $\mathrm{\Omega }$ is perturbed independently on the two interacting arms in Fig. 1 coupling the auxiliary spin qubit to the two register qubits as ${\mathrm{\Omega }}_1=\mathrm{\Omega }+{\delta }_1$ and ${\mathrm{\Omega }}_2=\mathrm{\Omega }+{\delta }_2$. The fidelity is plotted as a function of dimensionless parameters $\frac{\mathrm{\Omega }}{{\delta }_i}$, $i=1,2$.

Figure 8

Fidelity of the special perfect entangler $U(C_0;\theta =\frac{\pi }{4})$ against dephasing due to the hyperfine interaction with nuclei. The fidelity is plotted as a function of dimensionless parameter $\lambda =\frac{{\tau }_{\text{hd}}}{{\tau }_{\text{op}}}=\frac{N}{\hslash A{\tau }_{\text{op}}}$, i.e., the ratio between the hyperfine decoherence time ${\tau }_{\text{hf}}$ and the gate operation time ${\tau }_{\text{op}}$. Here, we have considered the square pulse function with amplitude $\mathrm{\Omega }$ for the scaling function $\mathrm{\Omega }\left(t\right)$ and that each electron spin interacts with two nuclear spins homogeneously, when all the coupling constants are the same.