Department of Physics, Tsinghua University

Beijing, P R China

Key Laboratory for Quantum Information and Measurements, Key Lab of MOE

Gui Lu Long

Workshop on Quantum Computation and Quantum Information,

Seoul, Nov.1-3

The Quantum Searching AlgorithmThe Quantum Searching Algorithm

清華大學物理系 龍桂鲁

Collaborators

• From Tsinghua University

Ph. D. Students

Y S Li(李岩松 ) H Y Yan(阎海洋 )

L Xiao(肖丽 )， F.G. Deng(邓富国） M.Sc. Students

C C Tu(屠长存 ), X S Liu(刘晓曙 )

W L Zhang(张伟林 )， H. Guo, Y. J. Ma• From University of Tennessee

Prof. Dr. Yang Sun(孙扬 )

I 、 Quantum Searching Algorithms

1. Structure of quantum search algorithm

2. Phase matching in quantum searching

3. SO(3) picture for quantum searching

4. Some misunderstanding about Grover algorithm

5. Phase matching for a general database

6. Zero failure rate Grover algorithm

7.Error tolerance in Grover algorithm

II 、 Realizations and related issues

1. NMR experimental realization2. The oracle in Grover algorithm3. Optimality theorem, Exponentially fast quantum search algorithms4. “hybrid” quantum computing - the Brschweiler algorithm5. 3 qubit NMR realization of Brschweiler algorithm6. Summary

• Separate quatum search engine from the quantum database

• Phase matching condition depends both on the quantum search engine and the quantum database

• Zero failure rate Grover algorithm can be achieved by replacing phase inversions with phase rotations of angles smaller than

• Quantum searching process is easier to understand in SO(3)-picture

SummarySummary

1. Structure of a quantum search algorithm

Grover’s quantum search algorithm

Quantum mechanics helps in searching a needle in a haystack, PRL 79(1997) 325.

It requires steps to search for an item from an unsorted data.

Classically, it requires N/2 steps.

)( NO

Unsorted database search is important

Finding the owner of a phone number, Deciphering DES like code (Brassard, Science 1997)

The hidden shift problem: (JJ. Twamley, J.Phys.A33 (2000) 8973.

The Hamiltonian circuit problem(.H Guo, G. L. Long, Y. Sun, Commun. Theor.Phys.35(2001)385)

The Simon Problem:Proc.of 35th Annual Symposium on the Foundations of Computer Sciences, pp.116-23

Quantum Counting: G.Brassard et al., Lecture Notes in Computer Science, Vol.1443,1998,pp.820

Procedure in Grover’s quantum search algorithmProcedure in Grover’s quantum search algorithm

It can be rewritten as:

where

)arcsin(

,||

1

11

N

iN ic

12101

0 NN

cW |cos|sin0|| 0

1st, prepare an even superposition of all basis states

One iteration in Grover algorithm consists of two actions (4 steps ）1 ） Inversion of the marked state |>2) Inversion about average Dij

a) Hadmard transfotmation b) Inversion of the state |0> c) Hadmard transfotmation

jiN

jiNDij

,12

,2

2nd, perform the following iteration O(N) times

10

例子： N=8

765432108

10

76543521082

1 21

765432108

111

765431121084

122

Mathematically, the operator for the quantum search algorithm can be written as,

；00 20 II

2II

IWWIG 0

In the space span by |> and |c>, G can be written as

One iteration is a rotation through 2 ， after j successive iterations ， the state vector becomes|j>=cos[(2j+1)]|c> +sin[(2j+1)]|>.

cccG 2cos2sin2sin2cos

cos2 2sin

sin2 2cosG

| |c|

|c

Nj

j

j

op 42

1

4

2)12(

1])12sin[(

For the maximum probability:

Note that (2j+1) may not be exactly / 2, the maximum probability is usually not 100%.

