A family of sure-success quantum algorithms for solving a generalized Grover search problem

4 Pages·0.1 MB·English

by  Chia-Ren Hu

A family of sure-success quantum algorithms for solving a generalized Grover search
problem
Chia-Ren Hu
Department of Physics, Texas A& M University, College Station, Texas 77843-4242
(Jan. 12, 2002)
This work considers a generalization of Grover’s search problem, viz., tofind any oneelement in a
setofacceptablechoiceswhichconstituteafraction f ofthetotalnumberofchoicesinanunsorted
data base. An infinite family of sure-success quantum algorithms are introduced here to solve this
problem, each memberfor a differentrange of f. Thenth memberof thisfamily involves n queries
ofthedatabase,andsothelowestfewmembersofthisfamilyshouldbeveryconvenientalgorithms
withintheirrangesofvalidity. TheevenmemberA2n ofthefamilycoverseverlargerrangeoff for
larger n, which is expected tobecome thefull range 0≤f ≤1 in thelimit n→∞.
2
0
PACS numbers:
0
2

n Quantum computing as a new powerful approach to at least for the members , , , but most-likely
2 4 6
a A A A
solve difficult computational problems is still in its in- also for all higher even members of the family, φ = 2θ
J
fancy. Only a handful useful algorithms have been pro- is an acceptable choice for φ, (ii) for each of the even
2
posedsofar. Mostofthemfallintotwocategories: Those members , , , and most-likelyalso for eachofall
1 A2 A4 A6
for factorizing large integers, and those for “searching a higherevenmembersofthefamily,therequiredvaluefor

1 needle in a haystack”, or finding the only acceptable el- θ for it to work is a unique function of f just inside the
v ement in a large unsorted data base. The main idea in boundary of its validity range of f, but the number of
9
the former category is due to Shor, [1] and in the lat- acceptable values of θ graduately increases to n deep in-
4
ter category is due to Grover. [2] Here we wish to con- sidethevalidityf-rangefor . Thealgorithmmember
0 A2n
1 sider a generalization of Grover’s search problem, viz., 1,ontheotherhand,requiresφ= 2θ,thenθ depends
A −
0 to find any one element in a set of acceptable choices uniquely on f within the validity range. No other odd
2 which form a fraction f of the total number of choices members have yet been analyzed. In all cases studied, I
0
in an unsorted data base of size N. [3] An infinite fam- find the required θ and φ to be independent of N, and
/
h ily n n = 1,2,3, of qunatum algorithms is intro- toonlydependonf. Thereisstrongindicationthatthis
{A | ···}
p duced here, each similar to one stage of Grover’s algo- statement is true for all members of the family.
-
rithm, except that, unlike Grover’s original algorithm, All members of this family n = 1,2,3, are
t n
n {A | ···}
which requires iteration to some optimal stage, which in achievedwithtwounitaryoperatorswhichgeneralizethe
a
generalis still nota sure-successalgorithm,[4]here each two corresponding operators introduced by Grover:
u
q member is anindependent sure-successalgorithmwithin In a Hilbert space spanned by a set of N orthonormal
: its range of validity. Each member of the family intro- basisstates i> n=1,2,3, ,N ,eachofwhichrep-
v {| | ··· }
duced here is characterized by an iteration number, n, resents one element of the data base, Grover introduced
i
X in the sense introduced in the originalGroveralgorithm. an unitary operator, which I shall denote as Fˆ , which
ν
r Thisnumber isalsothe number oftimes the databaseis changes the sign of the ν’s amplitude C in any quan-
a N ν
queried. Here weonlyanalyzefourmembersofthis fam- tum state Ψ >= C i >. This operator is gener-
| i=1 i|
ily, correspondingto the iterationnumbers 1,2, 4,and 6. alized to the operaPtor Fˆ(a), which introduces the extra
φ
We find that A1 is valid for 0.25 ≤ f ≤ 1.0; A2 is valid phasefactor eiφ toeachofthe amplitudes Cν ν a ,
for 0.095491502 f 0.65450849 ; is valid for − { | ∈ }
···≤ ≤ ··· A4 where a denotes the set of acceptable elements in the
0.030153689 f 0.88302222 ; and is valid
··· ≤ ≤ ··· A6 data base. [5] Mathematically,
for 0.014529091 f 0.94272801 . These results
strongly indicate··t·h≤at by≤using of··e·ver larger n, an Fˆ(a) Iˆ (eiφ+1) ν ><ν . (1)
A2n φ ≡ − | |
ever larger range of f can be covered which in the limit Xν∈a
of n → ∞ approaches the full range 0 ≤ f ≤ 1, but a where Iˆ N i >< i is the identity operator. For
generalproof has not yet been obtained. The validity or ≡ i=1 | |
φ=0,andaPcontainingonlyoneelementν,thisoperator
non-validity of this statement, and the properties of the
reduces to the operator F introduced by Grover.
