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VOLUME 80, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 A PRIL 1998

Experimental Implementation of Fast Quantum Searching

Isaac L. Chuang, 1,* Neil Gershenfeld, 2 and Mark Kubinec 31 IBM Almaden Research Center K10 D1, 650 Harry Road, San Jose, California 95120

2Physics and Media Group, MIT Media Lab, Cambridge, Massachusetts 021393College of Chemistry, D7 Latimer Hall, University of California, Berkeley, Berkeley, California 94720-1460

(Received 21 November 1997; revised manuscript received 29 January 1998 )

Using nuclear magnetic resonance techniques with a solution of chloroform molecules we implementGrovers search algorithm for a system with four states. By performing a tomographic reconstructionof the density matrix during the computation good agreement is seen between theory and experiment.This provides the rst complete experimental demonstration of loading an initial state into a quantumcomputer, performing a computation requiring fewer steps than on a classical computer, and thenreading out the nal state. [S0031-9007(98)05850-5]

PACS numbers: 89.70.+c, 03.65.w

The study of computation in quantum systems beganwith the recognition of the theoretical possibility [13].This was followed by a series of results leading up toproofs that a quantum computer requires fewer operationsthan a classical computer for problems including factoring[4] and searching [5,6]. Appreciation of the powerof quantum computing was quickly tempered by therealization that preserving quantum coherence made theimplementation of practical quantum computers appear tobe unlikely [79].

Two recent developments have changed that conclusion.The rst is the recognition that quantum error correctioncan be used to compute with imperfect computers [10,11].And the second is that it is possible to decrease theinuence of decoherence by computing with mixed-stateensembles rather than isolated systems in a pure state.This can be done by introducing extra degrees of freedom[12] using quantum spins [13], space [14], or time [15]to embed within the overall system a subsystem whichtransforms like a pure state. We apply these ideas here inthe rst experimental realization of a signicant quantumcomputing algorithm, using nuclear magnetic resonance(NMR) techniques to perform Grovers quantum searchalgorithm [5,6].

Classically, searching for a particular entry in anunordered list of N elements requires ON attempts.The list could be stored as a table, such as nding a nameto go along with a phone number in a phone book, orcomputed as needed, like testing possible combinations

to unlock a padlock. Grovers surprising result is that aquantum computer can obtain the result with certainty in

Op N attempts.The simplest interesting application of Grovers algo-rithm is the N 4 case, which can be posed as follows:on the set x 0,1,2,3 a function f x 1 except atsome x0 , where f x0 2 1. How many evaluations of f are required to determine x0? In the worst case, x0 hasa uniform probability of being either 0, 1, 2, or 3, andso the average number of evaluations required classically

is 9 4 2.25 . With a quantum computer using Groversalgorithm, this is reduced to a single evaluation. We haveexperimentally implemented this case using molecules of chloroform as a quantum computer, and conrmed the pe-riodic behavior expected of the algorithm.

The algorithm works by representing x as a pair of two-state quantum systems. We take these to be the spinsof the carbon and hydrogen nuclei, writing j " j1and j # j0 . The function f x is implemented as aunitary transform that ips the phase of the x0 element.If the operator corresponding to x0 3 is applied to thesuperposition jc 0 j00 1 j01 1 j10 1 j11 2 theresult is j00 1 j01 1 j10 2 j11 2. Measurementof this state is not useful because each answer occurswith equal probability. Grovers algorithm amplies thecorrect answer by following the conditional ip with asecond operation that inverts each state about the mean.Applied to a superposition Pk a kjk this step gives a newstate Pk b kjk with b k 2a k 1 2 a , where a is themean value of a k . For N 4 and x0 3 the resultof the conditional ip followed by the inversion aboutthe mean is the state jc 1 j11 , providing the answerimmediately. For general N , about p p N 4 repetitions of these two steps are required to nd x0 [16].

Further iteration of the ip and inversion operationsleads to a periodicity in the state. Let U be the unitarytransform which does these two operations, so that

jc n U njc 0 is the state after the n th iteration. Boyeret al. have shown that the amplitude x0 jc n sin 2n 1 1 u , where u arcsin 1

p N ; this periodi-city arises from the nite size of the system and the uni-

tarity of U . For N 4 the theoretical expectation is thesequence j11 jc 1 2 jc 4 jc 7 2 jc 10 . . . , aperiod of 6 (or 3 if the overall sign is disregarded).

