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 Algorithm(II)The Quantum Searching Algorithm(II)

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 )

II 、 Realizations and related issues

1. NMR experimental realization2. Oracle--an example3. Optimality theorem, exponentially fast quantum search algorithms4. “hybrid” quantum computing - the Brschweiler algorithm5. 3 qubit NMR realization of Brschweiler algorithm6. Summary

• 2 qubit NMR realization of generalized search algorithm

• Grover algorithm is optimal for solely QC.

• Hybid QC, DNA+QC, can achieve exponential speedup

• Blackbox: a complicated computable function

• 3 qubit NMR realization of Bruschweiler algorithm

SummarySummary

1. NMR realization of generalized quantum searching algorithmStandard Grover algorithm has been realized

in 2 qubit NMR system:

J. A. Jones et al, Nature 393 (1998) 341

I. L. Chuang et al, Phys. Rev. Lett. 80 (1998) 3408

L. P. Fu et al, Chin. J. Magn. Res. 16 (1999) 341

in 3 qubit NMR system:

L. M. K. Vandersypen et al, Appl.Phys. Lett. 76 (2000) 646

G.L.Long, H.Y.Yan,Y.S.Li , L. Xiao, C.C.Tu, J.X.Tao, H.M.Chen, M.L.Liu, X.Zhang, J.Luo, X.Z.Zeng

Experimental NMR realization of a generalized quantum search algorithm,

Physics Letters A 286(2001) 121.

Working media ： H2 PO3 。

The experiments were performed on a Bruker 500MHz AM NMR

Qubit 1

Qubit 2

8

The parameters for the 2 qubit system are:

J-coupling constant 647.451 Hz

Frequencies: 500 MHz for 1H (spin A)

220 MHz for 31P(spin B)

Temporal average method is used to obtain the effective pure state needed for the QC

Knill, Chuang, Laflamme, Phys. Rev. A 57(98)3348

IP :0

221

222/121 :

A

JAB

JB XYXYP

IP :0

221

222/122 :

B

JAB

JA XXXYP

Pulse sequences for preparing the pseudo-pure state

2 sets of experiments were performed ：==/2 （ Phase matching) ； =/2 ， = 3/2(phase mismatching)

JBBAA

JBBAA YXYXYXYXG 2/22/2/2/2/2/)2(2/2/22

Features:

Non-90 pulses

Delay pulses need not be 1/4J

12

The searching have been performed to 10 iterations. At each step, the density matrix of the system is constructed. Only 12, 34 matrix elements(from spin A spectra) and 13,24 matrix elements(from spin B) can be measured.

44342414

34332313

24232212

14131211

To get all the matrix elements of the density operator, one has to perform II, IX, IY, XI, XX, XY, YI, YX, YY, then measure the spectra.

YY(90 degree pulse along Y for A, along Y for B)The measurement has to be done for A and B respectively. In all, 9 2 spectrum measurements have to be carried out.

Together with temporal average method, the total number of measurements for each density matrix reconstruction is:

9 2 3=54

Then, area integration of the spectrum were performed to get the real part and the imaginary part.

State tomography:

Chuang, Gershenfeld, Kubinec and Leung, Proc. R. Soc. Lond A 454, 447 (1998)

Good agreement between experiment and the data is obtained.

It is demonstrated that when phase matching is satisfied, the marked state can be found with a high probability, when phase matching is not satisfied, the probability of finding the marked state is very low.

Reconstruction of the density matrix can be simplified from 18(x 3) read-outs to only 5(x 3) read-outs without significant loss in the accuracy.

In Quantum optics

U. Leonhardt, Phys. Rev. Lett., 74 (1995) 4101U. Leonhardt, Phys. Rev., A53 (1996) 2998.

Cavity QED

R.Walser,J.I.Cirac, P. Zoller, Phys Rev Lett 77 (1996) 2658

single spin(pure or mixed)J.P.Amiet and S.Weigert,J.Phys.A 31(1998) L543J.P.Amiet and S.Weigert,J.Phys.A 32 (1999) 2777J.P.Amiet and S.Weigert,J.Phys.A 32 (1999) L269

23G.L.Long et al, to appear J. Optics B: (2001) last issue

2. The oracle

Soonchil Lee at EQIS’01 workshop: All implementation of Grover algorithm have not used an oracle. The marked state is presumed, such as 11 etc.

