8
Approaching five-bit NMR quantum computing R. Marx Institut fu ¨r Organische Chemie, Johann-Wolfgang-Goethe-Universita ¨t, Marie-Curie-Straße 11, D-60439 Frankfurt, Germany A. F. Fahmy Biological Chemistry and Molecular Pharmacology, Harvard Medical School, 240 Longwood Avenue, Boston, Massachusetts 02115 John M. Myers Gordon McKay Laboratory, Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138 W. Bermel Bruker Analytik GmbH, Silberstreifen, D-76287 Rheinstetten, Germany S. J. Glaser Institut fu ¨r Organische Chemie und Biochemie, Technische Universita ¨t Mu ¨nchen, Lichtenbergstraße 4, D-85748 Garching, Germany ~Received 14 May 1999; revised manuscript received 11 October 1999; published 15 June 2000! Nuclear-magnetic-resonance ~NMR! quantum computation is a fruitful arena in which to develop and dem- onstrate an enhanced capability for quantum control over molecular systems, regardless of the prospects, which may be limited, for building a quantum computer superior to a conventional computer for any computing task. We demonstrate a five-bit NMR quantum computer that distinguishes among various functions on four bits, making use of quantum parallelism, an example of the Deutsch-Jozsa problem. Its construction draws on the recognition of the sufficiency of linear coupling along a chain of nuclear spins, the synthesis of a suitably coupled molecule containing four distinct nuclear species, and the use of a multichannel spectrometer. Radio- frequency pulse sequences are described to execute controlled-NOT gates on two adjoining spins while leaving the other three spins essentially unaffected. PACS number~s!: 03.67.Lx, 89.80.1h, 75.10.Jm I. INTRODUCTION While quantum computers of two bits have been imple- mented @1#, as have nuclear-magnetic-resonance ~NMR! quantum computers of three bits @2#, extending the number of bits has not proved easy. We report the implementation of an NMR quantum computer having five bits, involving the use of a linear coupling pattern @3#, synthesis of a molecule having five usable spin-active nuclei with predominantly lin- ear spin-spin coupling, and the development of radio- frequency ~rf! pulse sequences to act as quantum logic gates for the molecule synthesized. Techniques to suppress un- wanted couplings between nuclear spins are described, as are techniques to avoid perturbing some nuclear spins while ma- nipulating others. Results are presented of a test of the five- bit computer on a problem of Deutsch and Jozsa to distin- guish one class of mathematical function from another @4#. II. DEFINITION OF AN n-BIT NMR COMPUTER An n-bit quantum computer is called on to do three things: ~1! accept an instruction to prepare a starting state and prepare that state; ~2! accept instructions for and imple- ment quantum gates ~from which more general unitary trans- formations of the state can be composed!; and ~3! measure the state and yield an outcome. The connection to computa- tion with classical computers depends on the recognition, due to Bennett @5#, that all classical computations can be made reversible. Any terminating reversible, classical com- putation is a permutation of the inputs, which is unitary, and thus belongs to the class of transformation performable on a quantum computer. ~For issues of possibly nonterminating programs, see @6#.! In theory, a variant of the quantum computer is the expectation-value quantum computer ~EVQC!, which in place of an outcome of a measurement yields the expectation value @7,8#. NMR quantum computing was born of the rec- ognition that an EVQC can be approximated by use of an NMR spectrometer containing a liquid sample, the molecules of which have n atoms with a nuclear spin of 1/2 ~and pos- sibly other atoms, either spinless or having spins not used! @7–9#. Because tumbling of the molecules decouples each molecule from all the others, the sample can be described by a density matrix for the nuclear spins of the atoms of a single molecule @10#, with only the spin degrees of freedom, corre- sponding to the desired Hilbert space of dimension 2 n . NMR spectrometers sense only the traceless part of the density matrix, so in place of matter in a pure state, an NMR com- puter can use a liquid sample described by a density matrix proportional to a sum of a pure state and any multiple of the unit matrix. Such a density matrix, called a pseudopure state @7#, plays a role in the five-bit quantum computer. Acting as an n-bit EVQC, a suitable NMR spectrometer allows the preparation of a pseudopure starting state, the pro- gramming and execution of rf pulse sequences that imple- ment quantum gates, and the determination of expectation values visible in NMR spectra. To perform the unitary op- erations required of a quantum computer, a sufficient set of PHYSICAL REVIEW A, VOLUME 62, 012310 1050-2947/2000/62~1!/012310~8!/$15.00 ©2000 The American Physical Society 62 012310-1

Approaching five-bit NMR quantum computing

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PHYSICAL REVIEW A, VOLUME 62, 012310

Approaching five-bit NMR quantum computing

R. MarxInstitut fur Organische Chemie, Johann-Wolfgang-Goethe-Universita¨t, Marie-Curie-Straße 11, D-60439 Frankfurt, Germany

A. F. FahmyBiological Chemistry and Molecular Pharmacology, Harvard Medical School, 240 Longwood Avenue, Boston, Massachusetts 0

John M. MyersGordon McKay Laboratory, Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02

W. BermelBruker Analytik GmbH, Silberstreifen, D-76287 Rheinstetten, Germany

S. J. GlaserInstitut fur Organische Chemie und Biochemie, Technische Universita¨t Munchen, Lichtenbergstraße 4, D-85748 Garching, Germany

~Received 14 May 1999; revised manuscript received 11 October 1999; published 15 June 2000!

