Toward a table-top quantum computer

by Y. MaguireE.N. Gershenfeld

Boyden

To investigate the scaling issues for quantumcomputation using thermalized ensembles ofspins, we are building a rudimentary quantumcomputer using nuclear magnetic resonance(NMR). We discuss the experimental issuesfacing the scalability of this technique includingthe number of qubits, issues of classical versusquantum behavior, and polarization. Due to theweak signals measured in NMR, the mostsignificant challenge to practical devices ispolarization enhancement. We present thearchitecture for a simple NMR spectrometer thatis table-top, low-cost, and software-radio drivenusing commodity electronics and unconventionalpermanent magnet designs. Optimizationsspecific to using this machine as a computationaldevice, such as probe design, compilation, andon-line control, are discussed.

W ith the quest for faster microcomputer chipsin the semiconductor industry, exponential in-

creases in speed must come at the exponentially in-creasing cost of some other resource. Advances inthe semiconductor industry have capitalized on tworesources: space, by shrinking the size of integratedcircuitry in semiconductor fabrication; and money,by the ever increasing investment in fabricationplants. It is likely that we will, in the not too distantfuture, reach fundamental physical limits: transistorswill be atomic in scale and memory will consist ofonly a handful of electrons. Time does not offer muchhelp as a resource, because light takes 100 picosec-onds (ps) to travel the 3 centimeters (cm) of a typ-ical chip. Exponentially large physical space or en-ergy are not realistically available resources. Peoplehave proposed DNA computers1 as a type of massiveparallelism to solve computational problems. This

solution does not scale either, because a computa-tionally hard problem requires exponential amountsof DNA, which can quickly require samples more mas-sive than the earth itself. 1 The dream of quantumcomputing is that quantum mechanics provides anuntapped, scalable exponential resource. Quantumcomputers can, in theory, factor numbers in expo-nentially fewer steps than classical computers (thuscracking modern cryptographic protocols), simulatequantum systems with exponentially fewer resourcesthan now possible, and search databases in 2(=N)queries rather than 2(N) on an ordinary machine.

The power of quantum computing begins with a pro-found observation. Whereas a classical bit can be inone of two states, 0 or 1, a quantum bit, or “qubit”(e.g., a spin-1/2 particle) can be in a weighted su-perposition of both states. For example,

au0& 1 bu1&

where a and b are complex numbers, and uau 2 1ubu 2 5 1. When measured with a readout operator,the qubit appears to collapse to state u0& with prob-ability uau 2 , and to state u1& with probability ubu 2 . Thestate of two qubits can be written as

uC& 5 Oi

ciui& 5 au00& 1 bu01& 1 cu10& 1 du11&

rCopyright 2000 by International Business Machines Corpora-tion. Copying in printed form for private use is permitted with-out payment of royalty provided that (1) each reproduction is donewithout alteration and (2) the Journal reference and IBM copy-right notice are included on the first page. The title and abstract,but no other portions, of this paper may be copied or distributedroyalty free without further permission by computer-based andother information-service systems. Permission to republish anyother portion of this paper must be obtained from the Editor.

IBM SYSTEMS JOURNAL, VOL 39, NOS 3&4, 2000 0018-8670/00/$5.00 © 2000 IBM MAGUIRE, BOYDEN, AND GERSHENFELD 823

Toward a table-topquantum computer

where uau 2 1 ubu 2 1 ucu 2 1 udu 2 5 1, and the prob-ability of measuring the amplitude of each state isgiven by the magnitude of its squared coefficient. Ingeneral, the state of n qubits is specified by 2 n11 21 real numbers—an exponential amount of informa-tion, relative to the number of physical particles re-quired. Most of these states are entangled—to cre-ate them requires some kind of interaction betweenthe qubits, and the qubits cannot be treated entirelyindependently from one another.

