Realization of quantum process tomography in NMR

PHYSICAL REVIEW A, VOLUME 64, 012314

Realization of quantum process tomography in NMR

Andrew M. Childs,1,2,3 Isaac L. Chuang,1 and Debbie W. Leung1,4,5

1IBM Almaden Research Center, San Jose, California 951202Physics Department, California Institute of Technology, Pasadena, California 91125

3Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 021394 Quantum Entanglement Project, ICORP, JST, Edward Ginzton Laboratory, Stanford University, Stanford, California 94305

5IBM T. J. Watson Research Center, Yorktown Heights, New York 10598~Received 6 December 2000; published 13 June 2001!

Quantum process tomography is a procedure by which the unknown dynamical evolution of an openquantum system can be fully experimentally characterized. We demonstrate explicitly how this procedure canbe implemented with a nuclear magnetic resonance quantum computer. This allows us to measure the fidelityof a controlled-NOT logic gate and to experimentally investigate the error model for our computer. Based on thelatter analysis, we test an important assumption underlying many models of quantum error correction, theindependence of errors on different qubits.

DOI: 10.1103/PhysRevA.64.012314 PACS number~s!: 03.67.2a, 06.20.Dk, 76.60.2k

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I. INTRODUCTION

Experimental characterization of the dynamical behavof open quantum systems has traditionally revolved arosemiclassical concepts such as coupling strengths, relaxrates, and phase coherence times@1,2#. However, quantuminformation theory tells us that so few parameters bareven begin to represent the full dynamics that quantumtems are capable of; for example, the evolution betweenfixed times of a two-level quantum system~a qubit!, coupledto an arbitrary reservoir, is described by 12 real parame@3#. For a two-qubit system, this number grows to 240, ain general, forn qubits it is 16n24n.

Of course, not all of these parameters are relevantevery physical process, and understanding the physicsboil down this huge parameter space to just a few impornumbers. On the other hand, when the physical originsquantum process arenot understood, or in question, it iinvaluable to know that all these parameters can, in princibe experimentally measured—albeit with exponential eff~in n)—using a procedure known asquantum process tomography. This method is a direct quantum extension of tclassical concept of ‘‘system identification,’’ which enablcontrol of dynamical systems and provides powerful msurement techniques for characterizing unknown systeFor closed systems, process tomography is the analog otermining the truth table of an unknown function; quantumechanically, this means the unitary transform involved iquantum computation.

The procedure for quantum process tomography has bdescribed in detail in the literature@3–5#. It relies upon theability to prepare a complete set of quantum statesr j asinput to the unknown processEt , and the ability to measurethe density matrices of the output quantum states fromprocessEt(r). As a result, one obtains a complete ‘‘blacbox’’ characterization ofEt . By systematically repeating thiprocedure for increasing process timet, one can also obtain acomplete master equation description of the process@6#.

This procedure seems to provide an ideal way to chaterize and test quantum gates on a small number of qubit

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for example, those realized using nuclear magneresonance~NMR! techniques@7,8#. In fact, NMR quantumprocess tomography has been performed on a single q@9#. However, to date, this has not been extensively practibecause of two problems: first, it is desirable to expliciinclude the maximally mixed state as one of ther j ’s, so as todirectly subtract its contribution, but such a state cannotprepared by unitary actions alone. Second, one can omeasure traceless observables, and thus one cannot othe entire output density matrix of a process.

Here, we show explicitly how these hurdles can be ovcome, and demonstrate the use of these methods in adding two outstanding issues in quantum computation: thedelity of a quantum logic gate, the controlled-NOT gate, andthe validity of the independent error model, which is wideassumed~though not strictly necessary! for quantum errorcorrection and fault-tolerant computation.

The paper is divided into two major sections, which dscribe the theory and our experiment. We begin with aview of quantum process tomography~QPT!; then we ex-plain our extensions to the basic procedure that enable Qwith NMR, and present theoretical models for decoherein NMR. We then describe our experiment and proceduand our two main experimental results.

