Abstract
Quantum searching for one of N marked items in an unsorted database of n items is solved in \(\mathcal {O}(\sqrt{n/N})\) steps using Grover’s algorithm. Using nonlinear quantum dynamics with a Gross–Pitaevskii-type quadratic nonlinearity, Childs and Young (Phys Rev A 93:022314, 2016, https://doi.org/10.1103/PhysRevA.93.022314) discovered an unstructured quantum search algorithm with a complexity \(\mathcal {O}( \min \{ 1/g \, \log (g n), \sqrt{n} \}) \), which can be used to find a marked item after \(o(\log (n))\) repetitions, where g is the nonlinearity strength. In this work we develop an quantum search on a complete graph using a time-dependent nonlinearity which obtains one of the N marked items with certainty. The protocol has runtime \(\mathcal {O}(n /(g \sqrt{N(n-N)}))\), where g is related to the time-dependent nonlinearity. We also extend the analysis to a quantum search on general symmetric graphs and can greatly simplify the resulting equations when the graph diameter is less than 4.
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References
Childs, A.M., Young, J.: Optimal state discrimination and unstructured search in nonlinear quantum mechanics. Phys. Rev. A 93, 022314 (2016). https://doi.org/10.1103/PhysRevA.93.022314
Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79(2), 325 (1997)
Bennett, C.H., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing. SIAM J. Comput. 26(5), 1510–1523 (1997)
Shenvi, N., Kempe, J., Whaley, K.B.: Quantum random-walk search algorithm. Phys. Rev. A 67(5), 052307 (2003)
Childs, A.M., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A 70(2), 022314 (2004)
Roland, J., Cerf, N.J.: Quantum search by local adiabatic evolution. Phys. Rev. A 65(4), 042308 (2002)
Meyer, D.A., Wong, T.G.: Nonlinear quantum search using the Gross–Pitaevskii equation. New J. Phys. 15(6), 063014 (2013)
Kahou, M.E., Feder, D.L.: Quantum search with interacting Bose–Einstein condensates. Phys. Rev. A 88(3), 032310 (2013)
Meyer, D.A., Wong, T.G.: Quantum search with general nonlinearities. Phys. Rev. A 89(1), 012312 (2014)
Abrams, D.S., Lloyd, S.: Nonlinear quantum mechanics implies polynomial-time solution for np-complete and# p problems. Phys. Rev. Lett. 81(18), 3992 (1998)
Knuth, D.E.: Big omicron and big omega and big theta. ACM Sigact News 8(2), 18–24 (1976)
Nelson, R.J., Weinstein, Y., Cory, D., Lloyd, S.: Experimental demonstration of fully coherent quantum feedback. Phys. Rev. Lett. 85, 3045–3048 (2000). https://doi.org/10.1103/PhysRevLett.85.3045
Lloyd, S.: Coherent quantum feedback. Phys. Rev. A 62, 022108 (2000). https://doi.org/10.1103/PhysRevA.62.022108
Grimsmo, A.L.: Time-delayed quantum feedback control. Phys. Rev. Lett. 115(6), 060402 (2015)
Wang, S., James, M.R.: Quantum feedback control of linear stochastic systems with feedback-loop time delays. Automatica 52, 277–282 (2015)
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de Lacy, K., Noakes, L., Twamley, J. et al. Controlled quantum search. Quantum Inf Process 17, 266 (2018). https://doi.org/10.1007/s11128-018-2031-6
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DOI: https://doi.org/10.1007/s11128-018-2031-6