Nuclear-magnetic-resonance~NMR! quantum computation is a fruitful arena in which to develop and dem-onstrate an enhanced capability for quantum control over molecular systems, regardless of the prospects, whichmay be limited, for building a quantum computer superior to a conventional computer for any computing task.We demonstrate a five-bit NMR quantum computer that distinguishes among various functions on four bits,making use of quantum parallelism, an example of the Deutsch-Jozsa problem. Its construction draws on therecognition of the sufficiency of linear coupling along a chain of nuclear spins, the synthesis of a suitablycoupled molecule containing four distinct nuclear species, and the use of a multichannel spectrometer. Radio-frequency pulse sequences are described to execute controlled-NOT gates on two adjoining spins while leavingthe other three spins essentially unaffected.

PACS number~s!: 03.67.Lx, 89.80.1h, 75.10.Jm

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I. INTRODUCTION

While quantum computers of two bits have been impmented @1#, as have nuclear-magnetic-resonance~NMR!quantum computers of three bits@2#, extending the numbeof bits has not proved easy. We report the implementationan NMR quantum computer having five bits, involving thuse of a linear coupling pattern@3#, synthesis of a moleculehaving five usable spin-active nuclei with predominantly lear spin-spin coupling, and the development of radfrequency~rf! pulse sequences to act as quantum logic gafor the molecule synthesized. Techniques to suppresswanted couplings between nuclear spins are described, atechniques to avoid perturbing some nuclear spins whilenipulating others. Results are presented of a test of the fibit computer on a problem of Deutsch and Jozsa to disguish one class of mathematical function from another@4#.

II. DEFINITION OF AN n-BIT NMR COMPUTER

An n-bit quantum computer is called on to do thrthings: ~1! accept an instruction to prepare a starting stand prepare that state;~2! accept instructions for and implement quantum gates~from which more general unitary transformations of the state can be composed!; and ~3! measurethe state and yield an outcome. The connection to comption with classical computers depends on the recognitdue to Bennett@5#, that all classical computations can bmade reversible. Any terminating reversible, classical co

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putation is a permutation of the inputs, which is unitary, athus belongs to the class of transformation performable oquantum computer.~For issues of possibly nonterminatinprograms, see@6#.!

In theory, a variant of the quantum computer is texpectation-value quantum computer~EVQC!, which inplace of an outcome of a measurement yields the expectavalue @7,8#. NMR quantum computing was born of the reognition that an EVQC can be approximated by use ofNMR spectrometer containing a liquid sample, the molecuof which haven atoms with a nuclear spin of 1/2~and pos-sibly other atoms, either spinless or having spins not us!@7–9#. Because tumbling of the molecules decouples emolecule from all the others, the sample can be describeda density matrix for the nuclear spins of the atoms of a sinmolecule@10#, with only the spin degrees of freedom, corrsponding to the desired Hilbert space of dimension 2n. NMRspectrometers sense only the traceless part of the dematrix, so in place of matter in a pure state, an NMR coputer can use a liquid sample described by a density maproportional to a sum of a pure state and any multiple ofunit matrix. Such a density matrix, called apseudopurestate@7#, plays a role in the five-bit quantum computer.

Acting as ann-bit EVQC, a suitable NMR spectrometeallows the preparation of a pseudopure starting state, thegramming and execution of rf pulse sequences that imment quantum gates, and the determination of expectavalues visible in NMR spectra. To perform the unitary oerations required of a quantum computer, a sufficient se

©2000 The American Physical Society10-1

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MARX, FAHMY, MYERS, BERMEL, AND GLASER PHYSICAL REVIEW A62 012310

quantum gates consists of all single-spin operations andcontrolled-NOT gates that act on one nuclear spin undercontrol of another nuclear spin. Single-spin gates are immented by selective rf pulses. Controlled-NOT gates betweennuclei having spin-spin coupling will be described, alowith techniques to avoid unwanted influences on other spA key feature of the present design of the NMR quantcomputer is the reliance on a chain of linear coupling anduse of swap gates to implement a controlled-NOT in which aspin j controls spink, wherej andk have no direct spin-spincoupling@3#. This allows use in NMR quantum computersa molecule having a simpler coupling pattern, and easesproblem of unwanted influences on spins.

