On the quantum adiabatic evolution with the most general system Hamiltonian

AbstractIn this paper, we study the problem that when quantum adiabatic evolution with the most general form of system Hamiltonian will get failed. Here the most general form means that the initial and final Hamiltonians are just designed according to the adiabatic theorem in quantum mechanics. As we will see, even in this most general model of quantum adiabatic evolution, it still exists the possibility that the quantum adiabatic computation can fail totally if some condition is satisfied, which implies the time complexity of the quantum algorithm is infinity. That is, here we propose a rather general criterion for judging whether a quantum adiabatic evolution is successful. This result largely extends the authors’ previous research on this topic, and it may be seen as a further important clue for us when designing quantum algorithms in the framework of adiabatic evolution for some practical problems.

[1]  Jun Zhang,et al.  Schedule path optimization for adiabatic quantum computing and optimization , 2016 .

[2]  New construction for a QMA complete three-local Hamiltonian , 2007, quant-ph/0612113.

[3]  Songfeng Lu,et al.  ON THE ADIABATIC EVOLUTION OF ONE-DIMENSIONAL PROJECTOR HAMILTONIANS , 2012 .

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

[5]  Fang Liu,et al.  Speedup in adiabatic evolution based quantum algorithms , 2012 .

[6]  M. Amin,et al.  Algorithmic approach to adiabatic quantum optimization , 2011, 1108.3303.

[7]  Daniel Nagaj.  Fast universal quantum computation with railroad-switch local Hamiltonians , 2009, 0908.4219.

[8]  B. Terhal,et al.  Universal adiabatic quantum computation via the space-time circuit-to-Hamiltonian construction. , 2014, Physical review letters.

[9]  Jérémie Roland,et al.  Anderson localization makes adiabatic quantum optimization fail , 2009, Proceedings of the National Academy of Sciences.

[10]  Johannes Bausch,et al.  Analysis and limitations of modified circuit-to-Hamiltonian constructions , 2016, Quantum.

[11]  Seth Lloyd,et al.  Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation , 2007, SIAM J. Comput..

[12]  Quntao Zhuang.  Increase of degeneracy improves the performance of the quantum adiabatic algorithm , 2014 .

[13]  Michael J. Dinneen,et al.  Solving the Hamiltonian Cycle Problem using a Quantum Computer , 2019, ACSW.

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

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

[16]  Daniel A Lidar,et al.  Simple proof of equivalence between adiabatic quantum computation and the circuit model. , 2007, Physical review letters.

[17]  Itay Hen,et al.  Exponential Complexity of the Quantum Adiabatic Algorithm for certain Satisfiability Problems , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Alan Mink,et al.  Diffusion Monte Carlo approach versus adiabatic computation for local Hamiltonians , 2017, 1709.03971.

[19]  S. Braunstein,et al.  On Models of Nonlinear Evolution Paths in Adiabatic Quantum Algorithms , 2013, 1301.1115.

[20]  A. Messiah.  Quantum Mechanics , 1961 .

[21]  N. Cerf,et al.  Quantum search by local adiabatic evolution , 2001, quant-ph/0107015.

[22]  Randy Kobes,et al.  Energy and efficiency of adiabatic quantum search algorithms , 2003 .

[23]  Lov K. Grover.  Quantum Mechanics Helps in Searching for a Needle in a Haystack , 1997, quant-ph/9706033.

[24]  S. Lloyd,et al.  Adiabatic and Hamiltonian computing on a 2D lattice with simple two-qubit interactions , 2015, 1509.01278.

[25]  B. Terhal,et al.  Space-time circuit-to-Hamiltonian construction and its applications , 2013, 1311.6101.

[26]  Jie Sun,et al.  On the General Class of Models of Adiabatic Evolution , 2016, Open Syst. Inf. Dyn..