Even in the case of large N, where high success rate in finding the marked state is expected by using the standard Grover’s algorithm, inevitable noises including decoherence and gate inaccuracies can significantly affect the efficiency of the algorithm. To overcome such demerit we therefore should either apply the fault-tolerant computation [174] to reduce gate imperfections and decoherence, or limit the size of the quantum database to depress the effect of the uncertainty of the phase inversion operations. In another way, we can also, if possible, consider implementing a modified algorithm which is itself robust against phase imperfections and or decoherence. Recently, Hu [175] introduced an interesting family of algorithms for the quantum search. Although these algorithms are more complicated than the standard Grover’s algorithm, they can be proved to be robust against imperfect phase inversions, so the limitation of the size of database can be greatly relieved. In what follows we therefore intend to analyze the algorithms introduced by Hu [175] in detail then show the robustness of the family in resisting the effect of imperfect phase inversions [176, 177].
If we denote the phase inversion of marked state Iτ = 1 + (eiφ−1) |τi hτ| and the phase
inversion of the initial state Is = 1 + (eiθ− 1) |si hs| , then the generalized Grover operator is given by G = IsIτ and in n iteration the unit probability for finding the marked state, viz., |hτ| Gn|si|2 = 1, is ensured if φ = θ. Instead of applying Gn on the initial state
|si, Hu [175] presented and utilized the operators A2n=(Is†Iτ†IsIτ)n and A2n+1 = GA2n to accomplish a quantum search with certainty, and named the former the even member and the latter the odd member of the family {An,n = 1, 2, ...} because they require even (2n) and odd (2n + 1 ) oracle calls in computation, respectively. The arrangement Is†Iτ†IsIτ will be shown to have cancellation effect on phase errors in each iteration of the algorithm A2n
and A2n+1 and as a whole can ensure the robustness against imperfect phase inversion.
Consider a two-dimensional Hilbert space spanned by the marked state |τi and the state |τ⊥i, which is orthogonal to |τi . The initial state, as a uniform superposition of all states, then can be express by |si = W |0i = sin(β) |τi + cos(β) |τ⊥i, where sin(β) ≡ pM/N and M is the number of the target states. The eigenvalues of the operator Is†Iτ†IsIτ are λ1,2 = cos(ω) ± i sin(ω) and the corresponding eigenvectors are computed
|λ1i = cos(x) |τi + i sin(x)ei(φ2−γ)|τ⊥i ,
|λ2i = i sin(x)e−i(φ2−γ)|τi + cos(x) |τ⊥i , (9.33)
where the rotation x and the related parameters are defined by
tan(x) = 2r sin(φ2) sin(θ2) sin(2β)
where sin(2x) and cos(2x) can be computed in use of the definition (9.34) and given by
sin(2x) = ( 1 − sin2(θ2) sin2(2β)
1 − sin2(θ2) sin2(φ2) sin2(2β))1/2, (9.38) cos(2x) = sin(θ2) cos(φ2) sin(2β)
(1 − sin2(θ2) sin2(φ2) sin2(2β))1/2. (9.39) When the quantum search is carried out by using the even member A2n, the component of the final state after n iterations of (Is†Iτ†IsIτ) in the basis state |τ⊥i is expressed by hτ⊥| A2n|si = REe+ iIMe, and accordingly the exact success rate in finding the marked state |τi then is given by
p = 1 − |hτ⊥| A2n|si|2 = 1 − (REe2+ IMe2), (9.40)
where
REe = cos(nω) cos(β) − sin(nω) sin(2x) cos(φ
2 − γ) sin(β), (9.41)
IMe= − sin(nω) sin(β)
(1 − sin2(θ2) sin2(φ2 sin2(2β)))1/2sin(θ + φ
2 ). (9.42)
It is clear that when IMe = 0, one obtains the n-independent phase matching condition, φ = −θ, for A2n, and the success rate then becomes
p = 1 − REe2 = 1 − cos2(nω − α), (9.43)