III. DESIGN OF TEST

The proof of the pudding is in the eating: the five-bNMR computer to be described was tested on the DeutJozsa problem for functions of four bits@4#, in the formdescribed in@11#, modified for efficiency with NMR as described by Jones and Mosca@12#. ~A recent simplification@13#, unused here, would permit working with functionsfive bits.! The problem is to decide whether a function prgram selected from a set of possible programs computeskind of function or another. Specifically, the problem isdistinguish programs for balanced functions from prografor constant functions, where the functions are from$0,1%4 to$0,1%. ~A function is constant if its value is independent ofargument, and is called balanced if the value for halfarguments is 1 while the value is 0 for the other half.! Thetest actually made was to distinguish between programsone constant and one balanced function, defined as follo

f 0~xW !5def

0 ~1!

and

f b~xW !5def

x1% x2% x3% x4 ~2!

for all xW , wherexW 5def

(x1, x2, x3, x4), and ‘‘% ’’ is additionmodulo 2. Also, the one-bit operations~spin-selective 90°and 180° pulses! and the controlled-NOT ~CNOT! gates thatwere used in the implementation were carefully tested.particular, for a large number of initial states, the resultspectra were compared with the theoretically predictedfects for parts of the implemented pulse sequences as wefor the complete unitary transformations that were impmented. The balanced function chosen,f b , has the niceproperty of being implementable also in classical reversgates with no work bits.

Used to solve this problem, a quantum computer is asource used both to specify the function under test anddetermine what it is. In order to separate these two uses,can view the quantum computer as used alternately bspecifierof the function and adecision maker, two players ofa game in which~A! the decision maker prepares the starti

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state;~B! the specifier runs the function program;1 and ~C!the decision maker makes a measurement independent ofunction program, and interprets the result to decidefunction class.

On an NMR quantum computer,~A! the decision makerstarts a play by using rf pulses and magnetic-field gradie~independent of the function to be specified! to put the liquidsample in the pseudopure state having a density matrix wa traceless part proportional to

r i5def

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in terms of the polarization operatorsI ka5( 1

2 11I kz) and I kb

5( 12 12I kz) usual to NMR@10,3#. Then the decision make

applies a unitary transformU90 by use of a hard 90°y pulsewhich for this particular state has the same effect asHadamard transform on each spin@12#,

r i →U90

r05U90r iU90†

516~ 12 11I 1x!~

12 11I 2x!

3~ 12 11I 3x!~

12 11I 4x!~

12 12I 5x!2 1

2 1. ~4!

~B! The specifier chooses a functionf from one of the setof functions undergoing test, heref 0 or f b , and runs thequantum version of a program to computef; this program isa sequence of gates, each a unitary transformation immented by a rf pulse sequence. The total program impments a unitary transformationU( f ), defined by its action onbasis vectorsuxW ,x5&:

U~ f !uxW ,x5&5uxW ,x5% f ~xW !&. ~5!

The transformU( f ) produces the density matrix with traceless part proportional tor f :

r0 →U~ f !

r f . ~6!

~C! The decision maker reads out the NMR spectruwhich depends onr f . The spectrum differs according twhetherf is balanced or constant, and thus tells the decismaker the function class, with only one function evaluatioa large saving over classical computation, which couldquire nine evaluations for functions of four bits.

In theory, for the casef 5 f 0 , U( f ) is specified to beU( f 0), which by Eqs.~1! and~5! turns out to be the identitymatrix, so one should haver f5r0. The spectrometer detectonly the terms of the right-hand side of Eq.~4! that are linearin I x , so for a spectrometer adjusted to give an upward p

1These two moves must be iterated for a classical computer,not in the quantum solution of the Deutsch-Jozsa problem, givthe quantum computer a large advantage over the clascomputer.

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APPROACHING FIVE-BIT NMR QUANTUM COMPUTING PHYSICAL REVIEW A62 012310

For the balanced function,f 5 f b @Eq. ~2!#, U( f b) is de-fined by U( f b)uxW ,x5&5uxW ,x1% x2% x3% x4% x5&. A unitaryoperator that is simpler to implement, that has the samefect on the fifth~value! bit, and that allows the distinctionbetween constant and balanced functions isU( f b), definedby

U~ f b!uxW ,x5&5ux1 ,x1% x2 ,x1% x2% x3 ,x1% x2% x3% x4 ,x1

% x2% x3% x4% x5&, ~7!

IV. REALIZATION AND TEST OF A FIVE-BIT NMRCOMPUTER

A five-bit NMR quantum computer requires a molecuhaving five spin-active nuclei, with long relaxation timeLarge separation of resonance frequencies of the nuclelows rapid selective control of the spins. For frequency seration, it is desirable to use different atomic species for dferent spins, which requires a multichannel NMspectrometer. In the preparation and detection step, theferent gyromagnetic ratios of the heteronuclear spins mustaken into account@14,15#. In our experiments, the preparation started each time from proton polarization, and hethe different gyromagnetic ratios had no influence~see Ap-pendix B 4!. In the detection process, the different sensitivof heteronuclear spins due to different gyromagnetic ra~and also due to different characteristics of the resonantcuits at different frequencies! can be taken into account btransferring proton polarization selectively to each hetenucleus and detecting the signal. The amplitude of this sigcan be used as a reference for this spin species. The lopolarization during the transfer from protons to each hetenucleus can be quantified by transferring the polarizatback to protons and comparing the remaining signal withoriginal proton signal.

