Abstract
Numerous sufficient conditions for adiabaticity of the evolution of a driven quantum system have been known for quite a long time. In contrast, necessary adiabatic conditions are scarce. Recently a practicable necessary condition well suited for many-body systems has been proved. Here we tailor this condition for estimating run times of adiabatic quantum algorithms. As an illustration, the condition is applied to the adiabatic algorithm for searching in an unstructured database (adiabatic Grover search algorithm). We find that the thus obtained lower bound on the run time of this algorithm reproduces \( \sqrt{N} \) scaling (with N being the number of database entries) of the explicitly known optimum run time. This is in contrast to the poor performance of the known sufficient adiabatic conditions, which guarantee adiabaticity only for a run time on the order of O(N), which does not constitute any speedup over the classical database search. This observation highlights the merits of the new adiabatic condition and its potential relevance to adiabatic quantum computing.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Born, Z. Phys., 40, 167 (1926).
M. Born and V. Fock, Z. Phys., 51, 165 (1928).
T. Kato, J. Phys. Soc. Jpn., 5, 435 (1950).
T. Albash and D. A. Lidar, Rev. Mod. Phys., 90, 015002 (2018).
P. Pfeifer and J. Frohlich, Rev. Mod. Phys., 67, 759 (1995).
W. K. A. Salem and J. Frohlich, Commun. Math. Phys., 273, 651 (2007).
S. Deffner and S. Campbell, J. Phys. A: Math. Theor., 50, 453001 (2017).
D. M. Tong, Phys. Rev. Lett., 104, 120401 (2010).
G. Rigolin and G. Ortiz, Phys. Rev. A, 85, 062111 (2012).
S. Boixo and R. D. Somma, Phys. Rev. A, 81, 032308 (2010).
Z.-Y. Wang and M. B. Plenio, Phys. Rev. A, 93, 052107 (2016)
M. Zhao and J. Wu, Phys. Rev. Lett., 106, 138901 (2011).
D. Comparat, Phys. Rev. Lett., 106, 138902 (2011).
D. M. Tong, Phys. Rev. Lett., 106, 138903 (2011).
D. Li and M.-H. Yung, New J. Phys., 16, 053023 (2014).
R. D. Somma, D. Nagaj, and M. Kieferova, Phys. Rev. Lett., 109, 050501 (2012).
O. Lychkovskiy, O. Gamayun, and V. Cheianov, Phys. Rev. Lett., 119, 200401 (2017).
D. Thouless, Phys. Rev. B, 27, 6083 (1983).
D. M. Gangardt and A. Kamenev, Phys. Rev. Lett., 102, 070402 (2009).
O. Lychkovskiy, O. Gamayun, and V. Cheianov, AIP Conf. Proc., 1936, 020024 (2018).
O. Gamayun, O. Lychkovskiy, and V. Cheianov, Phys. Rev. E, 90, 032132 (2014).
M. Schecter, D. M. Gangardt, and A. Kamenev, Phys. Rev. E, 92, 016101 (2015).
O. Gamayun, O. Lychkovskiy, and V. Cheianov, Phys. Rev. E, 92, 016102 (2015).
J. Roland and N. J. Cerf, Phys. Rev. A, 65, 042308 (2002).
E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, “Quantum computation by adiabatic evolution,” arXiv quant-ph/0001106 (2000).
E. Farhi, J. Goldstone, S. Gutmann, et al., “A quantum adiabatic evolution algorithm applied to random instances of an np-complete problem,” Science, 292, 472 (2001).
T. D. Kieu, “A new class of time–energy uncertainty relations for time-dependent Hamiltonians,” arXiv:1702.00603 (2017).
T. D. Kieu, “The travelling salesman problem and adiabatic quantum computation: an algorithm,” arXiv:1801.07859 (2018).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lychkovskiy, O. A Necessary Condition for Quantum Adiabaticity Applied to the Adiabatic Grover Search. J Russ Laser Res 39, 552–557 (2018). https://doi.org/10.1007/s10946-018-9751-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10946-018-9751-z