Path Independence in Adiabatic Quantum Computing for Hadamard Gate
DOI:
https://doi.org/10.5614/j.math.fund.sci.2014.46.1.3Keywords:
adiabatic quantum computation, extra Hamiltonian, Hadamard gate, nonlinear interpolation, time complexity, variational principleAbstract
The computation time in adiabatic quantum computing (AQC) is determined by the time limit of the adiabatic evolution, which in turn depends on the evolution path. In this research we have used the variational method to find an optimized path. For the simplest case involving a single qubit and for the most general path involving one or more independent interpolating functions, the result is path independent. This result does not change when there is an extra Hamiltonian term. We have also applied these two scenarios in AQC to a Hadamard gate. Adding an extra Hamiltonian gives a non-trivial result compared to the normal AQC, however it does not result in a speed-up. Moreover, we show that in these two scenarios we can choose an arbitrary path provided that it satisfies the boundary conditions.References
Nielsen, M.A. & Chuang, I.L., Quantum Computation and Quantum Information 10th Anniversary Edition, Cambridge University Press, 2010.
Farhi, E., Goldstone, J., Gutmann, S.& Sipser, M., Quantum Computation by Adiabatic Evolution, arXiv:quant-ph/0001106v1, 2000.
Perdomo, A., Truncik, C., Tubert-Brohman, I., Rose, G.& Aspuru-Guzik, A., On Construction of Model Hamiltonians for Adiabatic Quantum Computation and Its Application to Finding Low-Energy Conformations of Lattice Protein Models, Physical Review A78 012320, 2008.
Garnerone,S., Zanardi, P.& Lidar, D.A., Adiabatic Quantum Algorithm for Search Engine Ranking, arXiv:1109.6546v5 [quant-ph], 2012.
Hogg, T., Adiabatic Quantum Computing for Random Satisability Problems, arXiv:quant-ph/0206059v2, 2004.
Zhang, Y.& Lu, S., Quantum Search by Partial Adiabatic Evolution, arXiv:1007.1528v1[cs.DS], 2010.
Roland, J.& Cerf, N. J., Quantum Search by Local Adiabatic Evolution, Phys. Rev. A65, 042308, 2002.
Ahrensmeier, D., Das, S., Kobes, R., Kunstatter, G.& Zaraket, H., Rapid Data Seacrh Using Adiabatic Quantum Computation, arXiv:quant-ph/0208107v1, 2002.
Ahrensmeier, D., Kobes, R., Kunstatter,G., Zaraket, H.& Das, S., Structured Adiabatic Quantum Search, arXiv:quant-ph/0208065v1, 2008.
Kieu, T.D., Quantum Adiabatic Computation and the Travelling Salesman Problem, arXiv:quant-ph/0601151v2, 2006.
Biamonte J.D. & Love P.J., Realizable Hamiltonians for Universal Adiabatic Quantum Computers, Phys. Rev. A78, 012352, 2008.
Aharonov, D., van Dam, W., Kempe,J., Landau, Z., Lloyd, S. & Regev, O., Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation, arXiv:quant-ph/0405098v2, 2005.
Mizel, A., Lidar, D.A.& Mitchell, M., Simple Proof of Equivalence between Adiabatic Quantum Computation and the Circuit Model, Phys. Rev. Lett.,99, 070502, 2007.
Perdomo, A., Venegas-Andraca, S.E.& Aspuru-Guzik, A., A Study of Heuristic Guesses for Adiabatic Quantum Computation, arXiv:0807.0354v4 [quant-ph], 2010.
Eryigit,R., Gunduc,Y.& Eryigit,R., Local Adiabatic Quantum Search with Different Paths, arXiv:quant-ph/0309201v4, 2004.
Jie,S., Song-Feng,L.U. & Braunstein,S.L., On Models Of Nonlinear Evolution Paths in Adiabatic Quantum Algorithms, arXiv:1301.1115v1 [quant-ph], 2013.
Kinjo, M. & Shimabukuro, K., Speed-up of Neuromorphic Adiabatic Quantum Computation by Local Adiabatic Evolution, 41st IEEE International Symposium on Multiple-Valued Logic, 2011.
Cao, X., Zhuang, J., Ning, X.-J.&Zhang W., Accelerating An Adiabatic Process by Nonlinear Sweeping, arXiv:1201.1902v1 [quant-ph], 2012.
Yousefabadi, N., Optimal Annealing Paths for Adiabatic Quantum Computation, Thesis, Dalhousie University Halifax, Nova Scotia, 2011.
Rezakhani, A.T., Kuo, W.J., Hamma, A., Lidar, D. A.& Zanardi, P., Quantum Adiabatic Brachistochrone, Phys. Rev. Lett. 103,080502, 2009.
Zhao, Y., Reexamination of the Quantum Adiabatic Theorem, Phys. Rev.,A 77, 032109, 2008.
McMahon, D., Quantum Computing Explained, John Wiley & Sons Inc., 2008.
