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]} The proposal was to use the intrinsic spin-1/2 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.

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, ^[3] 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. ^[4]

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.

Experimental realizations

Spin qubits mostly have been implemented by locally depleting two-dimensional electron gases in semiconductors such a gallium arsenide, ^{[5] [6]} and germanium. ^[7] Spin qubits have also been implemented in other material systems such as graphene. ^[8] A more recent development is using silicon spin qubits, an approach that is e.g. pursued by Intel. ^{[9] [10]} The advantage of the silicon platform is that it allows using modern semiconductor device fabrication for making the qubits. Some of these devices have a comparably high operation temperature of a few kelvins (hot qubits) which is advantageous for scaling the number of qubits in a quantum processor. ^{[11] [12]}

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. 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
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. Xue, Xiao; Patra, Bishnu; van Dijk, Jeroen P. G.; Samkharadze, Nodar; Subramanian, Sushil; Corna, Andrea; Paquelet Wuetz, Brian; Jeon, Charles; Sheikh, Farhana; Juarez-Hernandez, Esdras; Esparza, Brando Perez; Rampurawala, Huzaifa; Carlton, Brent; Ravikumar, Surej; Nieva, Carlos; Kim, Sungwon; Lee, Hyung-Jin; Sammak, Amir; Scappucci, Giordano; Veldhorst, Menno; Sebastiano, Fabio; Babaie, Masoud; Pellerano, Stefano; Charbon, Edoardo; Vandersypen, Lieven M. K. (2021-05-13). "CMOS-based cryogenic control of silicon quantum circuits". Nature. 593 (7858): 205–210. doi: 10.1038/s41586-021-03469-4 . ISSN   0028-0836.
10. "What Intel is Planning for the Future of Quantum Computing: Hot Qubits, Cold Control Chips, and Rapid Testing - IEEE Spectrum".
11. Yang, C. H.; Leon, R. C. C.; Hwang, J. C. C.; Saraiva, A.; Tanttu, T.; Huang, W.; Camirand Lemyre, J.; Chan, K. W.; Tan, K. Y.; Hudson, F. E.; Itoh, K. M.; Morello, A.; Pioro-Ladrière, M.; Laucht, A.; Dzurak, A. S. (2020-04-16). "Operation of a silicon quantum processor unit cell above one kelvin". Nature. 580 (7803): 350–354. doi: 10.1038/s41586-020-2171-6 . ISSN   0028-0836.
12. Camenzind, Leon C.; Geyer, Simon; Fuhrer, Andreas; Warburton, Richard J.; Zumbühl, Dominik M.; Kuhlmann, Andreas V. (2022-03-03). "A hole spin qubit in a fin field-effect transistor above 4 kelvin". Nature Electronics. 5 (3): 178–183. doi: 10.1038/s41928-022-00722-0 . ISSN   2520-1131.

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.