Spin qubit quantum computer

The spin qubit quantum computer is a quantum computer based on controlling the spin of charge carriers (electrons and electron holes) in semiconductor devices.^[1] The first spin qubit quantum computer was first proposed by Daniel Loss and David P. DiVincenzo in 1997,^[1][2] also known as the Loss–DiVicenzo quantum computer.^{[citation needed]} The proposal was to use the intrinsic spin-½ degree of freedom of individual electrons confined in quantum dots as qubits. This should not be confused with other proposals that use the nuclear spin as qubit, like the Kane quantum computer or the nuclear magnetic resonance quantum computer.

Spin qubits so far have been implemented by locally depleting two-dimensional electron gases in semiconductors such a gallium arsenide,^[3][4] silicon^[5] and germanium.^[6] Spin qubits have also been implemented in graphene.^[7]

Loss–DiVicenzo proposal

 

A double quantum dot. Each electron spin S_L or S_R 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,^[8] 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.^[9]

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.

See also

• Kane quantum computer
• Quantum dot cellular automaton

References

1. Vandersypen, Lieven M. K.; Eriksson, Mark A. (2019-08-01). "Quantum computing with semiconductor spins". Physics Today. 72 (8): 38. Bibcode:2019PhT....72h..38V. doi:10.1063/PT.3.4270. ISSN 0031-9228. S2CID 201305644.
2. Loss, Daniel; DiVincenzo, David P. (1998-01-01). "Quantum computation with quantum dots". Physical Review A. 57 (1): 120–126. arXiv:cond-mat/9701055. Bibcode:1998PhRvA..57..120L. doi:10.1103/physreva.57.120. ISSN 1050-2947.
3. Petta, J. R. (2005). "Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum Dots". Science. 309 (5744): 2180–2184. Bibcode:2005Sci...309.2180P. doi:10.1126/science.1116955. ISSN 0036-8075. PMID 16141370. S2CID 9107033.
4. Bluhm, Hendrik; Foletti, Sandra; Neder, Izhar; Rudner, Mark; Mahalu, Diana; Umansky, Vladimir; Yacoby, Amir (2010). "Dephasing time of GaAs electron-spin qubits coupled to a nuclear bath exceeding 200 μs". Nature Physics. 7 (2): 109–113. doi:10.1038/nphys1856. ISSN 1745-2473.
5. Wang, Siying; Querner, Claudia; Dadosh, Tali; Crouch, Catherine H.; Novikov, Dmitry S.; Drndic, Marija (2011). "Collective fluorescence enhancement in nanoparticle clusters". Nature Communications. 2 (1): 364. Bibcode:2011NatCo...2..364W. doi:10.1038/ncomms1357. ISSN 2041-1723. PMID 21694712.
6. Watzinger, Hannes; Kukučka, Josip; Vukušić, Lada; Gao, Fei; Wang, Ting; Schäffler, Friedrich; Zhang, Jian-Jun; Katsaros, Georgios (2018-09-25). "A germanium hole spin qubit". Nature Communications. 9 (1): 3902. arXiv:1802.00395. Bibcode:2018NatCo...9.3902W. doi:10.1038/s41467-018-06418-4. ISSN 2041-1723. PMC 6156604. PMID 30254225.
7. Trauzettel, Björn; Bulaev, Denis V.; Loss, Daniel; Burkard, Guido (2007). "Spin qubits in graphene quantum dots". Nature Physics. 3 (3): 192–196. arXiv:cond-mat/0611252. Bibcode:2007NatPh...3..192T. doi:10.1038/nphys544. ISSN 1745-2473. S2CID 119431314.
8. 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
9. Barenco, Adriano; Deutsch, David; Ekert, Artur; Josza, Richard (1995). "Conditional Quantum Dynamics and Logic Gates". Phys. Rev. Lett. 74 (20): 4083–4086. arXiv:quant-ph/9503017. Bibcode:1995PhRvL..74.4083B. doi:10.1103/PhysRevLett.74.4083. PMID 10058408. S2CID 26611140.

External links

• QuantumInspire online platform from Delft University of Technology, allows building and running quantum algorithms on "Spin-2" a 2 silicon spin qubits processor.