ν
odd members of this family, will be discussed in a future
AsecondunitaryoperatorintroducedbyGroveristhe
work. All members ofthis family ofalgorithmsare char-
“inversionaboutthemean”operator,whichcanbe writ-
acterized by two phase parameters, θ and φ. These two
ten in the form:
parameters are individually adjusted in order to make
each member a sure-success algorithm. I find that (i) Oˆ [(2/N) δi,j] i><j . (2)
≡ − | |
Xi,j
1
I generalize it to the set a, since probability is conservedby unitary oper-
ations. Belowweshowhowthis isdone explicitly forthe
Oˆθ ≡Xi,j [(2cosθ/N)−eiθδi,j]|i><j|, (3) fuoluatremaebmobuteraslAl e1v,eAn2m,Aem4,baenrsdA6. AoffttehretfhaamtiIlyw,illelasvpiencg-
2n
A
the oddmembers higher thanthe firstto be discussedin
which reduces to Grover’s “inversion about the mean”
operator if θ = 0. That Oˆ is unitary can be easily a later work.
θ
Consider first the algorithm member . One has the
verified. Itisalsoeasytoshowthatitisthemostgeneral A1
identity:
unitary operator of the form [(A+Bδ ] i >< j ,
i,j i,j | |
if one disregards an unimportaPnt overall phase factor. I Oˆ Fˆ(a) Φ >=[2cosθ(1 f feiφ) eiθFˆ(a)]Φ > .
am not aware of any earlier published work introducing θ φ | 0 − − − φ | 0
this umitary operator. (4)
Since Fˆ(a) and Oˆ are both complex operators, I also
φ θ Since the operator Fˆ(a) is equivalent to an identity op-
need their hermitian conjugate operators,F(a)† and Oˆ†, φ
φ θ eratorin the subspace correspondingto all unacceptable
which are also the inverse operators of Fˆφ(a) and Oˆθ, re- elements of the data base, sure success of this algorithm
spectively. Actually they are simply Fˆ(a) and Oˆ . is achieved by demanding
−φ −θ
Before any algorithm is applied, every element in the
2cosθ(1 f feiφ) eiθ =0. (5)
databaseshouldberegardedastohaveequalprobability − − −
ofbeingthe rightchoice. Groverrepresentedthisfactby
which has the solution φ= 2θ, and
starting with the quantum state: −
θ =(1/2)cos−1[(1/2f) 1]. (6)
N −
Ψ >=(1/√N) i>,
| 0 | Note that if (φ, θ) is a solution, then ( φ, θ) is also
Xi a solution. This is true for all higher m−emb−ers of the
i.e., the state with every C = 1/√N, so that the family also, and one can easily see why. Equation (6)
i
has solution only for 1/4 f 1, which is the va-
probability of finding any element of the data base is
≤ ≤
C 2 = 1/N. The quantum algorithm he introduced is lidity range of this algorithm. Within this range, I
i
|tor|epreatedlyapplytheunitaryoperatorproductOˆFˆ n have plotted θ as a function of f in Fig. 1 assuming
ν
θ > 0. The following special cases are of interest: (i)
times on the state Ψ >, followed by a measurement to
0
|
cause the state to collapse to one of the basis states. He
showedthatwhennisofanoptimalvalueoftheorderof 1.2
