nLab adiabatic quantum computation

Idea

By adiabatic quantum computation one means models of quantum computation on parameterized quantum systems where the quantum gates are unitary transformations on a gapped (and possibly topologically ordered) ground state which are induced, via the quantum adiabatic theorem, by sufficiently slow movement of external parameters.

Often the term adiabatic quantum computation is used by default for optimization problems (“quantum annealing”, see the references below).

On the other hand, the possibly most prominent example of adiabatic quantum computation is often not advertized as such (but see CLBFN 2015), namely topological quantum computation by adiabatic braiding of defect anyons (whose positions is the external parameter, varying in a configuration space of points). This is made explicit in Freedman, Kitaev, Larsen & Wang 2003, pp. 6; Nayak, Simon, Stern & Freedman 2008, §II.A.2 (p. 6); and Cheng, Galitski & Das Sarma 2011, p. 1; see also Arovas, Schrieffer, Wilczek & Zee 1985, p. 1 and Stanescu 2020, p. 321; Barlas & Prodan 2020.

The following graphics shows this with labelling indicative of momentum-space anyons:

(graphics from SS22)

References

In optimization – quantum annealing

Review with focus on optimization problems (quantum annealing):

• Catherine C. McGeoch, Adiabatic Quantum Computation and Quantum Annealing: Theory and Practice Synthesis Lectures on Quantum Computing [doi:10.2200/S00585ED1V01Y201407QMC008))]

• Tameem Albash, Daniel A. Lidar, Adiabatic Quantum Computing, Rev. Mod. Phys. 90 (2018) 015002 [arXiv:1611.04471, doi:10.1103/RevModPhys.90.015002]

• Erica K. Grant and Travis S. Humble, Adiabatic Quantum Computing and Quantum Annealing, Oxford research Encyclopedia (20200) [doi:10.1093/acrefore/9780190871994.013.32]

• Atanu Rajak, Sei Suzuki, Amit Dutta, Bikas K. Chakrabarti, Quantum Annealing: An Overview [arXiv:2207.01827]

See also:

• Wikipedia, Adiabatic quantum computation

• Minjae Jo, Michael Hanks, M. S. Kim, Divide-and-conquer embedding for QUBO quantum annealing [arXiv:2211.02184]

• Yusuke Kimura, Hidetoshi Nishimori, Rigorous convergence condition for quantum annealing, J. Phys. A: Math. Theor. 55 (2022) 435302 [arXiv:2207.12096, doi:10.1088/1751-8121/ac9dce]

Proof that quantum annealing is universal for quantum computation:

• Dorit Aharonov, Wim van Dam, Julia Kempe, Zeph Landau, Seth Lloyd, Oded Regev, Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation, SIAM Review 50 4 (2008) [arXiv:quant-ph/0405098, doi:10.1137/080734479]

A more high-brow mathematical desription via “tangle machines”:

• Avishy Y. Carmi, Daniel Moskovich, §5 of: Tangle Machines, Proc. R. Soc. A 471 (2015) 20150111 [arXiv:1404.2862, doi:10.1098/rspa.2015.0111]

Geometric phase gates

References which consider quantum gates operating by (nonabelian) geometric Berry phases due to adiabatic parameter movement:

• Paolo Zanardi, Mario Rasetti, Holonomic Quantum Computation, Phys. Lett. A 264 (1999) 94-99 [doi:10.1016/S0375-9601%2899%2900803-8, arXiv:quant-ph/9904011 ]

• Jonathan A. Jones, Vlatko Vedral, Artur Ekert, Giuseppe Castagnoli, Geometric quantum computation using nuclear magnetic resonance, Nature 403 (2000) 869–871 [doi:10.1038/35002528]

• Giuseppe Falci, Rosario Fazio, G. Massimo Palma, Jens Siewert & Vlatko Vedral, Detection of geometric phases in superconducting nanocircuits, Nature 407 355–358 (2000) [doi:10.1038/35030052]

• L. M. Duan, J. I. Cirac, P. Zoller, Geometric Manipulation of Trapped Ions for Quantum Computation, Science 292 (2001) 1695 [arXiv:quant-ph/0111086, doi:10.1126/science.1058835]

In topological quantum computation

Specifically, references which make explicit that topological quantum computation with anyons is a form of adiabatic quantum computation:

• Daniel P. Arovas, Robert Schrieffer, Frank Wilczek, Anthony Zee, Statistical mechanics of anyons, Nuclear Physics B 251 (1985) 117-126 (reprinted in Wilczek 1990, p. 173-182) $[$doi:10.1016/0550-3213(85)90252-4$]$

• Michael Freedman, Alexei Kitaev, Michael Larsen, Zhenghan Wang, pp. 6 of Topological quantum computation, Bull. Amer. Math. Soc. 40 (2003), 31-38 (arXiv:quant-ph/0101025, doi:10.1090/S0273-0979-02-00964-3, pdf)

• Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, Sankar Das Sarma, §II.A.2 (p. 6) of: Non-Abelian Anyons and Topological Quantum Computation, Rev. Mod. Phys. 80 1083 (2008) $[$arXiv:0707.1888, doi:10.1103/RevModPhys.80.1083$]$

• Meng Cheng, Victor Galitski, Sankar Das Sarma, Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors, Phys. Rev. B 84 (2011) 104529 $[$arXiv:1106.2549, doi:10.1103/PhysRevB.84.104529$]$

• Chris Cesare, Andrew J. Landahl, Dave Bacon, Steven T. Flammia, Alice Neels, Adiabatic topological quantum computing, Phys. Rev. A 92 (2015) 012336 $[$arXiv:1406.2690, doi:10.1103/PhysRevA.92.012336$]$

• Tudor D. Stanescu, p. 321 of: Introduction to Topological Quantum Matter & Quantum Computation, CRC Press 2020 (ISBN:9780367574116)

• Yafis Barlas, Emil Prodan, Topological braiding of non-Abelian mid-gap defects in classical meta-materials, Phys. Rev. Lett. 124 (2020) 146801 [arXiv:1903.00463, doi:10.1103/PhysRevLett.124.146801]

• E. Macaluso, T. Comparin, L. Mazza, and I. Carusotto, Fusion Channels of Non-Abelian Anyons from Angular-Momentum and Density-Profile Measurements, Phys. Rev. Lett. 123 266801 (2019) [arXiv:1903.03011, doi:10.1103/PhysRevLett.123.266801]