Our experiments used a 0.5 milliliter, 200 millimolarsample of Carbon-13 labeled chloroform (CambridgeIsotopes) in d 6 acetone. Data were taken at roomtemperature with a Bruker DRX 500 MHz spectrometer.The coherence times were measured to be T 1 20 sec

3408 0031-9007 98 80(15) 3408(4)$15.00 1998 The American Physical Society

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and T 2 0.4 sec for the proton, and T 1 21 sec andT 2 0.3 sec for the carbon (the large ratio is due toC-Cl relaxation), and the coupling was measured to beJ 215 Hz. All resonance lines from other nuclei andthe solvent were far from the region of interest in thisexperiment. In the rotating frame of the proton (atabout 500 MHz) and carbon (at about 125 MHz), theHamiltonian for this system can be approximated as [17]

H 2p hJI zAI zB 1 P f A t I f A 1 P f B t I f B 1 H env ,(1)

where I f A and I f B are the angular momentum operatorsin the f direction for the proton ( A) and carbon ( B),and H env represents the coupling to the environment,responsible for the decoherence. P f A and P f B describethe strength of radio-frequency (rf) pulses which areapplied on resonance to perform single-spin rotations toeach of the two spins. These rotations will be denoted as X exp i p I x 2 for a 90rotation about the x axis, andY exp 2 ip I y 2 for a 90rotation about 2 y , with asubscript specifying the affected spin.

We used temporal labeling [15] to obtain the signalfrom the pure initial state

jc in j002664

1000

3775

(2)

by repeating the experiment three times, cyclically per-muting the j01 , j10 , and j11 state populations beforethe computation and then summing the results.

The calculation starts with a Walsh-Hadamard trans-form W , which rotates each quantum bit (qubit) from

j0

to j0 1 j1 p 2, to prepare the uniform superpositionstate

jc 0 W jc in12

2664

1 1 1 11 2 1 1 2 11 1 2 1 2 11 2 1 2 1 1

3775

2664

1000

3775

12

2664

1111

3775

. (3)

Note that W H A H B , where H X 2Y (pulses ap-plied from right to left) is a single-spin Hadamardtransform.

The operator corresponding to the application of f xfor x0 3 is as

C2664

1 0 0 00 1 0 00 0 1 00 0 0 2 1

3775

. (4)

This conditional sign ip, testing for a Boolean stringthat satises the AND function, is implemented by

using the coupled-spin evolution which occurs whenno rf power is applied. During a time t the systemundergoes the unitary transformation exp 2p iJI zAI zB tin the doubly rotating frame. Denoting a t 1 2J (2.3 millisecond) period evolution as the operator t , wend that C Y A X AY AY B X BY B t (up to an irrelevantoverall phase factor).

An arbitrary logical function can be tested by a network

of controlled- NOT and rotation gates [13,18], leaving theresult in a scratch pad qubit. This qubit can then beused as the source for a controlled phase-shift gate toimplement the conditional sign ip, if necessary reversingthe test procedure to erase the scratch pad. In ourexperiment these operations could be collapsed into asingle step without requiring an extra qubit.

The operator D that inverts the states about their meancan be implemented by a Walsh-Hadamard transform W ,a conditional phase shift P , and another W :

D WPW W 2

664

1 0 0 00 2 1 0 0

0 0 2 1 00 0 0 2 1

3

775

W 12

2664

2 1 1 1 11 2 1 1 11 1 2 1 11 1 1 2 1

3775

(5)

This corresponds to the pulse sequence PY A X AY AY B X BY B t .

Let U DC be the complete iteration. The state afterone cycle is

jc 1 UW

jc 0

j11

2

664

00

01

3

775. (6)

A measurements of the systems state will now give withcertainty the correct answer, j11 . For further iterations,jc n U njc 0 ,

jc 212

2664

2 12 12 1

1

3775 j

c 312

2664

2 12 12 12 1

3775

jc 42664

000

2 1

3775

. (7)

We see that a maximum in the amplitude of the x0 state

j11 recurs every third iteration.Like any computer program that is compiled to a mi-crocode, the rf pulse sequence for U can be optimizedto eliminate unnecessary operations. In a quantum com-puter this is essential to make the best use of the avail-able coherence. Ignoring irrelevant overall phase factors,and noting that H X 2Y also works, we can simplify U

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by removing sequential rotations which cancel each otherout, to get

U X AY A X BY B t X AY A X BY B t x0 3 . (8)

The other possible cases are obtained by changing thesigns of the rst two X rotations,

U 8

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the other possible values of x0 . Measuring each densitymatrix required 9 3 3 27 experimental repetitions, ninefor the tomographic reconstruction and three for the purestate preparation. Both of these operations were performedas tests of the computation, but neither was necessary. Inour experiment, starting from the thermal state the maxi-mum population can be identied in a single iteration, withthe result obtained from a single output spectrum. In the

general N case, readout of log N expectation value mea-surements would be required, and good inputs for Groversalgorithm can be distilled in a number of steps polynomialin log N [15].