Our NMR experiments were performed using a BrukAvance 400 spectrometer with five independent rf channand a quadruple probe with inverse design~QXI! probe~H,C-F, N!. The lock coil was also used for deuterium decopling utilizing a lock switch. A linear path of spin-spin couplings is sufficient for all computations@3#. Given the avail-ability of a five-channel spectrometer, we chose as‘‘hardware’’ of our NMR quantum computing experimenthe molecule BOC-(13C2-15N-2D2

a-glycine! fluoride, whereBOC stands fortert-butoxycarbonyl. The molecule containan isolated coupling network consisting of five nuclei, eahaving spin 1/2: the amide1H, the 15N, the aliphatic13Ca,the carbonyl13C8, and the19F nuclear spin~see Fig. 1!. Forsimplicity, we will refer to these spins~and the corresponding bits! as 1, 2, 3, 4, and 5, respectively. All spins a

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heteronuclear, except for Ca and C8 which, however, have arelatively large chemical shift difference. The five-spin sytem is well isolated from the protons of the BOC protectigroup, which are separated by more than four chembonds. In addition, the deuterium spins~D! that are attachedto Ca can be fully decoupled from the spins of interest usistandard heteronuclear decoupling techniques@16–18#. Thesubstance was synthesized starting from commercially avable 13C- and15N-labeled glycine~see Appendix A! and wasdissolved in deuterated dimethyl-sulfoxide (DMSO-D6).NMR experiments were performed at a magnetic fieldabout 9.4 T and a sample temperature of 27°C. The expmentally determinedT2 relaxation times for spins 1–5 wer250 ms, 490 ms, 450 ms, 590 ms, and 260 ms, respectivResonance frequenciesnk and scalar coupling constantsJklare summarized in Table I. Except for theJ23 coupling con-stant of 13.5 Hz, the spin chain is connected by one-bcoupling constantsJk$k11% larger than 60 Hz. In the multiplerotating frame~see Ref.@10# and Appendix B! the precessionfrequency of each individual spin is 0, which considerabsimplifies implementation, because only coupling terms nto be considered~and manipulated!.

The experimental implementation of the propagaU( f 0) corresponding tof 0 is trivial because by Eq.~5! thepropagator is the unit operator, implemented by doing noing. In contrast, the construction of the pulse sequenceimplement the series ofCNOT gates that define the unitartransformationU( f b) of Eq. ~7! for the balanced functionf b@Eq. ~2!# requires attention. The goal is to create robust pu

FIG. 1. Schematic representation of BOC(13C2-15N-2D2

a-glycine! fluoride. The atoms (1H, 15N, 13Ca, 13C8,and 19F) that form the five-spin system of interest are printedbold face.

TABLE I. Resonance frequenciesnk , chemical shiftsdk , one-bond coupling constantsJk$k11% , and nonzero two-bond couplingconstantsJk$k12% of the five-spin system used. No resolved three-four-bond coupling constants were observed.

n15400,133, 001.6 Hz (d157.51 ppm)n2540, 547, 895.3 Hz (d2575.54 ppm)n35100, 616, 858.0 Hz (d3541.05 ppm)n45100, 629, 089.1 Hz (d45162.61 ppm)n55376, 510, 545.5 Hz (d5531.92 ppm)J12594.1 Hz J23513.5 HzJ34565.2 Hz J455366.0 HzJ1352.7 Hz J35567.7 Hz

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MARX, FAHMY, MYERS, BERMEL, AND GLASER PHYSICAL REVIEW A62 012310

sequence elements that minimize the effects of experimeimperfections. The pulse sequence elements shown in2~a!–2~d! were designed specifically for the coupling topoogy of our five-spin system to implement the unitary opetors corresponding to~CNOT)12, ~CNOT)23, ~CNOT)34, and~CNOT)45.2 During theseCNOT gates, which act on two directly coupled spinsk andl, only the couplingsJkl are active,while the effect of all other couplings in the spin system arefocused by cyclic pulse sequences@10,19,20#. Figure 3shows schematically the pulse sequence actually used fo

propagatorU( f b) for the balanced functionf b ; this se-quence benefited from applying simple rules for pulse ccellation ~see Appendix B 2!.

The NMR implementation of the Deutsch-Jozsa algoritstarts with the preparation of the pseudopure stater i of Eq.~3!. The preparation of such a pseudopure state by a sipulse sequence requires a nonunitary transformation ofthermal equilibrium density operator@7,21#. This can beachieved using spatial averaging@7,3# or temporal averaging@7,15#. In the basis formed by Cartesian product operat@10#, r i can be expressed as a linear combination of 31 tethat consist ofz spin operators only:

2During each pulse sequence shown in Fig. 2 only the couplingJkl

is active, which is required in order to implement~CNOT)kl . Theeffects of other nonzero couplings~see Table I! are effectivelyeliminated, except forJ13 in the sequence implementing~CNOT)45

@Fig. 2~d!#. Although it would be straightforward to remove thcoupling also, this would require additional pulses, which canavoided because in our spin system the couplingJ1352.7 has anegligible effect during the relatively short durationD4551/(2J45)51.39 ms of this gate.