II. THEORY

A. Quantum-process tomography

Quantum process tomography, as introduced by Chuand Nielsen@3#, can be summarized as follows. A generquantum operation on a quantum stater is a superoperatorE,a linear, trace-preserving, completely positive map@10–12#.We consider only the case whereE(r) has the same dimension asr. A common form forE is the operator-sum representation,

E~r!5(k

AkrAk† , ~1!

where(kAk†Ak5I . One can easily demonstrate the transf

mation to a fixed-basis expansion of the quantum opera

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CHILDS, CHUANG, AND LEUNG PHYSICAL REVIEW A64 012314

such that the information about the process lies in a secoefficientsxmn instead of a set of operators. In this form,Emaps an initial density matrixr according to

E~r!5(m,n

xmnAmrAn† . ~2!

In this expression, theAm are a basis for operators on thspace of density matrices, andx is a positive Hermitian ma-trix with elementsxmn . In the following, we will omit ex-plicit summations and adopt the convention that repeaindices are summed over. Because of the restriction thEmust preserve the trace ofr, x containsN42N2 independentparameters for anN-dimensional system~for n spins, N52n).

Let the N2 matricesr j be a basis for density matriceApplying E to this basis gives

E~r j !5l jkrk . ~3!

Using quantum state tomography@13#, one can experimentally determinel jk , which fully specifiesE. To determinexfrom l, we defineb by

Amr jAn†5b jk

mnrk , ~4!

which is fully specified by the choice ofAm and r j . Thenone can show that

b jkmnxmn5l jk . ~5!

We may think ofb as a matrix andl andx as vectors, withmn a composite column index andjk a composite row in-dex; thenbxW 5lW . Using the pseudoinversek of b ~alsocalled the Moore-Penrose generalized inverse!, which maybe computed independent of measurement results, we h

xW 5klW . ~6!

B. QPT in NMR

Implementing QPT in high-temperature NMR presetwo main obstacles. First, the part of the system that canmanipulated by unitary operations is but a small deviatfrom a maximally mixed state. Second, one can only msure traceless observables, and the scaling of these obables with respect to the traceful part of the system isimmediately known. Fortunately, both of these hurdles cbe overcome.

The Hamiltonian for solution NMR is well approximateby

H52\v j

Zj

21hJjk

ZjZk

4, ~7!

where

Zj5S 1 0

0 21D ~8!

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is the Pauli operator for thej th spin along the direction of themagnetic field,v j is its Larmor frequency, and the sign othe first term is chosen so that the ground state is spinThe second term represents a small scalar coupling betwspins known as theJ coupling.

The initial state of an NMR quantum computer is ththermal equilibrium state with density matrix

r`5e2H/kBT

tr~e2H/kBT!5cI1D` , ~9!

wherec522n for a system ofn spins@so that tr(r`)51# andD` is traceless. In general, we refer to the traceless parany density matrixr as the deviation density matrix. If weneglect the small coupling between spins and consider htemperature with respect to the Larmor frequencies, the elibrium density matrix is approximately

r`'cS I 1\v jZj

2kBT D . ~10!

By performing unitary operations on the system, we mrotate thev jZj deviation, allowing us to prepare a linearindependent set of inputsr j . Because all of the interestininformation will lie in this deviation, we would like to beable to prepare an input with no deviation~i.e., the maxi-mally mixed state!, so that we may directly subtract the cotribution of the identity. Such an input clearly cannot bprepared by unitary means. However, the maximally mixstatecan be prepared from the thermal state by applyingp/2 pulse followed by a uniformly spatially varying rf pulseDifferent parts of the sample experience different amountsrotation, so that the ensemble is effectively dephased tomaximally mixed state.

