Holonomic quantum computation with electron spins in quantum dots

With the help of the spin-orbit interaction, we propose a scheme to perform holonomic single-qubit gates on the electron spin confined to a quantum dot. The manipulation is done in the absence (or presence) of an applied magnetic field. By adiabatic changing the position of the confinement potential, one can rotate the spin state of the electron around the Bloch sphere in semiconductor heterostructures. The dynamics of the system is equivalent to employing an effective non-Abelian gauge potential whose structure depends on the type of the spin-orbit interaction. As an example, we find an analytic expression for the electron spin dynamics when the dot is moved around a circular path (with radius $R$) on the two dimensional electron gas (2DEG) and show that all single-qubit gates can be realized by tuning the radius and orientation of the circular paths. Moreover, using the Heisenberg exchange interaction, we demonstrate how one can generate two-qubit gates by bringing two quantum dots near each other, yielding a scalable scheme to perform quantum computing on arbitrary $N$ qubits. This proposal shows a way of realizing holonomic quantum computers in solid-state systems.

Figure 1

A set of metallic gates, deposited on top of the heterostructure, control the quantum dot position in the ($x$,$y$) plane. The gates that define the current dot position are highlighted in a darker color. The superimposed gates are separated from each other by an insulating layer. The measurement site is used to initialize and read out the spin state of the quantum dot with the help of additional controls (not shown here). By applying time-dependent voltages to the gates, one is able to move the quantum dot along a desired trajectory, thus forming, for instance, a Wilson loop (dashed line).

Figure 2

Same as in Fig. 1, except that the quantum dot is defined using a single layer of metallic gates deposited on top of a ring. The ring can be obtained out of a heterostructure by means of etching or defined with the help of an atomic force microscope using the oxidation technique of Refs. [27, 28].

Figure 3

(a) Trajectory of the quantum dot center ${\mathit{r}}_0(t)$ and the evolution of the spin state due to displacement. The spin state changes when going from ${\mathit{r}}_0$ to ${\mathit{r}}_0+\delta {\mathit{r}}_0$, due to the infinitesimal transformation [Eq. (A17)]. Since the directions of $\delta {\mathit{r}}_0$ can be different in different parts of the curve, the infinitesimal transformations do not commute with each other and have to be ordered along the path of integration. (b) Two oriented circles with the same radii $R$ but different centers (with their corresponding holonomic rotations at $P$) show the typical paths for the quantum dot on the 2DEG. (c) Moving the dot along the circles will rotate the spin of the electron around the $\mathit{\eta }$ axis. For instance, blue (1) and red (2) circles in (b) correspond to $\varphi =0$ and $\varphi =-\frac{\pi }{2}$, respectively.

Figure 4

Schematic of the two-qubit rotation setup. The confined electrons are brought together at the intersection of the circular paths (with their potential profile shown in the inset at this point), where they interact via Heisenberg exchange interaction. The (holonomic) single-qubit operations are done at position $P$ ($P^{\prime }$) while the (exchange) two-qubit gates are performed at position $G$ ($G^{\prime }$).