√N, All C 2 will be very close to zero except the par-
i
ticular on|e C| 2, correspondingto the desirable element 1
ν
| |
ν in Grover’ssearchproblem,which will be very close to
0.8
unity. However,exceptfor some specialvalues of N, one
walilllonthoetrobCtia2in. eTxhaucst Gunriotvyerfoarlg|Corνi|t2h,manids ienxagcetnezrearlonfoort θ (rad) 0.6
| |
a sure-success alorithm, even in theory, when potenial 0.4
implementation errors are not taken into account. We
generalize Grover’s algorithm to a family of sure-success 0.2
algorithms,eachmember ofwhichischaracterizedby an
integer n. Denoting these member algorithms as , 0
{An} 0 0.2 0.4 0.6 0.8 1
then the even [(2n)th] member 2n are defined as ap- f
{A }
plyingtheunitaryoperatorproductΛˆ ≡Oˆθ†Fˆφ(a)†OˆθFˆφ(a) FIG.1. Plotted is θ versusf for thealgorithm A1.
ntimestothestate Ψ >,followedbythesamemeasure-
0
|
mentusedintheGroveralgorithm. The odd[(2n+1)th] For f = 1/4, I find φ = θ = 0, and the operators re-
member ,is to applythe unitary operatorprod- duce to those introduced by Grover, and this algorithm
2n+1
{A }
uct Oˆ Fˆ(a)Λˆn to the state Ψ >, before the same mea- becomes a special case of Grover’s algorithm. (ii) For
θ φ | 0
surement is made. Thus makes n queries of the f = 1/3, I find φ = π/3 and θ = π/6. (iii) For
data base. “Sure success”{oAfnea}ch of these algorithms is f = 1/2, I find φ = ±π/2, and θ = ∓π/4. (iv) For
± ∓
achievedbyadjustingthetwoparametersθandφsothat f = 2/3, I find φ = 104.477 ◦ = 0.580430...π and
± ··· ±
all C 2,with inotbelongingto the seta ofthe general- θ = 52.2387 ◦ = 0.290215 π. Finally, (v) for
ized| Gi|rover search problem introduced here, are exactly f =1∓, I find φ=··· 2π/∓3, and θ =···π/3, but in this case
± ∓
zero. All C 2 with i a will then be exactly equal to the operator product Oˆ Fˆ(a) acting on Φ > simply re-
| i| ∈ θ φ | 0
1/(fN), where fN N is the number of elements in produces Φ >.
a 0
≡ |
2
Next, let us consider the second member . One has That is,
2
A
the identity:
A A B∗e−2iθ A
Λˆ|Φ0 > ={[(2cosθ)2|(1−f −feiφ)|2−e2iθ] (cid:18)Bnn++11(cid:19)=(cid:18)B11 −−e1−2iθ (cid:19)(cid:18)Bnn(cid:19) . (12)
(2cosθ)e−iθ(1 f feiφ)Fˆ(a)† Φ >
− − − φ }| 0 Thus A2 = B1 4 [2cos(2θ) + e2iθ]B1 2 + e4iθ and
≡(A1−B1Fˆφ(a)†)|Φ0 > . (7) B2 = [|B1|2 −| 2c|os−(2θ)]B1. To ensure| su|re-success for
this algorithm, one needs to require A B = 0. It is
2 2
−
(Note that A = B 2 e2iθ.) Thus to ensure that this easy to show that
1 1
| | −
is asure-successalgorithm,one needsonly demandA
1
− Im(A B )=[B 2 2cos(2θ)]Im(A B ). (13)
B = 0. The imaginary part of this condition can be 2 2 1 1 1
1 − | | − −
written as
Ishallconsiderinafutureworkthepossibilityofsatisfy-
ingthis equationbysetting the firstfactorequaltozero.