The second difficulty can be handled by calibrating mesurements to the known initial state of the system. By sttomography, we can measureD5dD` , where the constantdis arbitrary — it depends in detail on the gain of the ampfiers, the efficiency of the rf coils, etc. But the initial deviation density matrix is known to bec(\v jZj /2kBT). Hencewe may calibrate our apparatus by determining the valud

that makes the measuredD/d closest to the theoreticalD` .In practice, we effect this calibration by normalizing all mesurements~integrals of spectral peaks! for spin j to a refer-ence measurement taken on that spin~a p/2 pulse applied tothe thermal state! multiplied by c(\v j /2kBT).

C. Decoherence primitives

Decoherence of a single NMR spin in solution is genally characterized by two well-known processes: amplitudamping and phase damping. Here, we present thesecesses, as well as generalizations to multiple spins. To fatate the description of multiple simultaneous processes,describe amplitude damping and phase damping as sgroups in terms of their generators.

Assuming that a quantum operation can be describednorm continuous one-parameter semigroup, given a supeeratorEt satisfying the stationarity and Markovity conditio

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REALIZATION OF QUANTUM PROCESS TOMOGRAPHY IN NMR PHYSICAL REVIEW A64 012314

EsEt5Es1t @14#, its generatorZ is defined as@15#

Z~r!5 limt→0

Et~r!2r

t. ~11!

The generators have the advantage that they easily desthe simultaneous action of multiple processes. In particuwe have the Lie product formula@15#

E t(112)5 lim

n→`

~E t/n(1)E t/n

(2)!n, ~12!

where

Z (112)5Z (1)1Z (2), ~13!

with obvious generalization to more than two processes.Furthermore, we may return to the superoperator form

exponentiating the generator@15#:

E5 limn→`

S 12t

nZD 2n

~14!

5eZt. ~15!

This exponentiation may be defined by its Taylor seriesthe case of interest where is bounded. To implement exnentiation in practice, it is useful to work with a manifestlinear representation, i.e., that of Eq.~3!. In this case, a superoperator and its generator can be represented as mathat have the property that composition of superoperacorresponds to matrix multiplication. Then the procedureEq. ~14! corresponds to matrix exponentiation.

We now consider the single-qubit versions of phaseamplitude damping. For further discussion of these pcesses, including operator-sum representations, see Ref.@16#.We write the density matrix of a qubit as

r5S r00 r01

r10 r11D . ~16!

Phase damping can be thought of as a consequence ofdom phase kicks. It acts on a qubit as

E tPD~r!5S r00 e2gtr01

e2gtr10 r11D , ~17!

for some damping rateg; i.e., it has a generator that acts

Z PD~r!52gS 0 r01

r10 0 D . ~18!

Ordering the matrix elements ofr in the vector(r00,r01,r10,r11)

T @which effectively defines the basis usein Eq. ~3!#, we have

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Z PD5S 0 0 0 0

0 2g 0 0

0 0 2g 0

0 0 0 0

D . ~19!

Generalized amplitude damping is a process whereby thebit may exchange energy with a reservoir at some fixed teperature. It acts on a qubit as

E tGAD~r!5S k1r001k2r11 e2Gt/2r01

e2Gt/2r10 k3r001k4r11D , ~20!

where k2[(12n)(12e2Gt), k3[n(12e2Gt), k1[12k3,andk4[12k2, with G a damping rate andn a temperatureparameter. Its generator acts as

ZGAD52GS n 0 0 n21

0 12 0 0

0 0 12 0

2n 0 0 12n

D . ~21!

Computing the fixed point of this process, we find that wmay interpret it as occurring at a temperature

kBT5DE

ln12n

n

, ~22!

where DE is the splitting between the ground and excitstates of the system.

Heuristically, we expect phase damping to describe theT2~dephasing! process of NMR. Similarly, generalized ampltude damping should describe theT1 process of NMR,whereby a spin relaxes to align with the magnetic field. Nfrom the geometry of the Bloch sphere that amplitude daming necessarily includes loss of phase information, so it acontributes to theT2 process. However, in typical systemT1 is much longer thanT2, so the contribution of amplitudedamping to dephasing is small.