where α = sin−1(sin(β) cos(φ/2 + γ)). The 100% success rate for the search problem can be achieved as by letting cos(nω −α) = 0. For a search with certainty, since n is a positive integer, one therefore has to expect the iteration number given by
ne(θ, β) = ⌈fe(θ, β)⌉ , (9.44)
and the function fe(θ, β) is given by
fe(θ, β) =
π
2 + α(θ, β)
ω(θ, β) . (9.45)
Given β, the function fe has its minimal value as θ = π (and φ = −π thereby), as if minimal oracle calls are demanded in the computation, we should have the optimal phase θop associated with
fe(θop, β) = ⌈fe(π, β)⌉ . (9.46)
For example, if given β = 1, we have ⌈fe(π, 1)⌉ = 1, and the optimal phase angle θop = π±1.304 follows in the algorithm using the even member A2n. In usual operation, however, the quantum database is large, i.e., sin(β) ≪ 1, and the phase θ = π and φ = −π are fixed, then the required iterations are estimated by n ∼ π/8β and by (9.46) the maximal success rate will be approximately evaluated
pmax ∼ 1 − β2, for θ = π,
which is the same result obtained as if the standard Grover algorithm is implemented.
That is, as the phase θ = π is fixed, the present algorithm (A2n) is equivalent to the standard algorithm (Gm) with even oracle calls required in the computation. Nevertheless, since in a real operation, imperfections in the phase inversions are inevitable. In what follows, we will show that the present algorithm is robust against small phase imperfections in a quantum computation and provides a maximal success rate that is similar to the one given above.
In the absence of decoherence and error correction, we considered constant phase errors causing the phase φ and θ to be φ = π + φe and θ = π + θe, where |φe| ≪ 1 and
|θe| ≪ 1. By introducing the constant phase imperfections, one then have the following
approximations, when β ≪ 1,
sin(2x) ∼ 1 − 1
2β2φ2e, cos(2x) ∼ βφe, cos(φ
2 − γ) ∼ −1 + 1
8(θe− φe)2, sin(φ
2 − γ) ∼ 1
2(θe− φe), ω ∼ 4β(1 − 1
8(θe2+ φ2e) + 4 3β2).
Then, since the errors are unknown in advance of the computation, the iteration number is also considered to be n ∼ π/8β, and we thus have cos(nω) ∼ π(θe2+ φ2e)/16 − 2πβ2/3 and sin(nω) ∼ 1. The approximation of REe and IMe accordingly are evaluate by
REe∼ β + π
16(θe2+ φ2e) −2
3πβ2, (9.47)
IMe∼ − 1
2β(θe+ φe).
The maximal success rate, in uses of expression (9.40)-(9.42), now is approximately derived by
pmax ∼ 1 − β2− (H.O.T.), (9.48)
where H.O.T. represents high order terms higher than second-degree in the small param-eters β, θeand φe. Expression (9.48) clearly tells that the reduction of the probability due to the introduction of the phase errors in fact can almost be neglected. Then, through it, we can see that the present algorithm is robust against systematic phase imperfections.
The analysis of the algorithm using the odd member A2n+1 can be undertaken by the same procedure as in analyzing the even member. In this case, we have
pmax = 1 − (REo2+ IMo2), (9.49)
where
Note that in this case the inequality 1 − 4 sin(θ/2)2 ≥ 0 should be demanded since then the meaningful requirement fo ≥ 0 can then be fulfilled. Given β, the function fo(θ, β) also has its minimal value at θ = π (then φ = π), as the optimal choice of the phase θop
should be estimated by
fo(θop, β) = ⌈fo(θ, β)⌉ , (9.52)
when minimal oracle calls are demanded in a search with certainty. For β = 1, the choice of the phase should be θop= φop = π ±1.870, for example. The standard Grover algorithm with odd oracle calls can be recovered when θ = φ = π is fixed. In usual operations, when phase imperfections are introduced, i.e., as θ = π + θo and φ = π + φo, where both θo and φo are small errors in the phases, they also produce almost negligible reductions in the success rate as given by an expression like Eq. (9.48).
