Universal Set of Quantum Gates for Double-Dot Spin Qubits with Fixed Interdot Coupling

We propose a set of universal gate operations for the singlet-triplet qubit realized by two-electron spins in a double quantum dot, in the presence of a fixed inhomogeneous magnetic field. All gate operations are achieved by switching the potential offset between the two dots with an electrical bias, and do not require time-dependent control of the tunnel coupling between the dots. We analyze the two-electron dynamics and calculate the effective qubit rotation angle as a function of the applied electric bias. We present explicit gate sequences for single-qubit rotations about two orthogonal axes, and a CNOT gate sequence, completing the universal gate set.

Figure 1

Dependence of the two-electron energy levels in a double dot on the bias $\epsilon$ for $\delta h=0$. Here, $S^{\prime }$ denotes the first excited singlet state. In the presence of a small fixed inhomogeneous field $\delta h$ (see inset), the precession axis depends on $\epsilon$, as illustrated by the Bloch spheres.

Figure 2

(a) The three-step sequence Eq. (5) for $X$ rotations on the Bloch sphere. (b) Rotation angles $\chi$, $\varphi$ as functions of $\theta$, producing rotations about $X$ by $\gamma =\pi$, $\pi /2$, $\pi /4$.

Figure 3

(a) Angle $\theta$ as a function of $\epsilon$ for different values of $\delta h$. Here, $U=4\mathrm{meV}$ and $t=5\mu \mathrm{eV}$. (b) Blue lines: maximum value of $\delta h$ as a function of $t$, for different error thresholds for $Z$ rotations. Orange lines: minimum value of $\delta h$ as needed for $X$ rotation, for different values of $U$.

Figure 4

A CNOT is performed by electrically biasing the control qubit, shifting its charge distribution toward the target qubit (upper panel) if it is in a singlet state (a), but leaving the charge distribution unchanged if it is in the triplet state (b). This leads to a conditional shift of the working point of the target qubit (lower panels) and to a conditional operation.