The simplest extension of these processes to a multispin system is to allow the individual processes to act inpendently on each qubit. However, this is far from the mgeneral scenario: the decoherence might be correlated.though we have been unable to find a suitable correlageneralization of amplitude damping, we have incorporatesimple model of correlated phase damping due to Zhouour analysis@17#. In this model, each spin experiencesrandom phase shift with zero mean. The amount of dampand the extent of correlation are defined by the covariamatrix of the phase shifts. By explicit calculation, one measily find that this leads to a generator of the form

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Z CPD5diag@0,2g2 ,2g1 ,2~g11g21g3!,

2g2,0,2~g11g22g3!,2g1 ,

2g1 ,2~g11g22g3!,0,2g2 ,

2~g11g21g3!,2g1 ,2g2,0], ~23!

whereg1 andg2 may be interpreted as rates for independphase damping on spins 1 and 2, andg3 may be interpretedas a rate for correlated damping. Indeed, in the caseg350,this model reproduces independent phase damping.

Our completed model has the generator

Z5Z J1Z CPD1Z 1GAD1Z 2

GAD, ~24!

where a subscript indicates which spin is acted on andZ J isthe generator corresponding to the Hamiltonian

HJ5hJZ1Z2

4. ~25!

In this Hamiltonian, the Zeeman terms are dropped from~7! because we work in a frame rotating at the Larmor fquencies of the two spins. To produce the model superoptor at any given time, we simply exponentiate Eq.~24!.

III. EXPERIMENT

We have implemented QPT on a system of two spinsan NMR apparatus. The experiments were performed atIBM Almaden Research Center using an Oxford Instrumewide-bore magnet and a 500-MHz Varian Unity Inova sptrometer with a Nalorac triple resonance HFX probe. Tsample was approximately 0.5 ml of 200 millimolar isotopcally labeled chloroform (13CHCl3) in d6-acetone, where thtwo qubits were the proton~first spin! and carbon~secondspin!. This volume was chosen to make the sample fashort, as this gave the most effective gradient pulse chateristics.

A. Results: Gate fidelities

One application of quantum-process tomography is toagnose the accuracy of a quantum computation. Whenapplies a series of gates to perform the unitary operationU,the computer will actually perform a superoperatorE, whichone hopes is close toU. That closeness can be capturedmany distance measures — for example, the minimum gfidelity, defined as

F5minuc&

Šc zU†E~ uc&^cu!U zc‹. ~26!

F represents the smallest possible overlap between aacted on byU and the same state acted on byE. Note thatminimizing over pure states is sufficient because the fideis convex and because an arbitrary density matrix canwritten as a convex sum of pure states.

Using QPT, we measured the minimum gate fidelity focontrolled-NOT operation, a well-known member of universgate sets, with

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UCNOT5S 1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

D ~27!

The NMR pulse sequence chosen to implement this gate

Y1X1Y1Y2dS 1

2JDY2X2 , ~28!

where time goes right to left,Xj denotes ap/2 pulse in thexdirection for spinj, a bar denotes an inverse pulse,d( ) de-notes a time evolution during which no pulses are appliandJ5215 Hz is the coupling constant for the scalarJ cou-pling between the spins. Procedures similar to those useRef. @18# were adopted to implement this pulse sequenThe pulse lengths were approximately 10220 ms, suffi-ciently fast that the effect ofJ coupling could be neglectedduring a pulse. Using quantum-process tomography, weterminedx for this process, effectively determiningE. Thisresult is shown in Fig. 1.

Given x, we may calculate gate fidelities. For thcontrolled-NOT gate, numerically minimizing over inpustates givesFCNOT50.8060.04.

This figure may be thought of as a rough benchmarkthe quality of gates that can be implemented using currNMR quantum computers. However, several caveats shobe addressed. Foremost, the preparation and readoutperformed to do the process tomography contribute to mof the error. Performing process tomography on a null coputation (U5I ), we find FI50.9060.03. The primarysources of error are most likely imperfect state preparaand imperfect state tomography due to imperfect pulse cbration and inhomogeneity of the rf fields@19#.