9.5 Hamiltonian and measuring time for analog quan-tum search
Several researchers have proposed other ways to solve the quantum search problem, such as the analog analogue version of the Grover’s algorithm [178–180] and the adiabatic evolution to quantum search [181–183]. The former is to be considered in this work.
It is proposed that the quantum search computation can be accomplished by controlled Hamiltonian time evolution of a system, obeying the Schr¨odinger equation
id |Ψ(t)i
dt = H |Ψ(t)i , (9.53)
where the constant ~ = 1 is imposed for convenience. Farhi and Gutmann [178] presented the time-independent Hamiltonian Hf g = Ef g(|wi hw| + |si hs|), where |wi is the marked state and |si denotes the initial state. Later, Fenner [179] proposed another Hamiltonian Hf = Efi(|wi hs| − |si hw|). Recently, Bae and Kwon [180] further derived a generalized quantum search Hamiltonian
Hg = Ef g(|wi hw| + |si hs|) + Ef(eiφ|wi hs| + e−iφ|si hw|), (9.54)
where φ is an additional phase to the Fenner Hamiltonian. Unlike the Grover algorithm, which operates on a state in discrete time, a quantum search Hamiltonian leads to the evolution of a state in continuous time, so the 100% probability for finding the marked state can be guaranteed in the absence of all kinds of imperfection occurring in a quantum operation. Both the Hamiltonian Hf g and Hf can help to find the marked state with 100%
success. However, Bae and Kwon [180] addressed that the generalized Hamiltonian Hg can accomplish the search with certainty only when φ = nπ is imposed, where n is arbitrary integer. In this work, however, we will show that the generalized Hamiltonian Hg can be derived by an analytical method, which is distinct to the one implemented by Bae and Kwon [180], and the same method will lead to arbitrary chosen phase φ, depending on
when the measurement on the system is undertaken and how large the system energy gap is provided. Since Hamiltonian-controlled system is considered, the energy-time relation will play an essential role in the problem. Therefore, the evaluation of the measuring time for the quantum search becomes crucially important. In this study, we will derive the general Hamiltonian for the time-controlled quantum search system first. Then the exact time for measuring the marked state will be deduced. Finally, the role played by the phase φ in the quantum search will be discussed, and both the measuring time and the system energy gap as variations with φ will be given [184].
Suppose that a two-dimensional, complex Hilbert space is spanned by the orthonormal set |wi, which is the marked state, and |w⊥i , which denotes the unmarked one. An initial state |si = |Ψ(0)i is designed to evolve under a time-independent quantum search Hamiltonian given by
H = E1|E1i hE1| + E2|E2i hE2| , (9.55)
where E1 and E2 are two eigenenergies of the quantum system, E1 > E2, and |E1i and
|E2i are the corresponding eigenstates satisfying the completeness condition |E1i hE1| +
|E2i hE2| = 1. The eigenstates can be assumed by
|E1i = eiαcos(x) |wi + sin(x) |w⊥i ,
|E2i = − sin(x) |wi + e−iαcos(x) |w⊥i . (9.56)
where x and α are two parameters to be determined later based on the required maximal probability for measuring the marked state. By the assumptions given in (9.56), the Hamiltonian can be written in the matrix form
H =
Ep + Eocos(2x) Eosin(2x)eiα Eosin(2x)e−iα Ep− Eocos(2x)
. (9.57)
where Ep = (E1+E2)/2 is the mean of eigenenergies and Eo = (E1−E2)/2 represents half
of the system energy gap. The major advantage of using the controlled Hamiltonian time evolution is that the marked state can always be searched with certainty in the absence of quantum imperfections. The crucial key of the present problem in turn is to decide when to measure the marked state by the probability of unity. So in what follows we will in detail deduce the relation between all the unknown appearing in the system and then evaluate the exact measuring time for finding the marked state with certainty.