Generalizationsa) More than one marked states (Grover, PRL 1998)Using the same procedure, inverts the sign of the amplitude of the marked states, m marked states can be found.

mm

321

1

15

cW |'cos|'sin0|| 0

)arcsin('

,|| 1

Nm

imN ic

16

In the space span by |> and |c>, G can be written as

One iteration is a rotation through 2’ ， after j successive iterations ， the state vector becomes|j>=cos[(2j+1)’]|c> +sin[(2j+1)’]|>.

cccG '2cos'2sin'2sin'2cos

'cos2 '2sin

'sin2 '2cos

G

| |c|

|c

17

The optimal iteration number is

m

N

m

NJ op 42

1

42

1

'4

Less steps are required …... Finding “a chain of needles”

b) Hadmard transformation replaced by arbitrary unitary transformations (Grover, PRL80, 1998)

IUUIG 1

Where |> is the marked state

|> is the prepared state, usually =0. Then

W I0 W is the inversion about average.

19

The initial state is taken as:

2|cos1|sin0|| 0 U

Where

sin=|<|U|0>|,

|1>=| >,

|2>=i|i><i|U|0>/cos

The number of required iteration is:

||4

U

Jop

Faster than standard Grover algorithm if |U|>1/N ?

c ） The initial distribution, is not evenly distributed:

12101

0 NN

1) D. Biron O. Biham, E. Biham, M. Grassl, D. A. Lidar, lecture notes in computer science, vol. 1509, 140 (Springer 1998). Also in /quant-ph/9801066,for standard Grover quantum algorithm

2) E. Biron, O. Biham, D. Biron, M. Grassl, D. A. Lidar and D. Shapira, Phys. Rev. A 63 (2001) 012310 for quantum search algorithm with arbitrary phase rotations.

22

In some literatures, the possible difference between the U’ in creating the initial state and the unitary transformation U in the quantum search engine is not paid attention. This causes some confusion in some literatures.

2|cos1|sin0|'| 000 U

d) The phase inversions can be replaced by arbitrary phase rotations satisfying a phase matching requirement:

1) G.L. Long, W.L.Zhang, Y.S.Li, L.Niu, Commun. Theor. Phys. 32 (99) 335；

2) G.L. Long, Y.S. Li, W.L.Zhang and L. Niu, Phys.Lett. A 262 (99) 27

2.Phase Matching in Quantum Searching 2|cos1|sin| 0

.||sin

,|cos

10|||

cos

12|

,|sin

10|||

sin

11|

20

0

0

kk

i ii

k kkkkk

U

UiUii

UU

The basis is determined by the search engine through U, the initial state is also chosen to be related to U.

25

IUUIG 10

Quantum search operator with arbitrary phase rotations:

1k kk

ieII

；00 10 ieII

D.Chi,J.Kim,Chaos Solitons Fractals 10 (1999) 1689, for marked states N/4, also in quant-ph/9708005

Brassard, Hoyer, Tapp, quant-ph/9802049, Quantum counting requires non- phase rotations。

The Simon Problem: Proc.of 35th Annual Symposium on the Foundations of Computer Sciences, pp.116-23

Non- phase rotations have been used in:

There were speculation that arbitrary phase rotations instead of the phase inversions, or phase rotation of the marked state instead of the phase inversion in the Grover algorithm may work in general, but with a smaller searching step.

j

j

i

i

j

j

i

i

j

j

B

A

e

eB

A

N

N

N

Ne

N

N

N

Ne

B

A

2cos2sin

2sin2cos

212

122

1

1

Using direct calculation, we found that the algorithm did not search in the way as expected: it fails totally!

2|1|| jjj AB

Replacing the phase inversion of the marked state

=/4

Pmax2.6%

Pmin0.36%

G.L. Long, W.L.Zhang, Y.S.Li, L.Niu, Commun. Theor. Phys. 32 (99) 335

Probability amplitude at the J+1 iteration

30

Now we change both phase inversions with arbitrary phase rotations:

；00 10 ieII

1k kk

ieII

It fails in general unless if the phase rotations satisfy the phase matching condition:

=

G L Long et al, Phys.Lett. A 262 (99) 27.

8th step amplitude vs and

Phase matching condition

=/2|Bj| versus and iteration number j. =/2

.

Rotates (2’), ’=sin(/2)

G. L. Long, C. C. Tu, Y. S. Li, W. L. Zhang and H. Y. Yan, An SO(3) picture for quantum searching, Journal of Physics A 34(2001) 861, also quant-ph/9911004

3. SO(3) picture for quantum searching3. SO(3) picture for quantum searching

Advantages: Quantum search process has a simple geometric picture. All calculations become simple using geometrical arguments.