Oracle blackbox can be understood in two ways:

1) Player A gives player B a blackbox. This blackbox contains the information of the marked state. It can perform a conditional phase change to an aucilla bit.

25

blackbox

2) The oracle is a computable function

Given a graph G, which has n vertices and m edges, the Hamiltonian circuit of G is defined as a loop that is composed of the edges of G , and the loop must traverse every vertex of G exactly once. Graph G=(V,E), where V is the vertex set of G, and E is the edge set of G, let E={e1,e2,… en}, V={v1,v2,…vm}.

27

Finding the Edge covering set of G

E’, a subset of E, that connect to every vertex in the graph

If it further satisfies a. |E’|=|V|, namely the number of vertices

of G is the same as the number of its edges

b. Each vertex is connected with just 2 edges of E’,

It is a Hamiltonian circuit.

28

For a given graph G, define m “edge Boolean variables” x1,x2…,xm corresponding to the m edges of G. x1=1 if it belongs to the edge covering set.define a clause Ci={xi1,xi2,…xik} for each vertex i, where the k Boolean variables are the k edges that is connected to vertex ei. If the truth value of the clause is 1, then there is at least one edge connected with the vertex.

29

If we can find a truth assignment of xk (k=1,,..m) that makes

)1(11 xjCx

nj

then we find a edge covering set of G(edge covering set is sub-set of edge set E, whose intersecting points are the vertex set V).

Every vertex should have its clause satisfied. Therefor is used.

30

Construct a series of unitary gates imposing on the m inputs, so we get a query function that is later used in the Grover algorithm.

Using Grover’s Algorithm, we can find the truth assignment that satisfies eq. (1), or fails to find an assignment satisfying (1) if there is no edge covering set.

This is actually a generalized SAT problem.

4. A Sample

1 2

3 4

x1

x2

x3

x4

4 edge Boolean variables: x1,x2,x3,x4

4 vertex Clauses:

C1={x1,x4}, C2={x1,x2}

C3={x2,x3}, C4={x3,x4}

32

(1) can be written as :

(x1+x4)(x1+x2)(x2+x3)(x3+x4)=1 (2)

Using the rules of Boolean algebra we can get

x1x3+x2x4=1 (3)

We can use (3) as a query function of Grover’s searching algorithm.

We construct the unitary gates series as following:

x1

x3

x2

x4

0

0

0

Query value bit

We can find that

(1,1,1,1), (1,0,1,0), (1,1,1,0),

(0,1,1,1), (0,1,0,1), (1,1,0,1), (1,0,1,1)

are appropriate truth assignments.

Checking them with the features of Hamilton circuit, we will find that only (1,1,1,1) is a Hamilton circuit.

3. The optimality theorem

Grover’s algorithm is the fastest search algorithm for a quantum computer.

Bennett, 1998(?)

C. Zalka, Phys. Rev. A60 (1999) 2746

There have been efforts to build an exponentially fast quantum search algorithm:

Chen and Diao, exponentially fast quantum search algorithm, quant-ph/0011109

Chen-Diao algorithm

Suppose N=22n, the query is

otherwise,0

20,1)(

)(2 jn

j

iif

i

iif

,0

,1)(

jj ffF fFn

Define auxiliary query

1st j bits 0,excluding 0..0

1

00 |

1|

N

i

iN

S

Starting from

For simplicity we assume the 1st bit of the marked state is not 0. kkSkkk SIISQS

k||| 1

kSk IIQk

11

||2iF

k

k

iiII

||2 kkS SSIIk

Starting from the evenly distributed state, after n iteration, the marked state will be found exactly

0

1

0|)(|| SIIS kS

n

kn k

The algorithm seems exponentially fast.

