All-electrical control of quantum gates for single heavy-hole spin qubits

In this paper several nanodevices which realize basic single heavy-hole qubit operations are proposed and supported by time-dependent self-consistent Poisson-Schrödinger calculations using a four band heavy-hole–light-hole model. In particular we propose a set of nanodevices which can act as Pauli $X$, $Y$, $Z$ quantum gates and as a gate that acts similar to a Hadamard gate (i.e., it creates a balanced superposition of basis states but with an additional phase factor) on the heavy-hole spin qubit. We also present the design and simulation of a gated semiconductor nanodevice which can realize an arbitrary sequence of all these proposed single quantum logic gates. The proposed devices exploit the self-focusing effect of the hole wave function which allows for guiding the hole along a given path in the form of a stable solitonlike wave packet. Thanks to the presence of the Dresselhaus spin-orbit coupling, the motion of the hole along a certain direction is equivalent to the application of an effective magnetic field which induces in turn a coherent rotation of the heavy-hole spin. The hole motion and consequently the quantum logic operation is initialized only by weak static voltages applied to the electrodes which cover the nanodevice. The proposed gates allow for an all electric and ultrafast (tens of picoseconds) heavy-hole spin manipulation and give the possibility to implement a scalable architecture of heavy-hole spin qubits for quantum computation applications.

Figure 1

Cross section of the nanodevice.

Figure 2

Modulus square of the components of the hole Luttinger spinor wave function: $|{\psi }_{HH}^{\uparrow }(x,y,t_0){|}^2$, $|{\psi }_{\text{LH}}^{\uparrow }(x,y,t_0){|}^2$, $|{\psi }_{\text{LH}}^{\downarrow }(x,y,t_0){|}^2$, and $|{\psi }_{\text{HH}}^{\downarrow }(x,y,t_0){|}^2$ in the spin up ground state in the presence of the DSOI interaction. It can be noticed that ${\psi }_{\text{HH}}^{\uparrow }(x,y,t_0)$ has the biggest contribution to the total wave function $\Psi (x,y,t_0)$, while the LH components ${\psi }_{\text{LH}}^{\uparrow }(x,y,t_0)$, ${\psi }_{\text{LH}}^{\downarrow }(x,y,t_0)$ are about 4 orders of magnitude smaller. In case of absence of the DSOI there are only two nonzero components ${\psi }_{\text{HH}}^{\uparrow }(x,y,t_0)$, ${\psi }_{\text{LH}}^{\downarrow }(x,y,t_0)$ and their modulus square looks identical as for those in the presence of DSOI as depicted in the above figure.

Figure 3

The time evolution of the expectation value of the HH spin components $s_x\left(t\right),s_y\left(t\right),s_z\left(t\right)$ (a), average position $x\left(t\right),y\left(t\right)$ of the hole wave packet (b), and the occupation probabilities $P_{|\text{HH}\uparrow \rangle }\left(t\right),P_{|\text{HH}\downarrow \rangle }\left(t\right),P_{|\text{LH}\uparrow \rangle }\left(t\right),P_{|\text{LH}\downarrow \rangle }\left(t\right)$ of the hole basis states (c), for the quantum Pauli $X$ (NOT) gate which is covered by the system of electrodes $e_1–e_5$ presented in (d). (c) The left (right) axis corresponds to the probability of finding the hole in the HH (LH) spin states. (d) The solid red line represents the hole trajectory (the orange arrow represents the direction of motion of the hole). The hole is initially confined under electrode $e_1$ and moves in the $+x$ direction. The HH spin qubit state is depicted on the Bloch spheres at times $t_0$, $t_A$, $t_B$, $t_C$, $t_D$ of the quantum gate operation cycle. The contour plots represent the charge density $\rho (x,y,t)$ at a few selected moments in time.

Figure 4

The same as Fig. 3, but for the quantum Pauli $Y$ gate which is covered by the system of electrodes $e_1–e_5$ presented in (d).

Figure 5

The same as Fig. 3, but for the quantum Pauli $Z$ gate which is covered by the system of electrodes $e_1–e_5$ presented in (d).

Figure 6

The same as Fig. 3, but for the quantum $U_S$ gate which is covered by the system of electrodes $e_1–e_5$ presented in (d).

Figure 7

The system of electrodes which covers the combo nanodevice (a). The electrodes are labeled with $e_1–e_{11}$. The interelectrode distance is about 7 nm. Fragment of the scalable architecture (b) consisting of four HH spin qubits on which the proposed quantum gates can be applied one by one and in an arbitrary sequence.

Figure 8

Time evolution of the HH spin components for the Pauli $X$ (a), $Y$ (b), $Z$ (c), and $U_S$ gate (d) realized by the gated combo nanodevice which is covered by the system of electrodes shown in Fig. 7a. The electrode voltage scheme which is responsible for initialization and control of a particular quantum logic operation can be found in Table . The corresponding occupation probabilities for the HH and LH spin states are depicted in (a${}^{\prime }$), (b${}^{\prime }$), (c${}^{\prime }$), and (d${}^{\prime }$). The hole trajectory for each quantum gate realized by the combo nanodevice can be found in Fig. 9 ($a_j$), where $j$ denotes the quantum gate.

Figure 9

The electrostatic potential distribution ${\varphi }_0(x,y,z_0)$ in the quantum well layer $\left(z_0\right)$ which comes from the presence of the electrodes to which a certain voltage $V_i$ is applied according to the scheme from Table , where $i=1,...,11$. The ${\varphi }_0(x,y,z_0)$ is plotted for four crucial moments of time of the quantum gate operation process: ($a_j$) $t\le t_0$, ($b_j$) $t_{\text{start}}<t<t_{\text{change}}$, ($c_j$) $t_{\text{change}}<t<t_{\text{stop}}$, ($d_j$) $t_{\text{stop}}<t$. The hole trajectory is depicted in ($a_j$), where $j$ denotes a particular quantum gate. The red isolines denote a positive electrostatic potential distribution while the blue lines a negative one.