Inexact and exact quantum searches with a preparation state in a three-dimensional subspace
- 140 Downloads
It is well known that exact quantum searches can be performed by the quantum amplitude amplification algorithm with some phase matching condition. However, recently it was shown that for some preparation states in a three-dimensional subspace, an exact search is impossible to accomplish. We show this impossibility derives from two sources: a problem of state restriction to a cyclic subspace and the solution of a linear system of equations with a \(k\)-potent coefficient matrix. Furthermore, using said system of equations, we introduce a class of preparation states in a three-dimensional space that, even though the quantum amplitude amplification algorithm is unable to find the target state exactly, the same system of equations implies modifications to the quantum amplitude amplification algorithm under which exact solutions in three-dimensional subspaces can be found. We also prove that an inexact quantum search in the 3-potent case can find the target state with high probability if the Grover operator is iterated a number of times inversely proportional to the uncertainty of said 3-potent coefficient matrix as an observable operator.
KeywordsQuantum amplitude amplification algorithm Grover algorithm Exact quantum search \(k\)-potent matrix Cyclic subspace
The authors would like to thank the anonymous reviewers for their helpful comments and suggestions.
- 7.Friedberg, S.H., Insel, A.J., Spence, L.E.: Linear Algebra, 4th edn. Prentice Hall, New Jersey (2003)Google Scholar
- 10.Grover, L.K.: Fixed-point quantum search. Phys. Rev. Lett. 95, 150501 (2005)Google Scholar
- 11.Harrow, A.W., Hassidim, A., Lloyd, S.: Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103, 150502 (2009)Google Scholar
- 12.Høyer, P.: Arbitrary phases in quantum amplitude amplification. Phys. Rev. A 62, 052304 (2000)Google Scholar
- 19.MacLane, S.: Categories for the Working Mathematician, 2nd edn. Springer, New York (1998)Google Scholar
- 21.Roland, J., Cerf, N.J.: Quantum search by local adiabatic evolution. Phys. Rev. A 65, 042308 (2002)Google Scholar