Grover’s algorithm provides a high probability in finding the object only for a large N. The probability will be lower as N decreases. Grover [163], however, also proposed that the Walsh-Hadamard transformation used in the original version can be replaced by almost any arbitrary unitary operator and the phase angles of rotation can be arbitrarily used as well, instead of the original π-angles. The utility of the arbitrary phase angles in fact can provide the possibility for finding the marked item with certainty, no matter whether N is large or not, if these angles obey a so-called matching condition.
Some typical literatures concerning with the matching condition will be mentioned here. Long et al. [164, 165] have derived the relation φ = θ, where φ and θ are the phases used in the algorithm, using an SO(3) picture. Høyer [166] , on the other hand, has proved a relation tan(φ/2) = tan(θ/2)(1 − 2/N), and claimed that the relation φ = θ is an approximation to this case. Recently, a more general matching condition has been derived by Long et al. [167] , also using the SO(3) picture . In the last article, however, only the certainty for finding the marked state is ensured. In fact a phase angle appearing in the amplitude of the final state after searching will remain. If the final state should be necessary for a future application, i.e., if it should interact with other states, this phase angle will be important for quantum interferences, but it can not be given in the SO(3) representation. We therefore intend to derive the matching condition in the SU(2) picture. In addition, we will also give a more concise formula for evaluating the number of the iterations needed in the searching and deduce the final state in a complete form as eiδ|τi, where |τi is the marked state. The optimal choice of the phase angles will be
discussed, too [168].
Suppose in a two-dimensional, complex Hilbert space we have a marked state |τi to be searched by successively operating a Grover’s kernel G on an arbitrary initial state |si.
The Grover kernel is a product of two unitary operators Iτ and Iη, given by
Iτ = I + (eiφ− 1) |τi hτ| , (9.1)
Iη = I + (eiθ − 1)U |ηi hη| U−1 ,
where U is an arbitrary unitary operator, |ηi is another unit vector in the space, and φ and θ are two phase angles. It should be noted that the phases φ and θ actually are the differences φ = φ2 − φ1 and θ = θ2 − θ1, where φ2, φ1, θ2, and θ1, as depicted in Refs.
[169, 170], denote the rotating angles to |τi, the vector orthogonal to |τi, U |ηi, and the vector orthogonal to U |ηi, respectively. The Grover kernel can be expressed in a matrix form as long as an orthonormal set of basis vectors is designated, so we simply choose
|Ii = |τi and |IIi = (U |ηi − Uτ η|τi)/l , (9.2)
where Uτ η = hτ| U |ηi and l = (1 − |Uτ η|2)1/2. Letting Uτ η = sin(β)eiα, we can write, from (9.2),
U |ηi = sin(β)eiα|Ii + cos(β) |IIi , (9.3)
and the Grover kernel can now be written
G = − IηIτ
= −
eiφ(1 + (eiθ− 1) sin2(β)) (eiθ − 1) sin(β) cos(β)eiα eiφ(eiθ− 1) sin(β) cos(β)e−iα 1 + (eiθ− 1) cos2(β)
. (9.4)
In the searching process, the Grover kernel is successively operated on the initial state
|si. We wish that after, say, m iterations the operation the final state will be orthogonal
to the basis vector |IIi so that the probability for finding the marked state |τi will exactly be unity. Alternatively, in mathematical expression, we wish to fulfill the requirement
hII| Gm|si = 0 , (9.5)
since then
|hτ| Gm|si| = |hI| Gm|si| = 1 . (9.6)
The eigenvalues of the Grover kernel G are
λ1,2 = −ei(φ+θ2 ±w) , (9.7)
where the angle w is defined by
cos(w) = cos(φ − θ
2 ) − 2 sin(φ 2) sin(θ
2) sin2(β) . (9.8)
The normalized eigenvectors associated with these eigenvalues are computed:
|g1i =
e−iφ2eiαcos(x) sin(x)
, |g2i =
− sin(x) eiφ2e−iαcos(x)
. (9.9)
In expression (9.9), the angle x is defined by
sin(x) = sin(θ
2) sin(2β)/p lm,
where
The initial state |si in this work is considered to be an arbitrary unit vector in the space and is given by
|si = sin(β0) |Ii + cos(β0)eiu|IIi . (9.11)
The requirement (9.5) implies that both the real and imagine parts of the term hII| Gm|si are zero, so, as substituting (9.10) and (9.11) into (9.5), one will eventually obtain the two equations:
− sin(mw) sin(φ
2 − α − u) sin(2x) sin(β0) + cos(mw) cos(β0) = 0, (9.12)
sin(mw) cos(φ
2 − α − u) sin(2x) sin(β0) − sin(mw) cos(2x) cos(β0) = 0. (9.13) Equation (9.13), by the definition of the angle x, will reduce to the matching condition
(sin(φ − θ
which is identical to the relation derived by Long et al. [167]:
tan(φ
2) = tan(θ
2)(cos(2β) + sin(2β) tan(β0) cos(α + u)
1 − tan(β0) tan(θ2) sin(2β) sin(α + u)). (9.15) Equation (9.12), under the satisfaction of the matching condition (9.14), or (9.15), will reduce to a concise formula for evaluating the number of iterations m:
cos(mw + sin−1(sin(β0) sin(φ
2 − α − u))) = 0. (9.16)
By equation (9.16), one can compute the number m
m = ⌈f⌉ , (9.17)
where ⌈ ⌉ denotes the smallest integer greater than the quantity in it, and the function f is given by
f =
π
2 − sin−1(sin(β0) sin(φ2 − α − u))
cos−1(cos(φ−θ2 ) − 2 sin(φ2) sin(θ2) sin2(β)). (9.18) It can also be shown that if the matching condition is fulfilled, then after m searching iterations the final state will be
Gm|si = eiδ|τi = ei[m(π+φ+θ2 )+Ω] |τi , (9.19) where the angle Ω is defined by
Ω = tan−1(cot(φ
2 − α − u)). (9.20)
The phase angle appearing in the amplitude of the final state will be important for quan-tum interferences if possibly the state should interact with other states in a future appli-cation, so we would had better remain it as the present form.