Also, in a long computation, experimental results suggthat there may be a significant cancellation of errors@18#.Typically, the major contribution to the error introduced ban individual gate will be largely due to systematic errorather than fundamentally irreversible decoherence. Ifpulse sequence exhibits some degree of symmetry, thesetematic errors may at least partially cancel. Thus the fide

FIG. 1. x matrices for the controlled-NOT gate. The matrix onthe left was experimentally measured by QPT, and that on the ris theoretical. Only the magnitudes of the matrix elementsshown; the phases also corresponded well between theory anperiment. The matrix element in the far left corner connectsII ↔II ,and the matrix elements are ordered asII ,XI, . . . ,YZ,ZZ.

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of a long sequence of gates cannot be deduced fromindividual gate fidelities, though one would certainly expeit to be subadditive.

Of course, the minimum gate fidelity is a pessimistic stadard; it may be more reasonable to consider theaveragegatefidelity. To numerically calculate the average fidelity, wmust be able to sample uniformly from quantum statescording to an appropriate probability measure. Although tproblem is subtle for the case of mixed states, therestraightforward choice for pure states: we take the unittransform of a fixed state, where the unitary operator is csen according to the Haar measure~the unique invariantmeasure on a Lie group!. Using the procedure describedRef. @20#, we calculated average fidelities with 105 randomlychosen unitaries, givingFCNOT

avg 50.955 and FIavg50.960.

Thus, the average fidelity is significantly better than tworst case; the error rate 12F improves by about one ordeof magnitude.

B. Results: Decoherence characterization

To characterize the decoherence occurring in the chloform system, we used QPT to measurex over the variousrelevant time scales. For an NMR spin, the relevant scare 1/(2J), T1, andT2 @21#. We thus sampled at times give

by ( 12 ) j1/(2J) for integersj P@0,4#, ( 1

2 ) j T1 for j P@21,4#,

and (23 ) j T2 for j P@0,9#, where J5215 Hz is the known

value of theJ coupling,T1520 s is a time scale on the ordeof T1, and T250.5 s is an intermediate time scale on torder ofT2.

By executing process tomography, we are able to deminex as a function of time. However,x is a large, complexcollection of numbers that cannot be easily interpreted.better understand the results, we fit the data to a modelcessxm , which hopefully provides a reasonable descriptiof the relevant physics. Specifically, we use the model tresults from exponentiating Eq.~24!.

For eachx, we determined the closest fit to our modelnumerically minimizing tr@(xm2x)†(xm2x)#, wherexm isa model superoperator derived from Eq.~24!. This figure ofmerit was on the order of 5% for most fits. The experimetally measuredx matrices as well as the corresponding fixm are shown in Fig. 2 for three delay times.

Fitting the various rate parameters ofxm as a function oftime allows us to characterize the process. These fitsshown in Figs. 3–5. The error bars in these plots are cputed numerically, and are derived solely from the statisof the measurements and the fitting procedure.

Our results are summarized in Table I. In addition to thedata, we found the correlated phase damping rateg3 to bezero to within statistical precision. Unfortunately, insufcient precision was available to determine the amplitudamping temperature parametersn j . This is not a significantdifficulty, as they are certain to be near 1/2 for higtemperature systems. The data are consistent with this v

For comparison, the relevant decoherence paramewere measured by standard techniques. By the inversrecovery technique, we foundT1518.5 s for the proton and

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T1521.1 s for carbon. These values agree roughly withtime scalesG j

21 for amplitude damping. Using the CarrPurcell-Meiboom-Gill sequence, we foundT254.7 s for theproton andT250.26 s for carbon. In an experiment thuses refocusing,T2 is the relevant time scale for phase cherence. However, because no refocusing was perforduring the process tomography, the time scaleT2* , includingthe effect of magnetic field inhomogenity, is more relevantour analysis. By fitting the free induction decay, we finT2* 50.86 s for the proton andT2* 50.20 s for carbon, val-ues that agree more closely with the time scalesg j

21 mea-sured by the QPT.