The longest computation, for n 7, took less than35 milliseconds, which was well within the coherencetime. The periodicity of Grovers algorithm is clearly seenin Fig. 1, with good agreement between theory and experi-ment. The large signal-to-noise ratio (typically better than104 to 1) was obtained with just single-shot measurements.Numerical simulations indicate that the 7%44% errorsare primarily due to inhomogeneity of the magnetic eld,magnetization decay during the measurement, and imper-fect calibration of the rotations (in order of importance).

These experimental results demonstrate the operation of a simple quantum computer that can load an initial state,perform a computation, and read out the answer. Whilethere is a long way to go from such a demonstrationto a system that can exceed the performance of thefastest classical computers, the experimental study of quantum computation has already come much farther inits short life than either early theoretical predictions orthe history of mature computing technologies would havesuggested. While scaling up to much larger systems posesdaunting challenges, many optimizations remain to betaken advantage of, including increasing the sample size,using coherence transfer to and from electrons, and opticalpumping to cool the spin system [19]. Furthermore,Grovers algorithm can be matched to convenient physicaloperations by performing generalized rapid search, whichuses transforms other than the Walsh-Hadamard [20].

The NMR system that we have described already has allof the components of a complete computer architecture,including the rudiments of compiler optimizations. Itcan implement a nontrivial quantum computation; thechallenge now is to accomplish a useful one.

We gratefully acknowledge the support of DARPAunder the NMRQC Initiative, Contract No. DAAG55-97-1-0341, and the MIT Media Labs Things That Think consortium. We thank Gilles Brassard, Lov Grover, andAlex Pines for helpful comments.

*Electronic address: [email protected][1] P. A. Benioff, Int. J. Theor. Phys. 21 , 177 (1982).[2] R. P. Feynman, Int. J. Theor. Phys. 21 , 467 (1982).[3] D. Deutsch, Proc. R. Soc. London A 400 , 97 (1985).[4] P.W. Shor, in Proceedings of the 35th Annual Sympo-

sium on Foundations of Computer Science, Santa Fe, NM,1994, edited by Sha Goldwasser (IEEE Computer Soci-ety Press, Los Alamitos, CA, 1994), pp. 124134; SIAMJ. Comput. 26 , 14841509 (1997).

[5] L. Grover, in Proceedings of the 28th Annual ACM Symposium on the Theory of Computation (ACM Press,New York, 1996), pp. 212219.

[6] L. K. Grover, Phys. Rev. Lett. 79 , 325 (1997).[7] W. G. Unruh, Phys. Rev. A 51 , 992 (1995).[8] I. L. Chuang, R. Laamme, P. Shor, and W. H. Zurek,

Science 270 , 1633 (1995).[9] G. M. Palma, K.-A. Suominen, and A. Ekert, Proc. R. Soc.

London A 452 , 567 (1996).[10] A. Steane, Proc. R. Soc. London A 452 , 2551 (1996).[11] A. R. Calderbank and P. W. Shor, Phys. Rev. A 54 , 1098

(1996).[12] I. Chaung and N. Gershenfeld, State Labeling for Bulk

Quantum Computation (unpublished).[13] N. Gershenfeld and I. L. Chuang, Science 275 , 350 (1997).[14] D. Cory, A. Fahmy, and T. Havel, Proc. Nat. Acad. Sci.

U.S.A. 94 , 1634 (1997).[15] E. Knill, I. Chuang, and R. Laamme, Effective Pure

States for Bulk Quantum Computation, Phys. Rev. A (tobe published).

[16] M. Boyer, G. Brassard, P. Hyer, and A. Tapp, LANLe-print quant-ph/9605034; Fortschr. Phys. (to bepublished).

[17] R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principlesof Nuclear Magnetic Resonance in One and Two Dimen-sions (Oxford University Press, Oxford, 1994).

[18] A. Barenco et al., Phys. Rev. A 52 , 3457 (1995).[19] I. L. Chuang, N. Gershenfeld, M. Kubinec, and D. Leung,

Proc. R. Soc. London A 454 , 447 (1998).[20] L. Grover, LANL e-print quant-ph/9711043 (1997).

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