# Loss–DiVincenzo quantum computer

A double quantum dot. Each electron spin SL or SR define one quantum two-level system, or qubit in the Loss-DiVincenzo proposal. A narrow gate between the two dots can modulate the coupling, allowing swap operations.

The Loss–DiVincenzo quantum computer (or spin-qubit quantum computer) is a scalable semiconductor-based quantum computer proposed by Daniel Loss and David P. DiVincenzo in 1997.[1] The proposal was to use as qubits the intrinsic spin-1/2 degree of freedom of individual electrons confined to quantum dots. This was done in a way that fulfilled DiVincenzo Criteria for a scalable quantum computer,[2] 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.

## Implementation of the two-qubit gate

The Loss–DiVincenzo quantum computer operates, basically, using inter-dot gate voltage for implementing Swap (computer science) 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:

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

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

From the pulsed Hamiltonian follows the time evolution operator

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

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

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

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

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

## References

1. ^ D. Loss and D. P. DiVincenzo, "Quantum computation with quantum dots", Phys. Rev. A 57, p120 (1998); on arXiv.org in Jan. 1997
2. ^ D. P. DiVincenzo, in Mesoscopic Electron Transport, Vol. 345 of NATO Advanced Study Institute, Series E: Applied Sciences, edited by L. Sohn, L. Kouwenhoven, and G. Schoen (Kluwer, Dordrecht, 1997); on arXiv.org in Dec. 1996