39

It is not exponentially fast, because the inversion about state |Sk>

||2 kkS SSIIk

has to be done with many queries.

WWII S 00

It contains no query.

11

||2iF

k

k

iiII

Contains one query.

40

0001 0011 ||2 SSSS IIIIISSII

It takes two queries. Together with the query in I1, the total number of query in the 2nd iteration is 3.

Continuing the process,

111 1

1

kSkSkSkSk

SkSkSS

IIIIIIIIQ

IIIIII

kkkK

kkkk

41

1

0 2

133

n

i

ni

The total number of queries is

It is slower than the Grover algorithm.

Details of the derivation is in

C C Tu and G L Long, quant-ph/0110098

42

4. Hybrid Quantum Computing

Brueschweiler combined DNA computing and quantum computing recently, and constructed an exponentially fast search algorithm,

R. Brueschweiler, Novel strategy for databse searching in Spin Liouville space by NMR ensemble computing, Phys. Rev. Lett. 85 (2000) 4815

The algorithm actually reads out each bit value of the marked item.

State is mapped on states in spin Liouville space

|01001|

nnininin IIIII 1321||

where

00

01)21(

2

1|| kzkkkk II

10

00)21(

2

1|| kzkkkk II

Value of the k-th-bit of the marked state is read out by the following procedure

First prepare the state . This is a mixed state representing N/2 item of the database:

10 II

32103210

32103210

33221010 ))((

IIIIIIII

IIIIIIII

IIIIIIII

Suppose the marked state is 100. Then apply the oracle to the mixed state, query takes place simultaneously to all the “number” state in the ensemble, then we have, after the query

32103210

32103210

IIIIIIII

IIIIIIII

The same state as before. Measuring the I0z

component, will give 4 relative unit.

Then prepare the state . This is a mixed state representing N/2 item of the database:

20 II

32103210

32103210

33211020 )()(

IIIIIIII

IIIIIIII

IIIIIIII

47

Then apply the oracle to the mixed state, we have, after the query

32103210

32103210

IIIIIIII

IIIIIIII

The same state as before. Measuring the I0z

component, will give 2 relative unit.

48

Similarly, we prepare , and apply the oracle function, and measure the z component of the aucilla qubit. It is also 2 unit.

30 II

bit # Ioz minus 3 yields

1 4 1

2 2 1

3 2 1

49

Uses the same amount of resources as quantum computer with effective pure state,

exponential gain in the speed.

Completion time is short, less demand on the decoherence.

Effective for general ensemble quantum computation.

50

5. Realization Bruschweiler’s algorithm in NMR homonuclear system

• the liquid sample

• Unitary transformation and pulse sequences

• Measurement and spectra analysis

Experimental system

• The structure of our sample in experiment

13C113C0

1H

1H

1H NH ＋2

1H

13C2

O

OH

52

pulse sequence

• Initial state made from the thermo equilibrium

The pulse sequence is

• Initial state made from equilibrium state

10 II

)0(

GradGrad yxy 1,01,02 )6

()4

()2

(

20 II

GradGrad yxy 2,02,01 )6

()2

()2

(

53

• Notations

• Grad is gradient field

022

1

J

012

1

J

The query U• The expression of U, 10 as marked state

10000000

00000100

00100000

00010000

00001000

01000000

00000010

00000001

xy11 )

2()

2(

The corresponding pulse sequence is

55

There is no need to measure I0z, just study the shape of the aucilla qubit spectrum. If the shape after the query remain the same as that before the query, then it is 1. If one of the pulse flips, it corresponds to 0. It is advantageous to generalize this “read-out” method into many qubit system. It is “topological”, thus error robustic.

I0I1before query

It is much easier to implement than the effective pure state ensemble quantum computation.

It is interesting to study ensemble quantum computation using mixed state.

60

2 qubit NMR realization of generalized search algorithmGrover algorithm is optimal for solely QC. Hybid QC, DNA+QC, can achieve exponential speedupBlackbox: a complicated computable function3 qubit NMR realization of Bruschweiler algorithm