Anderson localization makes adiabatic quantum optimization fail

Understanding NP-complete problems is a central topic in computer science (NP stands for nondeterministic polynomial time). This is why adiabatic quantum optimization has attracted so much attention, as it provided a new approach to tackle NP-complete problems using a quantum computer. The efficiency of this approach is limited by small spectral gaps between the ground and excited states of the quantum computer’s Hamiltonian. We show that the statistics of the gaps can be analyzed in a novel way, borrowed from the study of quantum disordered systems in statistical mechanics. It turns out that due to a phenomenon similar to Anderson localization, exponentially small gaps appear close to the end of the adiabatic algorithm for large random instances of NP-complete problems. This implies that unfortunately, adiabatic quantum optimization fails: The system gets trapped in one of the numerous local minima.

[1]  Lszl Babai.  Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004 , 2004, STOC.

[2]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[3]  Martin Horvat,et al.  Exponential complexity of an adiabatic algorithm for an NP-complete problem , 2006 .

[4]  E. M.,et al.  Statistical Mechanics , 2021, Manual for Theoretical Chemistry.

[5]  October I.  Physical Review Letters , 2022 .

[6]  G. L. Hofacker,et al.  J. E. Mayer und M. Goeppert‐Mayer: Statistical Mechanics, 2nd Ed., John Wiley & Sons Ltd., London, New York, Sydney, Toronto 1977. 491 Seiten, Preis: £ 17.50, $ 29.– , 1978 .

[7]  Edward Farhi,et al.  Quantum adiabatic algorithms, small gaps, and different paths , 2009, Quantum Inf. Comput..

[8]  F. Krzakala,et al.  Simple glass models and their quantum annealing. , 2008, Physical review letters.

[9]  J. I. Latorre,et al.  Simulation of many-qubit quantum computation with matrix product states (6 pages) , 2005, quant-ph/0503174.

[10]  Guilhem Semerjian,et al.  First-order transitions for random optimization problems in a transverse field , 2009 .

[11]  Edward Farhi,et al.  HOW TO MAKE THE QUANTUM ADIABATIC ALGORITHM FAIL , 2005 .

[12]  Ben Reichardt,et al.  The quantum adiabatic optimization algorithm and local minima , 2004, STOC '04.

[13]  Physical Review , 1965, Nature.

[14]  Rémi Monasson,et al.  THE EUROPEAN PHYSICAL JOURNAL B c○ EDP Sciences , 1999 .

[15]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[16]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[17]  A. Young,et al.  Size dependence of the minimum excitation gap in the quantum adiabatic algorithm. , 2008, Physical review letters.

[18]  Seth Lloyd,et al.  Adiabatic quantum computation is equivalent to standard quantum computation , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[19]  V. Choi,et al.  First-order quantum phase transition in adiabatic quantum computation , 2009, 0904.1387.

[20]  M. Sipser,et al.  Quantum Computation by Adiabatic Evolution , 2000, quant-ph/0001106.

[21]  E. Farhi,et al.  A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem , 2001, Science.

[22]  Umesh V. Vazirani,et al.  How powerful is adiabatic quantum computation? , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[23]  L. Zdeborová,et al.  Phase diagram of the 1-in-3 satisfiability problem. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  T. Hogg.  Adiabatic Quantum Computing for Random Satisfiability Problems , 2002, quant-ph/0206059.

[25]  P. Anderson,et al.  A selfconsistent theory of localization , 1973 .

[26]  Akademii︠a︡ medit︠s︡inskikh nauk Sssr.  Journal of physics , 1939 .

[27]  P. W. Anderson,et al.  Local moments and localized States. , 1978, Science.

[28]  P. Anderson.  Absence of Diffusion in Certain Random Lattices , 1958 .