Workshop on Quantum Computation and Quantum Information Seoul

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 1

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) 2

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 Br schweiler algorithm 5. 3 qubit NMR realization of Br schweiler algorithm 6. Summary 3

• • • 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 4

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 5 (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. 6

Working media：H 2 PO 3。 The experiments were performed on a Bruker 500 MHz AM NMR Qubit 1 Qubit 2 7

The parameters for the 2 qubit system are: J-coupling constant 647. 451 Hz Frequencies: 500 MHz for 1 H (spin A) 220 MHz for 31 P(spin B) 8

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 9

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2 sets of experiments were performed： • = = /2（Phase matching)； • = /2， = 3 /2(phase mismatching) Features: Non-90 pulses Delay pulses need not be 1/4 J 11

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. 12

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. 13

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, 14 447 (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. 15

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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) 4101 U. Leonhardt, Phys. Rev. , A 53 (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) L 543 J. P. Amiet and S. Weigert, J. Phys. A 32 (1999) 2777 22 J. P. Amiet and S. Weigert, J. Phys. A 32 (1999) L 269

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

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. 24

blackbox 25

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={e 1, e 2, … en}, V={v 1, v 2, …vm}. 26

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. 27

For a given graph G, define m “edge Boolean variables” x 1, x 2…, xm corresponding to the m edges of G. x 1=1 if it belongs to the edge covering set. define a clause Ci={xi 1, xi 2, …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. 28

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. 29

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. 30

4. A Sample 1 x 1 2 x 4 x 2 3 4 x 3 4 edge Boolean variables: x 1, x 2, x 3, x 4 4 vertex Clauses: C 1={x 1, x 4}, C 2={x 1, x 2} C 3={x 2, x 3}, C 4={x 3, x 4} 31

(1)(1) can be written as : (2) (x 1+x 4)(x 1+x 2)(x 2+x 3)(x 3+x 4)=1 (2) (3)Using the rules of Boolean algebra we can get (4) x 1 x 3+x 2 x 4=1 (3) (5)We can use (3) as a query function of Grover’s searching algorithm. (6)We construct the unitary gates series as following: 32

x 1 x 3 x 2 x 4 0 0 0 Query value 33 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, 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. 34

3. The optimality theorem Grover’s algorithm is the fastest search algorithm for a quantum computer. Bennett, 1998(? ) C. Zalka, Phys. Rev. A 60 (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 35

Chen-Diao algorithm Suppose N=22 n, the query is Define auxiliary query 1 st j bits 0, excluding 0. . 0 36

Starting from For simplicity we assume the 1 st bit of the marked state is not 0. 37

Starting from the evenly distributed state, after n iteration, the marked state will be found exactly The algorithm seems exponentially fast. 38

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. 39

It takes two queries. Together with the query in I 1, the total number of query in the 2 nd iteration is 3. Continuing the process, 40

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 41

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 42

The algorithm actually reads out each bit value of the marked item. State mapped on states in spin Liouville space is where 43

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: 44

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 I 0 z component, will give 4 relative unit. 45

Then prepare the state. This is a mixed state representing N/2 item of the database: 46

Then apply the oracle to the mixed state, we have, after the query The same state as before. Measuring the I 0 z component, will give 2 relative unit. 47

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 48

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. 49

5. Realization Bruschweiler’s algorithm in NMR homonuclear system • the liquid sample • Unitary transformation and pulse sequences • Measurement and spectra analysis 50

Experimental system • The structure of our sample in experiment 1 H 1 H 1 H 13 C 0 13 C 1 13 C 2 NH＋2 OH 1 H O 51

pulse sequence • Initial state made from thermo equilibrium The pulse sequence is • Initial state equilibrium state made from 52

• Notations • Grad is gradient field 53

The query U • The expression of U, 10 as marked state The corresponding pulse sequence is 54

There is no need to measure I 0 z, 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. 55

efore query 56

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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. 59

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 60