Universal Quantum Computation through Control of Spin-Orbit Coupling

We propose a method for quantum computation which uses control of spin-orbit coupling in a linear array of single electron quantum dots. Quantum gates are carried out by pulsing the exchange interaction between neighboring electron spins, including the anisotropic corrections due to spin-orbit coupling. Control over these corrections, even if limited, is sufficient for universal quantum computation over qubits encoded into pairs of electron spins. The number of voltage pulses required to carry out either single-qubit rotations or controlled-Not gates scales as the inverse of a dimensionless measure of the degree of control of spin-orbit coupling.

Figure 1

Four quantum dots forming two neighboring logical qubits, 12 and 34. The dots lie in the (001) plane and are aligned along the [110] direction. The spin-orbit induced spin precession axis is parallel to the $[1\overline{1}0]$ direction. Exchange gates between spins within a logical qubit are used for single-qubit rotations. Two-qubit gates are carried out using exchange gates acting on spins 2 and 3.

Figure 2

Rotation axes in the pseudospin space of two neighboring spins. The wedge lying in the plane perpendicular to $x$ and sweeping out the angle ${\theta }_m$ contains rotation axes which can be achieved using time-symmetric pulses and control of spin-orbit coupling. Successive $\pi$ rotations about ${\widehat{\mathbf{n}}}_1$ and ${\widehat{\mathbf{n}}}_2$, with ${\widehat{\mathbf{n}}}_1·{\widehat{\mathbf{n}}}_2=\cos \theta$, result in a $2\theta$ rotation about the $x$ axis. The effect of errors in the rotation angles, ${\delta }_1$ and ${\delta }_2$, on the net rotation axis is also shown. Here ${\widehat{z}}^{\prime }\parallel ({\widehat{\mathbf{n}}}_1+{\widehat{\mathbf{n}}}_2)$ and ${\widehat{y}}^{\prime }={\widehat{z}}^{\prime }\times \widehat{x}$.

Figure 3

Proposed CNOT construction. Each line corresponds to a logical qubit. $U(\Lambda ,\Phi )$ is defined in (10) with $\Lambda =(2n+1)\pi$. The value of $\Phi$ depends on the procedure used to carry out the CNOT. $H=({\sigma }_x+{\sigma }_z)/\sqrt{2}$ is a Hadamard gate and $R_x(\psi )$ is a single-qubit rotation about the $x$ axis through an angle $\psi$ equal modulo $2\pi$ to $(\Phi +\Lambda )/2$.