FIG. 2. Individual NMR quantum gates. Pulse sequencements are shown that create the unitary operators of~a! ~CNOT)12,~b! ~CNOT)23, ~c! ~CNOT)34, and ~d! ~CNOT)45. In this schematicrepresentation, 90° and 180° pulses are shown as narrow linesrectangles, respectively. 90° rotations around thez axis are indi-cated by dashed lines. As discussed in Appendix B 2, 180° puthat can be eliminated are indicated by white rectangles~cf. Fig. 3!.Deuterium decoupling is applied during the entire sequences~notshown!. The total durationtkl of each sequence is given by~a!t1251/(2J12)55.31 ms, ~b! t2351/(2J23)537.04 ms, ~c! t34

51/(2J34)57.69 ms, and~d! t4551/(2J45)51.39 ms.

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1 (k, l ,m,n

sn8I kzI lzI mzI nz216I 1zI 2zI 3zI 4zI 5z , ~8!

wheresn521 if n55 andsn51 otherwise. It is straightfor-ward to create each of these terms from the thermal equrium density operator, using standard building blockshigh-resolution NMR@22#. In principle, temporal averagingcould be realized by repeating steps~A!–~C! of the game forall 31 terms in Eq.~8! and by summing up the resultinspectra. However, because currently available NMR sptrometers require a distinct experiment to detect each sspecies (1H, 15N, 13C, and 19F) ~see Appendix B 3!, a totalof 124 NMR experiments would be required for each funtion f in order to include all terms in the temporal averaginA detailed analysis shows that of the 31 terms that constithe pseudopure stater i , only the five linear termsI kz and thefour bilinear terms 2I kzI $k11%z are transformed into detectable operators by the propagatorU90 ~to creater0) followedby the propagatorsU( f 0) or U( f b), as the case may be~seeTable II and Appendix C!. As pointed out previously@2#,preparing just the linear termsI kz suffices in some cases othe Deutsch-Jozsa problem to distinguish constant from

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FIG. 3. NMR pulse sequence to realize the balanced functiointerest. Complete pulse sequence used in the experiments to im

ment the unitary transformationU( f b) @Eq. ~7!# corresponding tothe balanced functionf b @cf. Eq. ~2!#. The durations of the indi-vidual quantum gates were chosen to be integer multiples oD51/un32n4u581.75 ms: t12565D, t235453D, t34594D, andt45517D ~see Appendix B 1!. The bell-shaped symbols labele‘‘e’’ and ‘‘g’’ represent selective e-SNOB 90° pulses@24# and se-lective Gaussian 180° pulses@25#, respectively. During the selective e-SNOB 90° pulses, spin 5 was actively decoupled by applya MLEV-4 @16–18# expanded sequence of 180° pulses to spin~boxes labeled ‘‘M’’!. The phasesh52wg12we , m522we , andj54wg12we , with we524° andwg5218°, correct for the non-resonant effect@26# created by the selective e-SNOB 90° anGaussian 180° pulses. The 90z

° rotations of the individual quantumgates~see Fig. 2! were absorbed in the phases of following pulsand corresponding shifts of the receiver phases for the individspins. Ticks are separated bytkl/4. The periodd has a duration oft12/42te/2, wherete is the duration of an e-SNOB 90° pulse~seeAppendix B 3!.

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APPROACHING FIVE-BIT NMR QUANTUM COMPUTING PHYSICAL REVIEW A62 012310

anced functions, because in these cases a balanced fungives a vanishing signal for at least one of the input spHowever, in the presence of experimental imperfections,desirable to identify a balanced function based on the sreversal of the signal of at least one of the input spins, rathan by the lack of a signal. For the special case ofbalanced functionf b that was chosen for this demonstratioexperiment, this can be achieved by including also the biear terms 2I kzI $k11%z as starting operators~see Table II!.

Samples described by these linear and bilinear terms or iwere prepared~see Appendix B 4! to demonstrate experimental control of the five-spin system and to execute caof the Deutsch-Jozsa algorithm. For each function (f 0 andf b) the following three sets of experiments were perform~see experimental spectra in Fig. 4!. Set 1~first row of curvesfrom the bottom in Fig. 4!: preparation of the linear termsI kz@with algebraic signs as specified in Eq.~8! and Table II#,application ofU90 and U( f ), and detection of spink for k51, . . . ,5; set 2~second row in Fig. 4!: preparation of thebilinear terms 2I kzI $k11%z @with algebraic signs as specifiein Eq. ~8! and Table II#, application ofU90 and U( f ), anddetection of spink for k51, . . . ,4; and set 3~third row inFig. 4!: preparation of the bilinear terms 2I $k21%zI kz @withalgebraic signs as specified in Eq.~8! and Table II#, applica-tion of U90 and U( f ), and detection of spink for k52, . . . ,5.

