. Each electron spin
SL or
SR define one quantum two-level system, or a spin
qubit in the Loss-DiVincenzo proposal. A narrow gate between the two dots can modulate the coupling, allowing
swap operations. The Loss–DiVicenzo quantum computer proposal tried to fulfill
DiVincenzo's criteria for a scalable quantum computer, namely: • identification of well-defined qubits; • reliable state preparation; • low decoherence; • accurate quantum gate operations and • strong quantum measurements. A candidate for such a quantum computer is a
lateral quantum dot system. Earlier work on applications of quantum dots for quantum computing was done by Barenco et al.
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: :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 \Delta E is much greater than \; kT • the pulse time scale \tau_{\rm s} is greater than \hbar / \Delta E , so there is no time for transitions to higher orbital levels to happen and • the
decoherence time \Gamma ^{-1} is longer than \tau_{\rm s}. k is the
Boltzmann constant and T is the temperature in
Kelvin. From the pulsed Hamiltonian follows the
time evolution operator :U_{\rm s}(t) = {\mathcal{T}} \exp\left\{ -i\int_0^t dt' H_{\rm s}(t') \right\}, where {\mathcal{T}} is the
time-ordering symbol. We can choose a specific duration of the pulse such that the integral in time over J(t) gives J_0 \tau_{\rm s} = \pi \pmod{2\pi}, and U_{\rm s} becomes the swap operator U_{\rm s} (J_0 \tau_{\rm s} = \pi) \equiv U_{\rm sw}. This pulse run for half the time (with J_0 \tau_{\rm s} = \pi /2) results in a
square root of swap gate, U_{\rm sw}^{1/2}. The "XOR" gate may be achieved by combining U_{\rm sw}^{1/2} operations with individual
spin rotation operations: :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 U_{\rm XOR} operator is a
conditional phase shift (controlled-Z) for the state in the basis of \mathbf{S}_{\rm L} + \mathbf{S}_{\rm R}. It can be made into a
CNOT gate by surrounding the desired target qubit with
Hadamard gates. ==Experimental realizations==