As previously mentioned, the QPT preparation stepnon-negligible for the controlled-NOT gate fidelity experi-ment. However, this is not the case for the decoherence msurements: the time scales are long compared to the timperform the QPT preparation, so small errors in the prepation are unimportant. For the experiments at short times

FIG. 2. x matrices for decoherence at various times. Matricesthe left are experimentally measured using QPT, and those onright are fits to Eq.~24!. Only the magnitudes of the matrix elements are shown, but their phases were also described well bymodel. From top to bottom,t50.065 s, 0.5 s, 20 s. Att50.065 s, the time was chosen to be an integer numberJ-coupling periods, which explains the lack ofII ↔ZZ andZZ↔ZZ terms in the corners. Att50.5 s, note that theIZ↔ZIterms arise from a combination ofJ coupling and independentphase damping, not from correlated phase damping. Att520 s,note that the nonzero antidiagonal terms found in the experimennot appear in the fit even though such terms may arise in ourplitude damping model. Due to the details of the fitting proceduthe optimal fit matches large elements closely but does not capthe detail of the small antidiagonal.

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measure the parameter describing unitary evolution (J), thepreparation is non-negligible, but the operation is simenough that we are still able to determineJ to within twostandard deviations.

IV. CONCLUSIONS

We have demonstrated the implementation of quanprocess tomography to characterize the dynamics of a tqubit NMR quantum computer. Such techniques shoprove useful in the future for diagnosing quantum informtion processing devices. However, we should stress thatto the exponential size ofx, QPT can only be used to chaacterize the dynamics of sufficiently small systems. For lasystems, one may only be able to perform QPT on a smpart of the total system, assuming independence from the

FIG. 3. Determination of the strength of the scalarJ couplingbetween spins.

FIG. 4. Determination of the phase damping ratesg j . Thecircles representj 51 ~proton!, and the squares representj 52 ~car-bon!. g3 is fitted to zero to within experimental uncertainty.

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of the system—an assumption which can, in turn, be checusing QPT.

Our measurements of gate fidelities underscore the dculty of implementing highly accurate logic gates in NMRThese fidelity measurements suggest a per-gate error rathe order of 1021, in agreement with previous implementations of quantum algorithms@18#. It has been suggested therror rates of 1022 can be achieved in solution NMR@22#.Indeed, if the previously discussed cancellation of errors ilong sequence of gates leads to aT2-limited computation~asobserved in@18#!, then an optimistic estimate ofT251 sandJ5100 Hz leads to an approximate error rate of 1022.Nevertheless, even this falls far short of the requirementsfault-tolerant quantum computing@23#: the least stringent estimates indicate a threshold for fault tolerance of about 1024

@24#. Clearly, the development of fault-tolerant NMR quatum computers will require substantially modified tecniques. One might construct an alternative model for fatolerance, perhaps based on topological properties@25#.However, with a conventional approach to fault toleranwe will either need significant improvement of gate fideliover what is currently achievable, specialization of fault terant protocols to the specific errors that occur in NMR,both.

In turn, analyses of the threshold for fault-tolerant quatum computation typically involve an assumption of indpendent errors. Although some special correlated errors

FIG. 5. Determination of the amplitude damping ratesG j . Thecircles representj 51 ~proton!, and the squares representj 52 ~car-bon!.

TABLE I. Dynamical time scales for13CHCl3, as measured byQPT. Recall that the proton is spin 1 and the13C nucleus is spin 2.g3 is fitted to zero to within experimental uncertainty.