The observed spectra shown in Fig. 4 correspond cloto the theoretical predictions~see Table II!. For the constant

TABLE II. List of initial Cartesian product operator terms ofr i

@Eqs.~3! and~8!# that give rise to detectable signals for at least oof the functionsf 0 or f b in the implemented version@11# of theDeutsch-Jozsa algorithm~see also Appendixes B and C!. The

propagatorsU( f 0)51 andU( f b) @Eq. ~7!# transformr0 @Eq. ~4!# to

r0 and rb , respectively@Eq. ~6!#. Only the underlined terms ofr0

andrb which contain single transverse spin operators corresponsingle quantum coherences that are detectable in a NMR exment.

r i r0 rb

I 1z I 1x 16I 1xI 2xI 3xI 4xI 5x

2I 1zI 2z 2I 1xI 2x I 1x

I 2z I 2x 8I 2xI 3xI 4xI 5x

2I 2zI 3z 2I 2xI 3x I 2x

I 3z I 3x 4I 3xI 4xI 5x

2I 3zI 4z 2I 3xI 4x I 3x

I 4z I 4x 2I 4xI 5x

22I 4zI 5z 22I 4xI 5x 2I 4x

2I 5z 2I 5x 2I 5x

01231

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function f 0, only the experiments of set 1 yield detectabsignals. For the balanced functionf b , the experiments of se1 yield a detectable signal only for spin 5, whereas for sp1–4 detectable signals are obtained only in the experimof set 2. As expected, only spurious signals are detectedthe experiments of set 3. The amplitude of these spurisignals is typically on the order of 4% compared to the fsignals. As expected, all the signals of spins 1–4 are posfor the constant function whereas the signal of spin 4 isverted by the propagatorU( f b) corresponding to the balanced function. Forf b the signal amplitudes reach only between 55% and 70% of the amplitudes found forf 0. Thissignal loss can be attributed mainly to relaxation and expmental imperfections during the sequence that implemeU( f b) ~Fig. 3!, which has an overall duration of 51.4 ms.

Through combined synthetic, analytic, and spectroscowork, a five-bit NMR quantum computer was built anshown to implement superposition, quantum interferenand designed unitary transformations. A nontrivial balanctest function f b @Eq. ~2!# was chosen for the experimentsolution of the Deutsch-Jozsa problem for five bits. For otbalanced functions the corresponding unitary transformatican be considerably more complicated to implement inpresent five-spin system because a larger number of loggates and swap operations is required. In addition, readcriteria may not be as straightforward as inspecting forverted lines. Given the relaxation properties of our samand the fact that experimental imperfections cannot be elinated completely, some of the other balanced test functwill be difficult or impossible to implement with sufficienprecision in our system without additional compensationcorrection schemes. However, to some degree, experimeimperfections and relaxation~decoherence! will be present in

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FIG. 4. Experimental results of the Deutsch-Jozsa algorithmthe constant functionf 0 and the balanced functionf b . Each sectioncorresponds to a spectral range of 30 Hz. The number of scans4 for 1H ~spin 1! detection, 8 for15N detection~spin 2!, 32 for 13Cdetection~spins 3 and 4!, and 4 for 19F ~spin 5!. For each spink,effectively decoupled spectra~see Appendix B! are shown for thefollowing terms ofr i @with algebraic signs as specified in Eq.~8!and Table II#: I kz for k51, . . . ,5 ~set 1!, 2I kzI $k11%z for k51, . . . ,4 ~set 2!, and 2I $k21%zI kz for k52, . . . ,5 ~set 3!. For f b

only, the signal of one of the four input spins~here spin 4! isnegative, which corresponds to the expected result of the DeutJozsa algorithm thatf b is in fact balanced, whereasf 0 is a constantfunction.

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MARX, FAHMY, MYERS, BERMEL, AND GLASER PHYSICAL REVIEW A62 012310

any attempt to realize quantum computing and these mvate the design of algorithms for quantum error correctiAlthough obstacles had to be overcome to realize a fiveNMR quantum computer, quantum computers with more bwill be built. Many interesting questions have been raisedfuture work pertaining to the constraints and opportunitfor linking molecular architecture, spectrometer design, aalgorithms for NMR quantum computing.

ACKNOWLEDGMENTS

S.J.G. acknowledges support by the Fonds der Chechen Industrie and the DFG. R.M. is supported by the Foder Chemischen Industrie and the Bundesministerium¨rBildung und Forschung~BMBF!. We thank C. GriesingerM. Grundl, R. Kerssebaum, B. Luy, R. Mayr-Stein, M. Kener, M. Reggelin, H. Schwalbe, and A. Tu¨chelmann for valu-able discussions and technical assistance. A.F.F. thankWagner~Harvard Medical School! for support and encouragement and acknowledges support from the Nationalence Foundation~Grant No. MCB-9527181!. J.M.M. thanksT. T. Wu ~Harvard University! for many critical insights.