The time evolution of the initial state is given by |Ψ(t)i = e−iHt|si. Therefore, the probability of finding the marked state will be P =
hw| e−iHt|si
2 = 1 −
hw⊥| e−iHt|si 2. Without loss of generality, let us consider the problem of searching one target from N unsorted items. The general form of the initial state considered in this study is given by
|si = eiusin(β) |wi + cos(β) |w⊥i , (9.58)
where sin(β) ≡ 1/√
N and u denotes the relative phase between the two components in the initial state. Note that the relative phase u may arise from a phase decoherence or an intended design during the preparation of the initial state. Now, because of e−iHt = e−iE1t|E1i hE1| + e−iE2t|E2i hE2|, using the expressions given in (56) and (58), we can deduce
hw⊥| e−iHt|si = e−iEpt((cos(β) cos(Eot) − sin(α − u) sin(2x) sin(β) sin(Eot)) +i(cos(2x) cos(β) − cos(α − u) sin(2x) sin(β)) sin(Eot)). (9.59)
To accomplish the quantum search with maximal probability, the time-independent term (cos(2x) cos(β) − cos(α − u) sin(2x) sin(β)) in (9.59) must vanish and thus the unknown x can be determined by
cos(2x) = sin(β) cos(α − u)
cos(γ) , or sin(2x) = cos(β)
cos(γ), (9.60)
where γ is defined by sin(γ) = sin(β) sin(α − u) . The probability for finding the marked
state then becomes
Usually, if the size of database N is large, then γ ≪ 1 and the marked state |wi will be measured at t = π/(2Eo) by a probability p = 1 − tan2γ ∼ 1, according to (9.61).
Expression (9.61) also indicates that, by letting cos2(E0t + γ) = 0, we can measure the marked state with unit probability, no matter how large N is, at the time instants
tj = (2j − 1)π/2 − sin−1(sin(β) sin(α − u)) Eo
, j = 1, 2, .... (9.62)
In what follows, let us only focus on the first instant t1 = (π/2 − sin−1(sin(β) sin(α − u)))/Eo. It is clear that a larger Eo, or equivalently a larger system energy gap, will lead to a shorter time for measuring the marked state with certainty. Meanwhile, as can be seen in (9.61), the probability for measuring the marked state varies with time as a periodic function whose frequency is the Bohr frequency Eo/π, so a larger Eo will also result in a more difficult control on the measuring time. In other words, the measuring time should be controlled more precisely for a higher Bohr frequency in the state evolution since then a small error in the measuring time will cost a serious drop of the probability.
However, the energy gap Eo depends on the size of database N, as will be mentioned later.
With the relations shown in (9.60), the present Hamiltonian now can be written by
H =
which is represented in terms of the energies Ep and Eo and the phase α. Alternatively,
if we let
then the Hamiltonian can also be expressed by
H =
which in turn is represented in terms of the energies Ef g and Ef and the phase φ. The Hamiltonian shown in (9.66) in fact can be expressed as Hg = Ef g(|wi hw| + |si hs|) + Ef(eiφ|wi hs| + e−iφ|si hw|), which is exactly of the same form as the Bae and Kwon Hamiltonian Hgshown in (9.54). However, Bae and Kwon [180] only consider the case u = 0 . In both the presentations (9.63) and (9.66) of the Hamiltonian H, the corresponding measuring time for finding the marked state |wi with certainty is at
t1 = Equation (9.67) indicates that when the phase difference α − u, or φ − u, is imposed and the energy gap Eo or the energies Ef and Ef g are provided, the measurement at
the end of a search should be undertaken at the instant t1. To discuss further, we first consider the case u = 0, i.e., the case where neither phase decoherence nor intended relative phase is introduced in the preparation of the initial state |si. If φ = nπ, or α = nπ, is imposed, then the present Hamiltonian reduces to that considered by Bae and Kwon [180] to serve for a search with certainty when the measurement is undertaken at t1 = π/(2Eo) = π/(2|(−1)nEf + Ef gsin(β)|). If Ef = 0, or if Eo = Ef gsin(β) and α = 0, is imposed, then the present Hamiltonian reduces to the Farhi and Gutmann Hamiltonian Hf g, which serves for a search with certainty at t1 = π/(2Eo) = π/(2Ef gsin β). Further, when Ef g = 0 and φ = π/2, or Ep = 0 and α = π/2 is chosen, the present Hamiltonian will reduce to the Fenner Hamiltonian Hf associated with the measuring time t1 = (π − 2β)/(2Eo) = (π − 2β)/(2Efcos β). In general, the phase φ, or α, in fact can be imposed arbitrary for a search with certainty as the condition u = 0 is imposed.