The matching condition (9.14), or (9.15), relates the angles φ, θ, β, β0 , and α + u for finding a marked state with certainty. If β, β0 and α + u are designated, then φ = φ(θ) is deduced by the matching condition. As φ(θ) is determined, we then can evaluate by (9.18) the value of f = f (φ(θ), θ) and consequently decide by (9.17) the number of iterations m. The functions φ(θ) and f (θ) for some particular designations of β, β0 and α + u have been shown in Figs. 9.1 and 9.2. These examples have schematically depicted that theoretically we can establish a tabulated chart of possible choices between all of the phases for finding a marked state with certainty. It is worth noticing that as α + u = 0 and β = β0, the matching condition recovers φ = θ automatically since then eq. (9.13) becomes an identity, and accordingly one has
f =
π
2 − sin−1(sin(φ2) sin(β))
2 sin−1(sin(φ2) sin(β)) , for φ = θ. (9.21) This is the case discussed in Ref. [165]; an example can be read by the straight line of unity slope for β=β0=10−4 and the corresponding f vs θ variation in Fig. 9.1. It can also be shown that the matching condition (9.14) will recover the relation considered by Høyer [166]:
tan(φ
2) = tan(θ
2) cos(2β), for cos(φ/2 − α − u) = 0. (9.22)
In Figs.9.1 and 9.2 we have shown by the cross marks some particular examples of this special case.
Observing Figs 9.1 and 9.2, one realizes that for every designation of β, β0 and α + u, the optimal choices for φ and θ is letting φ = θ = π, since then the corresponding f is minimum under the fact df /dθ = (∂f /∂φ)(dφ/dθ) + ∂f /∂θ = 0, for φ = θ = π. We thus denote the optimal value of m by
mop= ⌈min(f)⌉ =
Figure 9.1: Variations of φ(θ) (solid) and f (θ) (broken), for α + u = 0, β0 = 10−4, and β = 10−4 (1), 10−2 (2), 0.5 (3) and 0.7 (4), respectively. The cross marks denote the special case of Høyer [166], while the entire cirles correspond to the optimal choices of φop
and θop for α + u = 0, β0 = 10−4 and β = 0.7. The solid straight line 1 corresponds the case φ = θ, while the solid curve 2 is only approximately close to the former.
Figure 9.2: Variations of φ(θ) (solid) and f (θ) (broken), for α + u = 0.1, β0 = 0.1, and β = 10−4 (1), 10−2 (2), 0.5 (3) and 0.7 (4), respectively. The cross marks denote the special case of Høye [166]. The solid curves 1 and 2 are very close, and both of them are only approximately close to the line φ = θ.
With the choice of mop, however, one need to modify the phases θ and φ(θ) to depart from π so that the matching condition is satisfied again. For example, if α + u = 0, β0 = 10−4 and β = 0.7 are designated, then the minimum value of f will be min(f ) = 0.56 . So we choose mop = 1 and the modified phases are θop = (1 ± 0.490)π and φop = (1 ± 0.889)π, respectively. This example has been shown by the marked entire circles in Fig.1. It is worth noticing again that under the choice of mop the modified φ and θ for the special case considered by Long [165] will be
φop= θop= ⌈min(f)⌉ = 2 sin−1(sin(4mopπ+2) sin(β) ),
where
mop=
π 2 − β
2β
.
This is in fact a special case in which the phases φop and θopcan be given by a closed-form formula.