APPENDIX A: SYNTHESIS OF MOLECULE

We purchased 250 mg of13C2-15N-glycine from MartekBiosciences Corporation~6480 Dobbin Road, ColumbiaMaryland 21045!. The labeled glycine was fully deuterateby treatment with a solution of NaOD in D2O at 140°C. Theproduct was dissolved in water for reprotonation whiletaining the deuterium atoms in thea position. The resulting13C2-15N-2D2

a-glycine was protected in a standard reactiwith di-tert-butyl-dicarbonate~BOC-anhydride! as reagent@29#. Finally, the carboxylic acid was converted with cyanric fluoride into the desired acyl fluoride: BOC(13C2-15N-2D2

a-glycine! fluoride @30#. The substance dissolved in DMSO-D6 at room temperature shows NMR spetra that weaken with a half-life of about a week, indicativereactions not yet determined. The solution was stable dustorage at a temperature of230°C.

APPENDIX B: NMR PULSE SEQUENCES

For the preparation of the elements of a pseudopure sand the implementation of quantum gates, robust rf pusequences are desirable. Pulse-sequence parametersnegligible experimental errors are the durations of rf puland of delays. In addition, the phases of rf pulses and ofreceiver can be controlled with negligible errors. The mimportant experimental imperfections are rf amplitude errthat result from miscalibrations and from the rf field inhomgeneity created by the rf coils. In addition to the usecompensating schemes, such as supercycles and comppulses@10#, experimental imperfections can be reduceddesigning pulse-sequence elements with a minimum numof rf pulses. For example, pulses to refocus frequency ofterms in homonuclear spin systems with different chemshifts can be eliminated by implementing the experimentsthe multiple-rotating frame in which the precession fr

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quency of each individual spin is 0~see Sec. B 1 below!.More generally, pulses can often be eliminated or replaby phase adjustments with negligible errors~see Sec. B 2!.For the available spectrometer, the experimental pulserameters are summarized in Sec. B 3. The preparation ofelements of the pseudopure stater i is discussed in Sec. B 4

1. Implementation of experiments in themultiple rotating frame

For heteronuclear spins with resonance frequenciesnkand n l in the laboratory frame, the spins are irradiatedresonance and the observed signals are demodulated bdetermined resonance frequencies. If only a single rf chanis available for several homonuclear spins, on-resonancradiation of several homonuclear spins can be achieved uphase modulation of the rf pulses. The reference phaseach pulse applied to spink must be adjusted such thatmatches the desired phase in the corresponding rotaframe ~vide infra!. In addition, the phases of the detectsignals need to be corrected for the relative phases thatbeen acquired by the respective rotating frames duringcourse of the experiment. In our case with the two homnuclear spins Ca ~spin 3! and C8 ~spin 4!, the transmitterfrequency of the carbon rf channel was set to the Ca reso-nance frequency. In order to simplify the combinationdifferent quantum gates, the durations of the pulse sequefor each gate were chosen to be integer multiples ofD51/un32n4u581.75 ms. Hence, the rotating frames aaligned at the end of each gate.

2. Simplifying pulse sequences

Some quantum gates, such as~CNOT)kl , requirez rota-tions of individual spins which can be implemented usicomposite rf pulses@23#. However, these pulses can bavoided if z rotations ~by anglew) are implemented by acorresponding negative rotation of the respective rotatframe of reference. In practice, this results in an additiophase shift~by angle2w) of all following rf pulses that areapplied to this spin and of the receiver phase of this spFurthermore, 180q° pulses~with arbitrary phaseq) are re-quired in some cases to refocus the evolution due toJ cou-plings. In order to undo the rotation caused by these puladditional 180q° or 1802q° pulses are often needed at thbeginning or at the end of these quantum gates. An appriate choice of the position and phaseq of these pulsesoften makes it possible to cancel two pulses from adjacgates~e.g., 180x° and 1802x° ) or to absorb a 180° pulse intthe phase of an adjacent 90° pulse~e.g., a 180x° pulse pre-ceded or followed by a 902x° pulse is equivalent to a singl90x° pulse!.

Even if rf pulses cannot be completely eliminated, taccumulation of small flip angle errors can be avoided bproper choice of pulse phases, which is common pracin the design of modern NMR multiple-pulse sequenc@10#. For example, the so-called MLEV-4 expansion~cycledeveloped by M. Levitt! @16–18# 180x° -1802x° -1802x° -180x°~used here, e.g., for spin 5 decoupling during spin

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APPROACHING FIVE-BIT NMR QUANTUM COMPUTING PHYSICAL REVIEW A62 012310

and spin-4-selective 90° pulses! is preferable to180x° -1802x° -180x° -1802x° or to 180x° -180x° -180x° -180x° .

3. Experimental pulse parameters

Due to their large frequency separation, selective pufor spins 1 (1H), 2 (15N), and 5 (19F) could be implementedby simple square pulses. The durations of 90° pulses w8.85 ms, 41 ms, and 11.75ms, respectively. For spins(13Ca) and 4 (13C8) the following shaped pulses with minmal durations and optimal selectivity were chosen basednumerical simulations and experimental optimizations: 9pulses were implemented as excitation pulse for selecexcitation for biochemical applications~e-SNOB! pulses@24#, not for the usual 270°, but for a 90° rotation withduration of 224ms; selective 180° pulses were implementas Gaussian pulses@25# with a duration of 250ms and atruncation level of 20%. The application of these shape-SNOB and Gaussian pulses on Ca has a nonresonant effec@26# on C8 which corresponds to experimentally determinz rotations ofwe524° andwg5218°, respectively. Con-versely, a shaped e-SNOB pulse applied to C8 leads to azrotation of2we for Ca. In all experiments these phase shiwere taken into account by adjusting the phases of thelowing pulse and the receiver phases~see Fig. 3!. ~Note thatthe phases of the two selective Gaussian 180° pulses apto spin 3 in the periodt45 are not corrected because theabsolute phases are arbitrary; cf. Appendix B 2.! During thespin-3- or spin-4-selective 180° pulses, the evolution duethe strongJ35 andJ45 couplings is automatically refocusedAs this is not the case for spin-3- or spin-4-selective 9pulses, spin 5 was actively decoupled during these pu~see Fig. 3!.