Im(A B )=(2fcosθ)[sin(φ θ) sinθ)]=0. (8)
1 1
− − − HereI concentrateonthe factthatdue to its secondfac-
so it can be satisfied with φ=2θ. (It is easy to see that torthisequationcanbesatisfiedbylettingφ=2θ. Then
cosθ = 0.) Then the real part of this condition reduces θ is given by
6
to
Re(A B )=1+8fµ2 48f(1 f)µ4
2 2
− − −
Re(A1 B1)=1+4fµ2 16f(1 f)µ4 =0 (9) 64f2(1 f)µ6+256f2(1 f)2µ8 =0. (14)
− − − − − −
where µ cosθ. It has the solution I have plotted θ as a function of f for this algorithm in
≡
Fig.3assumingθ >0. Itisseenthatsolutionexistsonly
1 1 4
θ = cos−1 [ 3+(4f 3)] , (10)
2 {4(1 f) rf − − }
− 1.2
This equation has solution only if 0.095491502
··· ≤ 1
f 0.65450849 . Within this range, I have plotted θ
≤ ···
as a function of f for this algorithm in Fig. 2, assuming
0.8
θ >0.
d)
1.2 θ (ra 0.6
0.4
1
0.2
0.8
0
d) 0 0.2 0.4 0.6 0.8 1
a 0.6
θ (r f
0.4 FIG.3. Plotted is θ versusf for thealgorithm A4.
0.2 for 0.030153689 f 0.88302222 , and that in
··· ≤ ≤ ···
the narrower range 0.25 f 0.58682408 A second
≤ ≤ ···
0 solution for θ appears for each f. It should be obvious
0 0.2 0.4 0.6 0.8 1
that this algorithmis validfor those values off only, for
f
which at leastone solution for θ exists, thus the larger f
FIG.2. Plotted is θ versus f for thealgorithm A2.
range is also the validity range of this algorithm.
Finally, let us consider the algorithm member .
toNsloigtehttlhyabtewloiwth0t.h1e. lagorithmA2 we cancoverf down Eq. (12) allows me to obtain A3 = |B1|6 −[4cos(2θA)+6
e2iθ]B 4 +2[cos(4θ)+1+e4iθ]B 2 e6iθ, and B =
ingNex3t,alnetduhsigchoenrsidoedrdthmeemalbgoerrsithfomr mfuetmurbeerdiAsc4u,slseiaovn-, {|B1||41−|4cos(2θ)|B1|2+[2cos(4θ|)1+| 1−]}B1. Thus I fi3nd
A
since they are deemed less important. I have first es- Im(A B )= [B 2 2cos(2θ)]2 1 Im(A B ).
tablished the following theorem: If Λˆn Ψ >= [A 3− 3 {| 1| − − } 1− 1
0 n
| −
B F(a)†]Ψ >, then (15)
n φ | 0
Again, I shall not consider here letting the first factor
Λˆn+1 Ψ > = [A A e−2iθB∗B ]
| 0 { 1 n− 1 n equaltozero. Thenagainφ=2θ fromIm(A3 B3)=0,
[B A e−2iθB ]F(a)† Ψ >. (11) and θ is given by −
− 1 n− n φ }| 0
3
Re(A B )=1+12fµ2 96f(1 f)µ4 is then the most convenient, since it requires the least
3 3
− − −
256f2(1 f)µ6+1280f2(1 f)2µ8 number of queries of the data base.
− − −
+1024f3(1 f)2µ10 4096f3(1 f)3µ12 =0. (16)
− − −
I have plotted θ as a function of f for this algorithm in
Fig.4assumingθ >0. Itisseenthatsolutionexistsonly
1.2 [1] P. W. Shor,SIAMJ. Computing 26, 1484 (1997).