However, if inevitable phase decoherence in the preparation of the initial state |si is considered, then the phase u must be assumed to be arbitrary. Accordingly, the proba-bility for finding the marked state will not be unity at all. For example, if following Bae and Kwon [180] by letting t1 = π/(2Eo), then we only have a probability for finding the marked state given by
p = 1 −cos2(β) sin2(β) sin2(u)
1 − sin2(β) sin2(u) . (9.68)
It is easy to show that the probability shown in (9.68) is always greater than or equal to the lower bound pmin = 1 −sin2(β) = 1 −1/N. Of course, if the nonzero phase u is introduced by an intended design, not an inevitable phase decoherence, then a search with certainty can be accomplished for an arbitrary φ, or α, when associated with the measuring time shown in (9.67). For example, if u = π/2 is the phase designated, the ideal measuring time should be t1 = (π − 2β)/(2Eo), which is the same as the Fenner’s t1. Again if the phase decoherence is introduced into the system and changes the phase from π/2 to an undesired u, then one eventually obtains a poor probability p = 1 − (1 + sin2(u) − 2 sin(u)) sin2(β).
0.1 0.2 0.3 0.4 0.5
Β
0.75 0.85 0.9 0.95 1
p
Figure 9.5: The variation of ¯p(β) for cases of Bae-Kwon(solid), Farhi-Gutmann(solid), and Fenner(broken) at the specific measuring times, t1,BK = t1,F G = π/(2Eo) and t1,F = (π − 2β)/(2Eo).
Moreover if the phase error occurs randomly in a quantum database, then we cannot be sure when to take a measurement, and the probability for finding the marked state even drops off seriously in some cases. For investigating the effect of the random uncontrollable parameter u on p at a fixed measuring time, we average over all possible values of p(β, u) about all arbitrary values of phase parameter u. Fig. 9.5 shows the variation of the mean probability ¯p with β for cases of Bae-Kwon, Farhi-Gutmann and Fenner at the specific measuring times, t1,BK = t1,F G = π/(2Eo) and t1,F = (π − 2β)/(2Eo), those Hamiltonian suggest in such a case. The same character of their proposals is that ¯p is sensitive to a phase decoherence as the database is small. The mean success probabilities of Bae-Kwon and Farhi-Gutmann are the same and always greater than the one of Fenner. Then the Hamiltonians presented by Bae and Kwon, and Farhi and Gutmann are more robust against the phase decoherence than the one proposed by Fenner especially for low values of N.