As commercial high-resolution NMR spectrometers acommonly not equipped with multiple receivers, it was npossible to simultaneously detect the signals of different sspecies. Moreover, the application of any given pulsequence required four different pulse programs becauserouting of the rf channels~for the creation of1H, 15N, 13C,19F, and 2D pulses! depends on the detected spin spec(1H, 15N, 13C, or 19F). Due to this technical limitation, eacspin species had to be detected in a separate experimenevery term of the initial density operatorr i . However, thismade it possible to use standard heteronuclear decouptechniques to simplify the detected signals and to signcantly increase the signal-to-noise ratio of the experimen

During spin 1 detection, spins 2 and 3 were decoupwith rf amplitudesn rf5gBrf /(2p) of 0.6 kHz and 0.4 kHz,respectively. During spin 2 detection, spins 1 and 3 wdecoupled with rf amplitudes of 2.3 kHz and 0.4 kHz, rspectively. During spin 3 detection, spins 1, 2, and 5 wdecoupled with rf amplitudes of 2.3 kHz, 0.6 kHz, and 2kHz, respectively, and during spin 4 detection, spin 5 wdecoupled with a rf amplitude of 2.0 kHz. In all these casthe wideband alternating phase low-power technique for zresidual splitting~WALTZ-16! decoupling sequence@16–18#was used. In principle, theJ34 coupling could also be effectively eliminated during detection of spin 3 or spin 4 usitime-shared decoupling. However, this was not possible w

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our experimental setup because more than five separachannels would have been required. From the resulting dblets ~with splitting J34) an apparent singlet was createdmerging the two doublet components@27#. During spin 5detection, spins 3 and 4 were simultaneously decoupleding a double-selective G3-MLEV sequence~G3 is a shapedinversion pulse consisting of three Gaussian pulses! @16–18#with a rf amplitude of 6 kHz. In addition, during all experments deuterium decoupling was applied using a WALTZ-sequence withn rf50.5 kHz. In order to approximate a constant sample temperature of about 27°C in spite of the ational sample heating effected by the decoupling sequen16 dummy scans were used prior to signal acquisitionspins 1 and 5, whereas 4 dummy scans were used priosignal acquisition of spins 2, 3, and 4. Nevertheless, the lwidths of the experimental signals shown in Fig. 4 weslightly increased by residual sample heating effects andperfections of the decoupling sequences.

4. Pulse sequences for the preparation of the terms ofr i

In order to improve the sensitivity of the experiments ato filter out signals from impurities in the sample, individuCartesian product operator terms ofr i were created usingsequential inensitive nuclei enhancement by polarizattransfer~INEPT! transfer steps@22# starting from 1H mag-netization, corresponding to the operatorI 1z . The termI 1zwas prepared from the thermal equilibrium density operaby applying spin 2, 3, 4, and 5 selective 90° pulses followby a pulsed field gradient of the static magnetic field. AX-filter element@28# was used to select1H spins that arecoupled to 15N. For the preparation of other terms ofr i~starting fromI 1z), the phasefa of the first 90° pulse ap-plied to spin 1 was subject to a two-step phase cycle.addition, the phasefb of the 90° pulses for the implementation ofU0 @see Eq.~4!# was also subject to an independephase cycle. Overall, this resulted in a four-step phase cwith the pulse phases fa5$0°,180°,0°,180°%, fb5$90°,90°,270°,270°% and the relative receiver phasef rec5$0°,180°,0°,180°%.

APPENDIX C: DENSITY OPERATOR TERMS

For all 31 Cartesian product operator terms inr i @Eq. ~8!#,the corresponding terms inr0 and rb can be derived in a

straightforward way. The transformationr0 ——→U( f b)

rb of theunitary operator corresponding to the balanced functionf b iscomposed of four consecutive unitary transformations cosponding to~CNOT)kl quantum gates:

r0 ——→~CNOT)12

——→~CNOT)23

——→~CNOT)34

——→~CNOT)45

rb .

The transformations of the individual~CNOT)kl gates can bederived using the following rules@2#:

I kx ——→~CNOT)kl

2I kxI lx ; 2I kxI lx ——→~CNOT)kl

I kx ; I lx ——→~CNOT)kl

I lx .

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MARX, FAHMY, MYERS, BERMEL, AND GLASER PHYSICAL REVIEW A62 012310

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