Spin qubit quantum computer

The Loss–DiVincenzo quantum computer operates, basically, using inter-dot gate voltage for implementing swap operations and local magnetic fields (or any other local spin manipulation) for implementing the controlled NOT gate (CNOT gate).

The swap operation is achieved by applying a pulsed inter-dot gate voltage, so the exchange constant in the Heisenberg Hamiltonian becomes time-dependent:

${\displaystyle H_{\rm {s}}(t)=J(t)\mathbf {S} _{\rm {L}}\cdot \mathbf {S} _{\rm {R}}.}$

This description is only valid if:

• the level spacing in the quantum-dot ${\displaystyle \Delta E}$ is much greater than ${\displaystyle \;kT}$;
• the pulse time scale ${\displaystyle \tau _{\rm {s}}}$ is greater than ${\displaystyle \hbar /\Delta E}$, so there is no time for transitions to higher orbital levels to happen and
• the decoherence time ${\displaystyle \Gamma ^{-1}}$ is longer than ${\displaystyle \tau _{\rm {s}}}$.

From the pulsed Hamiltonian follows the time evolution operator

${\displaystyle U_{\rm {s}}(t)=T\exp \left\{-i\int _{0}^{t}dt'H_{\rm {s}}(t')\right\}.}$

We can choose a specific duration of the pulse such that the integral in time over ${\displaystyle J(t)}$ gives ${\displaystyle J_{0}\tau _{\rm {s}}=\pi ({\text{mod}}2\pi ),}$ and ${\displaystyle U_{\rm {s}}}$ becomes the swap operator ${\displaystyle U_{\rm {s}}(J_{0}\tau _{\rm {s}}=\pi )\equiv U_{\rm {sw}}}$.

The XOR gate may be achieved by combining ${\displaystyle {\sqrt {\text{swap}}}}$ (square root of swap) operations with individual spin operations:

${\displaystyle U_{\rm {XOR}}=e^{i{\frac {\pi }{2}}S_{\rm {L}}^{z}}e^{-i{\frac {\pi }{2}}S_{\rm {R}}^{z}}U_{\rm {sw}}^{1/2}e^{i\pi S_{\rm {L}}^{z}}U_{\rm {sw}}^{1/2}.}$

This operator gives a conditional phase for the state in the basis of ${\displaystyle \mathbf {S} _{\rm {L}}+\mathbf {S} _{\rm {R}}}$.