Group theory, su(2) is isomorphic to so(3)

A rotation in su(2) corresponds to a rotation in so(3) by the following

is an su(2) transformation. Ru is the transformation in SO(3). is an arbitrary vector in 3 dimensional space. is the Pauli matrices. These are well-known from textbooks.

xRx

xuxu

u

'

)'()( 1

xu

||

r

However, the correspondence between state vectors in SU(2) and SO(3) took us time, and we found that the polarization vector is the quantity to relate them:

1) State vector, the wave function of the QC register is represented by a unit vector with one end fixed at the origin.

The initial position is nearly at z= -1,

and the marked state is at

z=+1.

Major points of the geometric picture

)2cos,0,2(sin),,( zyx

2) A Grover search iteration is a rotation about an axis through an angle. The task of the Grover rotation is to rotate the state vector from -z axis to +z axis.

The axis of rotation

)2csc()2

cot()2

cos()2cot(

1

)2

(cot

]sinsin)2cos())2

(sin2

1cos

2

1)(2(sin

coscos)3)4(cos(4

1arccos[

22

The rotational angle is

39

0

1

0

For standard Grover algorithm

)1

arcsin(4N

Rotational axis

Angle of rotation

40

3) The probability for finding the marked state is the state vector projection onto the z-axis

2

)1( z

P

0.0,1 Pz

5.0,0 Pz

0.1,1 Pz

41

The probability for finding the marked state at any iteration is

)sin()())cos(1)(()cos( 000 jrljrlljrr nnnj

Where is the normalized vector of the axis. The probability for finding the marked state is

nl

2

)1( z

P

y axis

The marked state

Th

e initial state

View from the y-axis

==/2

==/2

==/2

==/2

Phase mismatching fails to reach the marked state

49

4. Misunderstandings of Grover algorithm

1) Dependence on the initial state

If the initial state has a large component in the marked state, then Grover’s algorithm requires less steps to find the marked state:

)1

(a

OJ

cba ||| 0

50

2) If the unitary transformation U, has a large matrix element, then there require less steps in searching the marked state:

||4

U

Jop

In both cases, Grover algorithm can exceeds the square root limit.

51

1) Confusion in relating the initial state with the unitary transformation:

Suppose the initial state is

c|cos|sin| 000 Using the standard Grover algorithm

；00 20 II 2II

IWWIG 0

52

cos2 2sin

sin2 2cosG

In the space span by |> and |c>, G can be written as

After j iterations, the state vector becomes

cjjj |)2cos(|)2sin(| 00

The speed is the same as the standard Grover algorithm, only differs in the starting point. No speedup!

53

This misunderstanding is caused by relating the initial state with the search operation. One should separate the quantum database from the quantum engine:

2|cos1|sin0|| 0 U

IUUIG 10

The quantum database(initial state) should not be related to quantum search engine.

54

2|cos1|sin0|'| 000 U

For instance, the initial state can be taken as:

Or mostly generally

110| 1100 Naaaa N

The marked state and the unmarked state are not tied together.

55

2) If initial state and the quantum search engine are related as above. The number of iteration can indeed be reduced! But it is not useful for searching purpose. It takes less steps in searching a particular marked item, but it takes more steps for searching other items. G.L. Long et al, PRA61(2000)042305

||4

U

Jop

1

0

20 1||

N

U

5. Phase matching condition for a general database

.||sin

,|cos

10|||

cos

12|

,|sin

10|||

sin

11|

20

0

0

kk

i ii

k kkkkk

U

UiUii

UU

The basis is determined by the search engine through U, the initial state is also chosen to be related to U. 2|cos1|sin| 000

ie

We take the basis states, arbitrary U, more than one marked item. U=W is most useful. U is used for generality.

