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Department of Physics, Tsinghua University Beijing, P R China

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Workshop on Quantum Computation and Quantum Information,

Seoul, Nov.1-3

The Quantum Searching Algorithm(II)

Gui Lu Long

清華大學物理系 龍桂鲁

Department of Physics, Tsinghua University

Beijing, P R China

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

- 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 realization

2. Oracle--an example

3. Optimality theorem, exponentially fast quantum search algorithms

4. “hybrid” quantum computing - the Brschweiler algorithm

5. 3 qubit NMR realization of Brschweiler algorithm

6. Summary

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

1. NMR realization of generalized quantum searching algorithm

Standard 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

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

Pulse sequences for preparing the pseudo-pure state

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

Features:

Non-90 pulses

Delay pulses need not be 1/4J

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.

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

G.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.

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 asa 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}.

- Finding the Edge covering set of G
- E’, a subset of E, that connect to every vertex in the graph
- If it further satisfies
- |E’|=|V|, namely the number of vertices of G is the same as the number of its edges
- Each vertex is connected with just 2 edges of E’,
- It is a Hamiltonian circuit.

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.

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

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.

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

x1

1

2

x4

x2

4

3

x3

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

4 vertex Clauses:

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

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

(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

Define auxiliary query

1st j bits 0,excluding 0..0

Starting from

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

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

The algorithm seems exponentially fast.

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

has to be done with many queries.

It contains no query.

Contains one query.

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

Continuing the process,

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

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

where

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:

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

The same state as before. Measuring the I0zcomponent, will give 4 relative unit.

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

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

The same state as before. Measuring the I0zcomponent, will give 2 relative unit.

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

bit # Ioz minus 3 yields

1 4 1

2 2 1

3 2 1

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.

5. Realization Bruschweiler’s algorithm in NMR homonuclear system

- the liquid sample
- Unitary transformation and pulse sequences
- Measurement and spectra analysis

1H

1H

O

1H

13C0

13C1

13C2

1H

NH＋2

OH

Experimental system

- The structure of our sample in experiment

pulse sequence

- Initial state made from the thermo equilibrium
The pulse sequence is

- Initial state made from
equilibrium state

- Notations
- Grad is gradient field

The query U

- The expression of U, 10 as marked state

The corresponding pulse sequence is

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.

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