Localisation at the target state

There are two approximate eigenvectors at the crossing, that is,|^ωmiwhich is used as start state and the other, the perturber state|^νλiwhose eigenphase crosses diagonal through the spectrum (figure 3.4) which has ideally a big overlap with the marked state|^sviof the search algorithm. To analyse if this is indeed the case, it is once more important to remember that b has been defined as the scalar product b= h^sv|^νλiⁱⁿ (3.3.5). For λ=1 and d≫1, the first derivative of g(λ)as calculated in (3.3.36) results in

h^sv|^νλi|λ=1≈

s 1

2+_2d^{1 . (3.3.47)}

This is in fact a crucial point as the approach in this thesis is to characterise U_λ by concentrating on the avoided crossings in the spectrum and to reduce the problem to a two dimensional subspace spanned by|^ωmi^and|^νλi^{. Since}|^ωmiis the starting state of the algorithm, and for m=0 also the uniform distribution, this result proves that the target state|^svihas a notable overlap with the other approximate eigenvector.

Only a relatively large b enables this approach to be appropriate since it proves that the perturber state|^svihas a sufficiently large contribution in the 2 dimensional subspace in question.

Although the overlap given in (3.3.47) does not seem much, it leads to a probability of nearly¹₂ which is in particular not d-dependent.

This result is notable since the algorithm for the m=0 crossing starts in the uniform distribution in position space. Half of the vertices have a distance to~_v= ~0 with even and half of them a distance with odd Hamming weight. Each application of Uλ=1

changes the parity of the Hamming weight; the amplitude at vertices in even distances are moved to vertices with odd distance and vice versa. To localise more than half of the probability on the target vertex would mean that there exists a time T such that after T applications of U_λ=1more than half of the probability is localised on vertices with an even distance, that is the target vertex|^viitself, which is impossible. As a consequence the search algorithm that starts in the uniform distribution cannot achieve a localisation probability higher than¹₂.

However, numerical simulations as shown in figure 3.2 suggest that|^νλiis localised on the hypercube somewhere near the target vertex|^vi. To specify this somewhere near the target vertex analytical, it is interesting to calculate the scalar producth^νλ|^U|^sviwhich gives a lower bound to the amplitude localised on the nearest neighbours.

Obviously, the state U|^sviis localised on the d nearest neighbours of the target vertex

since the state|^svihas been propagated once. An ad hoc argument predicts that the probability at the nearest neighbours is equal or larger than the probability on the marked vertex. It is known from numerics (figure 3.2) that the localisation at the target vertex holds for a few time steps. It can be deduced that the amplitude on the target vertex will propagate to the nearest neighbours when Uλis applied once and some of the amplitude of the nearest neighbours propagates to the target vertex to replace the amplitude that was shifted away. To keep the probability nearly constant the amplitude localised on the nearest neighbours cannot be less than b.

However, the calculation is straight forward and yields

h^νλ|^U|^svi =^be−iλπ−^12−d−1

which provides a more rigorous proof for the ad hoc argument discussed above. In the last step, the sum formula (3.3.11) and e^ig(λ) =e^iωm have been used. Note that U|^svi^is only one state out of d²states localised on the nearest neighbours and the probability that is actually localised on the nearest neighbours is very likely to extend the probability in U|^svi^.

For the main crossing at λ=1 the result is remarkable since the amplitude of|^νλi^{in the} two orthogonal states|^svi^{and U}|^sviis, up to a phase, equal to b and since it approaches

√1

2for d→∞. The probability P, to measure|^νλ=1ieither at the marked vertex|^vi^or on its nearest neighbours is

P=2b²= ¹−^2−d

1+_4d¹ −³·^2−d−1+ O^{e−d8d (3.3.51)}

and approaching 1 if d→∞. Therefore the state|^νλican be considered localised on the marked vertex and its nearest neighbours. A measurement once the search has localised in|^νλigives either the target vertex or one of its nearest neighbours with a probability close to 1. Note that for the crossings where m 6= ^{0, g′}(λ) ≈ ^π2 is known from the numerics only and the result has not been verified analytically.

Notably, the contribution on the nearest neighbours is not an arbitrary state but U|^svi^. As a consequence, if the outcome of the measurement is not the target vertex itself, the coin state at the measured vertex points into the direction of the target vertex as can be verified by the definition of the shift operator S in (3.1.3). Therefore the target vertex can be identified and confirmed after the measurement with at most two more oracle queries first on the vertex determined by the measurement and, if this fails to be the target state, on the neighbouring vertex of this vertex singled out by the coin state.

So far, it has been shown that the two dimensional subspace contains the starting state and a state localised at the target vertex. To prove that the search algorithm on the hypercube indeed succeeds to localise at the marked vertex in a timeO^√1_N^, it remains to show that there exists an integer T, such that U_λ^T|^ωmihas a strong overlap with|^νλi and to calculate the time T.