Implementation of the two-qubit gate

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 ${\displaystyle \tau _{\rm {s}}.}$

${\displaystyle k}$ is the Boltzmann constant and ${\displaystyle T}$ is the temperature in Kelvin.

From the pulsed Hamiltonian follows the time evolution operator

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

where ${\displaystyle {\mathcal {T}}}$ is the time-ordering symbol.

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

This pulse run for half the time (with ${\displaystyle J_{0}\tau _{\rm {s}}=\pi /2}$) results in a square root of swap gate, ${\displaystyle U_{\rm {sw}}^{1/2}.}$

The "XOR" gate may be achieved by combining ${\displaystyle U_{\rm {sw}}^{1/2}}$ operations with individual spin rotation operations:

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

The ${\displaystyle U_{\rm {XOR}}}$ operator is a conditional phase shift (controlled-Z) for the state in the basis of ${\displaystyle \mathbf {S} _{\rm {L}}+\mathbf {S} _{\rm {R}}}$.^[2]: 4 It can be made into a CNOT gate by surrounding the desired target qubit with Hadamard gates.