[2] L. V.Grover, Phys.Rev.Lett. 79, 325 (1997).
1 [3] This generalized Grover search problem has been con-
sidered before. [See, for example, R. M. Gingrich, C. P.
0.8 Williams,andN.J.Cerf,Phys.Rev.A61,052313(2000);
P. H6oyer, Phys. Rev. A 62, 052304 (2000); C. Zalka,
d)
θ (ra 0.6 Lanl-eprint/quant-ph/9902049; G. Brassard, P. H6oyer,
M. Mosca, and A. Tapp, Lanl-eprint/quant-ph/0005055.]
0.4 However, none of them appear to solve this problem sim-
ply by theidea presented in this work.
0.2 [4] Grover’s original algorithm is not a sure-success one, but
there are at least three revisions of it to make it a sure-
0 success algorithm. [See G. L. Long, Phys. Rev. A 64,
0 0.2 0.4 0.6 0.8 1
022307 (2001),andtwoearlierreferencescitedtherein.C.
f
Zalka in Lanl-eprint/quant-ph/9902049 has also outlined
FIG.4. Plotted is θ versus f for thealgorithm A6.
anotherrevisionwithoutgivingexplicitdetails.]However,
since they are all aiming at solving the original Grover’s
for 0.014529091 f 0.94272801 , which is the
··· ≤ ≤ ··· search problem, they are not identical to the idea pre-
validity range of this algorithm. In the narrower range
sented in this work.
0.12574462 f 0.78403237 a second solution [5] This unitary operator has been introduced in many ear-
··· ≤ ≤ ···
for θ appears for each f, and in the even narrowerrange lier works to solve either theoriginal Grover search prob-
0.32269755 f 0.56026834 a third solution for lemoritsgeneralization discussedhere.[See,forexample,
θ appears fo·r··e≤ach f≤. ··· P. H6oyer, Phys. Rev. A 62, 052304 (2000); G.-L. Long,
A trend is clearly established by the above study of W. L. Zhang, Y. S. Li, and L. Niu, Lanl-eprint/quant-
the first three even members. It strongly suggests that ph/9904077; G.-L. Long, Y. S. Li, W. L. Zhang, and
L. Niu, Lanl-eprint/quant-ph/9906020; G.-L. Long, Y.
forallevenmembers,(i)φ=2θisalwaysavalidsolution,
S. Li, W. L. Zhang, and C. C. Tu, Lanl-eprint/quant-
with θ depending on f, but not on N; (ii) the f-range
ph/9910076; G.-L. Long, L. Xiao, and Y. Sun, Lanl-
in which at least one θ value exists becomes ever larger
eprint/quant-ph/0107013.]However,noneofthemappear
if of ever larger n is considered, with the n
A2n → ∞ to have combined it with the unitary operator given in
limit being very likely the full range 0 f 1; (iii) in
Eq. 3 to solve either search problem.
≤ ≤
general the number of valid choices for θ increases to n
deep inside the validity f-range for . General proofs
2n
A
of these statements have not yet been obtained.
Insummary,aninfinitefamilyofsure-successquantum
algorithms is introduced here for solving the generalized
Groversearchproblemoffindinganyoneelementofaset
of acceptable choices which constitute a fraction f of all
elements in an unsorted data base. This is achieved by
two unitary operators each containing a phase parame-
ter. Theseoperatorsaregeneralizationsofthetwoopera-
torsintroducedbyGroverforhisoriginalsearchproblem.
The twophaseparametersareadjustedforeachmember
of the family to ensure its sure-success, which is found
possible only within a different f-range for each mem-
ber of the algorithm family. An infinite sub-family (the
“even” members) appears to have the property that the
validity f-range of a lower member is totally embedded
insidethatofahighermember,withthelimitbeingvery
likely the full range 0 f 1. As long as f is within
≤ ≤
the validity range, the lowest member of the sub-family
4