Tunneling and speedup in quantum optimization for permutationsymmetric problems
Tunneling is often claimed to be the key mechanism underlying possible speedups in quantum optimization via quantum annealing (QA), especially for problems featuring a cost function with tall and thin barriers. We present and analyze several counterexamples from the class of perturbed Hamming weight optimization problems with qubit permutation symmetry. We first show that, for these problems, the adiabatic dynamics that make tunneling possible should be understood not in terms of the cost function but rather the semiclassical potential arising from the spincoherent pathintegral formalism. We then provide an example where the shape of the barrier in the final cost function is short and wide, which might suggest no quantum advantage for QA, yet where tunneling renders QA superior to simulated annealing in the adiabatic regime. However, the adiabatic dynamics turn out not be optimal. Instead, an evolution involving a sequence of diabatic transitions through many avoidedlevel crossings, involving no tunneling, is optimal and outperforms adiabatic QA. We show that this phenomenon of speedup by diabatic transitions is not unique to this example, and we provide an example where it provides an exponential speedup over adiabatic QA. In yet another twist, we show that a classical algorithm, spinvector dynamics, ismore »
 Authors:

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 Univ. of Southern California, Los Angeles, CA (United States). Dept. of Physics and Astronomy and Center for Quantum Information Science and Technology
 Univ. of Southern California, Los Angeles, CA (United States). Dept. of Physics and Astronomy and Center for Quantum Information Science and Technology; Univ. of Southern California, Marina del Rey, CA (United States). Information Sciences Inst.
 Univ. of Southern California, Los Angeles, CA (United States). Dept. of Physics and Astronomy, Dept. of Electrical Engineering, Dept. of Chemistry and Center for Quantum Information Science and Technology
 Publication Date:
 Grant/Contract Number:
 AC0500OR22725
 Type:
 Published Article
 Journal Name:
 Physical Review. X
 Additional Journal Information:
 Journal Volume: 6; Journal Issue: 3; Journal ID: ISSN 21603308
 Publisher:
 American Physical Society
 Research Org:
 UTBatelle, LLC, Oak Ridge, TN (United States)
 Sponsoring Org:
 USDOE Office of Science (SC)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; tridiagonal transition matrices; mean passage times; model
 OSTI Identifier:
 1267545
 Alternate Identifier(s):
 OSTI ID: 1299215
Muthukrishnan, Siddharth, Albash, Tameem, and Lidar, Daniel A. Tunneling and speedup in quantum optimization for permutationsymmetric problems. United States: N. p.,
Web. doi:10.1103/PhysRevX.6.031010.
Muthukrishnan, Siddharth, Albash, Tameem, & Lidar, Daniel A. Tunneling and speedup in quantum optimization for permutationsymmetric problems. United States. doi:10.1103/PhysRevX.6.031010.
Muthukrishnan, Siddharth, Albash, Tameem, and Lidar, Daniel A. 2016.
"Tunneling and speedup in quantum optimization for permutationsymmetric problems". United States.
doi:10.1103/PhysRevX.6.031010.
@article{osti_1267545,
title = {Tunneling and speedup in quantum optimization for permutationsymmetric problems},
author = {Muthukrishnan, Siddharth and Albash, Tameem and Lidar, Daniel A.},
abstractNote = {Tunneling is often claimed to be the key mechanism underlying possible speedups in quantum optimization via quantum annealing (QA), especially for problems featuring a cost function with tall and thin barriers. We present and analyze several counterexamples from the class of perturbed Hamming weight optimization problems with qubit permutation symmetry. We first show that, for these problems, the adiabatic dynamics that make tunneling possible should be understood not in terms of the cost function but rather the semiclassical potential arising from the spincoherent pathintegral formalism. We then provide an example where the shape of the barrier in the final cost function is short and wide, which might suggest no quantum advantage for QA, yet where tunneling renders QA superior to simulated annealing in the adiabatic regime. However, the adiabatic dynamics turn out not be optimal. Instead, an evolution involving a sequence of diabatic transitions through many avoidedlevel crossings, involving no tunneling, is optimal and outperforms adiabatic QA. We show that this phenomenon of speedup by diabatic transitions is not unique to this example, and we provide an example where it provides an exponential speedup over adiabatic QA. In yet another twist, we show that a classical algorithm, spinvector dynamics, is at least as efficient as diabatic QA. Lastly, in a different example with a convex cost function, the diabatic transitions result in a speedup relative to both adiabatic QA with tunneling and classical spinvector dynamics.},
doi = {10.1103/PhysRevX.6.031010},
journal = {Physical Review. X},
number = 3,
volume = 6,
place = {United States},
year = {2016},
month = {7}
}