J 22063 Hzg1

21 0.7360.03 sg2

21 0.2160.01 sG1

21 12.760.3 sG2

21 22.460.7 s

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easier to correct than independent errors, specialized qtum codes exploiting full knowledge of the error processare necessary. Correlations that have complicated strucgenerally increase the requirements of error correction,the exact nature of these requirements has yet to be analyThus, for the experimental implementation of fault-toleracomputers, it is important to understand the error modedetail, and in particular, the extent to which the indepedence assumption holds. The results presented in Sec.show that this modelis well understood for the13CHCl3NMR system, and furthermore, that the errors are uncolated. The data are fit well by a combination of phase daming and independent generalized amplitude damping, wthe phase damping essentially uncorrelated. Thus theresimple computational systems in which an uncorrelated emodel is reasonable.

However, note that there are systems that do exhibit crelated decoherence, namely those that exhibit ‘‘cross re

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ation’’ ~in the terminology of NMR!. For example, thesimple two-spin proton-carbon system provided by sodiformate is known to possess cross relaxation@26#. Investiga-tion of such systems by QPT might be interesting not ofor the purpose of quantum computing, but also for studycross relaxation in the context of conventional NMR.

ACKNOWLEDGMENTS

We thank Constantino Yannoni for sample preparatand Lieven Vandersypen and Matthias Steffen for expmental assistance. We also thank all of the above as weMark Sherwood and Xinlan Zhou for numerous enlightenidiscussions. During final preparation of the manuscrA.M.C. was supported by the Fannie and John Hertz Fodation. D.W.L. was partially supported by IBM and NTwhile at Stanford, and by the NSA and ARDA under ARContract No. DAAG55-98-C-0041.

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tions of Quantum Theory, Lecture Notes in Physics Vol. 19~Springer-Verlag, Berlin, 1983!.

@13# State tomography for infinite-dimensional systems was pposed in K. Vogel and H. Risken, Phys. Rev. A40, 2847~1989!; see@16# for a review of the simpler finite-dimensionacase.

@14# We use multiplicative notation to stand for compositionsuperoperators. In other words,E (1)E (2)5E (1)°E (2).

@15# E. B. Davies,Quantum Theory of Open Systems~AcademicPress, London, 1976!; One-Parameter Semigroups~AcademicPress, London, 1980!.

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@16# M. A. Nielsen and I. L. Chuang,Quantum Computation andQuantum Information~Cambridge University Press, Cambridge, 2000!.

@17# X. Zhou ~private communication!.@18# L.M.K. Vandersypen, M. Steffen, M.H. Sherwood, C.S. Ya

noni, G. Breyta, and I.L. Chuang, Appl. Phys. Lett.76, 646~2000!.

@19# In an earlier experiment, we achieved the fidelitiesFCNOT

50.9160.04, FI50.9360.04. Unfortunately, we were unablto reproduce these results.

@20# K. Zyczkowski and M. Kus´, J. Phys. A27, 4235 ~1994!; M.Pozniak, K. Zyczkowski, and M. Kus´, ibid. 31, 1059~1998!.

@21# The time scalesv j21 are not relevant because we work in

rotating frame, using a pair of local oscillators tuned to mathe well-known Larmor frequencies.

@22# D.G. Cory, R. Laflamme, E. Knill, L. Viola, T.F. Havel, NBoulant, G. Boutis, E. Fortunato, S. Lloyd, R. Martinez,Negrevergne, M. Pravia, Y. Sharf, G. Teklemariam, Y.Weinstein, and W.H. Zurek, e-print quant-ph/0004104~2000!.

@23# P. W. Shor, inProceedings, 37th Annual Symposium on Fudamentals of Computer Science~IEEE Press, Los Alamitos,1996!, p. 56.

@24# D. Gottesman, Ph.D. thesis, California Institute of Technolo~1997!; J. Preskill, Proc. R. Soc. London, Ser. A454, 385~1998!.

@25# A. Yu. Kitaev, e-print quant-ph/9707021.@26# C.L. Mayne, D.W. Alderman, and D.M. Grant, J. Chem. Ph

63, 2514~1975!.

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Realization of quantum process tomography in NMR