57

IUUIG 10

Quantum search engine with arbitrary phase rotations and arbitrary unitary operations

1k kk

ieII

；00 10 ieII

Rotation axis ln

The marked state rf

Th

e initial state r

o

(rf ro). ln=0

59

Using the geometric picture of the quantum search algorithm, it is derived that the phase matching condition is

2tan2sinsintan1

2tan 0

2sincostan)2cos(

2tan 0

=

G L Long, L. Xiao, Y. Sun, submitted PRA, quant-ph/01

60

For usual database: 2|cos1|sin0|| 0 U

2sincostan)2cos(

2tan 0

=

2tan2sinsintan1

2tan 0

=0

=

61

It is shown that Hoyer’s phase condtion (PRA 62 (01) 052304):(a is the successful rate)

2|cos1|sin| 000 ie

a212

tan2

tan

satisfy the general phase matching condition, since his input data has the general form

0, 0,

G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, quant-ph/0005055.

2

cot2cos

2sin)12(cot

i

eJ

i

op

2|)12cos(1|)12sin(| 0 opop jj

Biham et al’s phase condition: PRA63(2001)012310

Biham’s initial state is different from an evenly distributed state,

111

11

01

NNNNN

110 110 Naaaa N

There is no phase matching condition for arbitrary initial state. However, for the “Difficult search problem limit”, there is a phase matching condition given by Biham et al.

64

Biham et al’s phase condition: PRA63(2001)012310

In the “difficult search problem limit”, we have

N>> N 1

)1(|)0('|),(|)0('| 2/1 OlWOk k

This is equivalent to |=U|0

thus the phase matching condition is

=

and

6. Zero failure rate Grover algorithmThe maximum probability for finding the marked state in Grover algorithm is not exactly 100%.

n 1 2 3 7 10 13 20

N 2 4 8 100 1000 104 106

Pmax 0.5 1.0 0.77 0.998 0.9996 1-10-6 1-10-6

We can improve this by replacing the phase inversions with smaller phase rotations.

G L Long, Phys. Rev. A 64(2001)022307, Grover algorithm with zero theoretical failure rate,

66

sin

2sin

x

)arccos(2 x

)arcsin()1(4)1( xJJ

The angle span by the initial state and the target state vectors is (using the SO(3) picture)

This should be an integer (J+1) number of the basic angle (the polarization vector lies just in the +z axis).

67

Rotation axis ln

The marked state rf

Th

e initial state r

o

4(J+1)=2

4

68

=

NJ

J

64sinarcsin2

sin

64sin

arcsin2

2

1

422

opJ

N

1arcsin

op J J

Real solution only for

Some examples of angles(in unit of )

n N (J 0) (J, 1) (J, 2)

1 2 (1, 0.5) (2, 0.2879) (3, 0.2038)

2 4 (1, 1.0) (2, 0.4241) (3, 0.2936)

3 8 (2, 0.677) (3, 0.4334) (4, 0.3268)

7 100 (8, 0.7480)

10 1000 (24, 0.8540)

13 104 (79, 0.9009)

Other zero error schemes

1. Run standard Grover Jop-1 iterations, then change the quantum search engine with different phase rotations determined by an equation.

G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, quant-ph/0005055

2. Change the initial state and modify the quantum search engine with a phase condition:

P. Hoyer, PRA 62 (01) 052304):

71

Systematic phase errors in the Grover algorithm cause a drop in the probability

222

2

max

4

'

'

NP

N

8

Where is phase inversion error. If Pmax=0.5

Random errors in the phase inversion has a minor effect.

7 、 Error tolerance in Grover algorithm:

Long et al, Phys. Rev. A61 (00) 042305) quant-ph/9910076

Systematic errors in the Hadmard transformation cause a shift in the optimal iteration number, causes a reduction in the success probability.

Random errors in Hadmard transformation causes a leakage in the 2-dim vector space, (U first iteration, V second iteration)

2|)1(sin1|cos

'2|sin'1|cos

2|sin1|cos|

1

1

1

VU

73

)1(cossin

)1(sincos

1

1

G

G can be approximated by

After j iteration, the amplitude of the marked state becomes

)sin()2

11( 1 j

j

At the optimal iteration

N 2

1 2

1

81

NP

For half success rate

These results have also been shown by other authors

P.Hoyer, Phys. Rev. A62(2000)052304

Biham et al, Phys. Rev. A63(2001) 012310

SummarySummary• Phase matching condition depends both on

the quantum search engine and the quantum database

• Zero failure rate Grover algorithm can be achieved by replacing phase inversions with phase rotations of angles smaller than

• Quantum searching process is easier to understand in SO(3)-picture