Now we proceed to give a brief review on the comparison between Ef g and Ef, which has been discussed in Ref. [185], and to recall the implication behind the analog quantum
search first presented by Farhi and Gutmann [178]. Suppose there is a (N − 1)-fold degeneracy in a quantum system and its Hamiltonian is read as H0 = E |wi hw|, then our assignment is to find the unknown state |wi. Since one does not yet know what |wi is, it is natural to add a well known Hamiltonian, HD = E |si hs|, such that the initial state of the system |si can be drove into |wi. The total Hamiltonian therefore becomes H = H0+ HD = E(|wi hw|+|si hs|), which is just the Hamiltonian of Farhi and Gutmann [178] Hf g. It can be simplified under the large database limit,
Hf g ≈ E(|wi hw| + |si hs|) + E sin(β)(|wi hs| + |si hw|). (9.69)
From it one can realize that the driving Hamiltonian induces transitions between |wi and
|si with a mixing amplitude O(E sin(β)), which causes |si to evolve to |wi. By Eq. (9.65), thus it is rational to assume Ef ∼ Ef gsin(β), and therefore the energy gap Eo should be proportional to sin β , or 1/√
N. The measuring time then is easily found to be t1 ∝√ N from Eq. (9.67). However, if consider the case Ef ≫ Ef g, like the extreme situation considered by Fenner [179], then we encounter with Eo ∼ Efcos(φ − u) and accordingly the measuring time t1 is independent of the size of database N. Therefore, in an usual case the assumption Ef ∼ Ef gsin(β) is reasonable.
An interesting phenomenon occurs when the critical condition Ef = Ef gsin(β) is considered. Fig. 9.6 shows the variations of t1 and Eo with the phase difference φ − u in such a case. It is observed that when φ−u = ±π the energy gap Eo becomes zero and then the eigenstates of the quantum search system correspond to the common eigenvalue E = E1 = E2 and become degenerate. In such case, the Hamiltonian becomes proportional to the identity 1(= |wi hw| + |w⊥i hw⊥|). Therefore, the initial state |si does not evolve at all and the probability for finding the marked state |wi indeed is the initial one, viz., p = sin2(β) = 1/N, which can also be deduced using Eq. (61). In other words, the quantum search system is totally useless as long as φ − u = ±π is imposed under the critical condition Ef = Ef gsin(β). When φ − u 6= ±π, both t1 and Eo are finite, as
-3 -2 -1 0 1 2 3 10
110
210
310
43 2 1
2 3
1
E O t 1
φ -u
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Figure 9.6: Variations of t1(φ − u) (broken) and Eo(φ − u) (solid), for β = 0.085 (1), β = 0.031 (2), and β = 0.0055 (3).
can be seen from Fig. 9.6, and therefore the quantum search system becomes efficient again and is capable of finding the marked state with certainty, especially when the phase difference is imposed around φ − u = 0. As a conclusion, for an efficient, useful quantum search system, the critical condition mentioned above should be avoided and in fact the reasonable condition Ef ∼ Ef gsin(β) is recommended to be imposed.
Experimental generation of hyperentangled photons and
experimental realization of one-way quantum computing
10.1 Introduction
Cluster states have recently received enormous attentions in the field of quantum infor-mation and are important for one-way universal quantum computing [31–33]. Moreover, with highly robustness they are also essential for quantum error correction codes and quantum communication protocols [187, 188]. Many efforts have been stepped toward generating and characterizing cluster states in linear optics [52, 53, 99, 189–192]. Recently the principal feasibility of one-way quantum computing model has been experimentally demonstrated through 4-photon cluster state successfully [52, 53, 62].
In this chapter we show an experimental realization of one-way quantum computing with a 2-photon 4-qubit cluster state. We develop and employ a bright cluster state source which produces a 2-photon state entangled both in polarization and spacial modes.
PHOTONS AND EXPERIMENTAL REALIZATION OF ONE-WAY QUANTUM COMPUTING
The Grover’s search algorithm is demonstrated with highly performences. The genuine four-partite entanglement and high fidelity of better than 88% for this cluster state are characterized and verified by measurement of an optimal entanglement witness with two local measurement settings. Inheriting the intrinsic two-photon character, compare with the one using multi-photon, our scheme promises a brighter source in quantum computing by more than 4 orders of magnitude, which offers a significantly high efficiency for optical quantum computing. It thus provides a simple and fascinating alternative to complement the usual multi-photon cluster state [54].