Designing quantum algorithms is made difficult bythe fact that it is impossible to extract the entire ex-ponential amount of information encoded in thestate of a quantum system: only a polynomial amountof information can be read out, since a measurementcollapses the exponential enormity of the state intoa handful of classical values, with probabilities of oc-currence given by the amplitudes as described in theprevious paragraph. Furthermore, the only compu-tational primitives allowed are unitary operators(which evolve the state of a system under Hamilto-nian dynamics) that are reversible and therefore con-serve probability. Nevertheless, a collection of won-derful algorithms have been discovered. In 1994,Peter Shor2 showed quantum computers could fac-tor large numbers into prime factors in polynomialsteps, compared to exponential steps3 on classicalcomputers. Prime factorization is an essential partof modern public key cryptographic protocols, par-amount to privacy in the electronic world. In 1997,Lov Grover4 demonstrated a search algorithm thatcould search an unsorted list for a single elementquadratically faster on a quantum computer than ona classical computer.

Introduction to NMR

Liquid state nuclear magnetic resonance (NMR) isthe manipulation of bulk degrees of freedom to ob-tain information about the underlying nuclear spinsin an applied magnetic field. In order to understandhow to build a quantum computer using NMR, webegin with the general requirements for a quantumcomputer.5

1. It starts in a known state (a pure state such as theground state).

2. It is able to efficiently represent any unitary map-ping from one state to another from a basic setof elementary unitary mappings.

3. It is able to concatenate these mappings to forma circuit, that is maintain quantum states longenough to represent any unitary mapping.

4. It performs projective measurement in order toread out the computation results.

To implement any of the above requires that we un-derstand the dynamics of nuclear spins when theyact as an ensemble. Quantum statistical mechanicsallows us to compute the equilibrium thermal stateof the liquid. Moreover, if we know the dynamics ofa single molecule, we can easily calculate the dynam-ics of the entire liquid. To begin, the dynamics ofquantum systems are described by operators thatmap one quantum state to another, uC&3 uF&. Forthe nuclear spins of our quantum computer, oper-ators are finite dimensional matrices. The Hamilto-nian * is a special operator that describes the com-plete dynamics of a molecule under time evolutionand also determines the energy eigenstates of thesystem. The Hamiltonian of a rapidly tumbling mol-ecule in a liquid containing n spin-1/2 nuclei with astrong magnetic flux density BY 5 B0z is given by

* 5 Oi51

n

\gi~B0 1 dBi!Iiz 1 Oi,j

2p\Jij Iiz Ijz

where I iz represents the z-axis spin operator for nu-clei i, g i is the gyromagnetic ratio of spin i, dBi de-scribes a local magnetic flux density adjustment foreach spin i due to the local chemical environment,and Jij describes the coupling, in Hertz, between spinsi and j. We assume for the discussion of the dynam-ics below that we are in the weak coupling limit whereug idBi 2 g jdBj u .. J ij . In equilibrium, the Ham-iltonian dictates that spin i precesses quickly aboutthe z-axis at a frequency g iB0/ 2p adjusted by thelocal term, gdBi / 2p, the so-called chemical shift.This precession frequency is called the Larmor fre-quency. This fast oscillation (usually at several MHz)is slowly modulated by the J-coupling (which usu-ally proceeds at a few Hz). These precession statesare the equilibrium states of the system, called theeigenstates. With the energy eigenstates, we are nowready to begin to understand the spin dynamics.Given a (2 n21)-dimensional basis ua& of single-mol-ecule eigenstates, we can write the state of a ther-mal liquid ensemble of molecules as a density ma-trix r, where the matrix element rab is given by

rab 5 cac*b

where the average is taken over the thermal ensem-ble, and cac*a is the probability of being in the stateua&. Given a time-evolution operator

MAGUIRE, BOYDEN, AND GERSHENFELD IBM SYSTEMS JOURNAL, VOL 39, NOS 3&4, 2000824

U~t1, t0! 5 e ~i*~t12t0!!/\

where * is the Hamiltonian of the quantum systemduring the period from time t0 to time t1 , the den-sity matrix evolves as

r~t1! 5 Ur~t0!U 21

To know the dynamics of the system for all futuretimes, we just need to know the initial state of thesystem. Quantum statistical mechanics provides theanswer with the thermalized (maximum entropy)density matrix

r~0! 5 e 2*/kBT/] 51] S1 1 O

i51

n \gi BkT

Iiz 1 . . .DThe partition function ] is the normalization factorsuch that tr(r(0)) 5 1. Because the Zeeman en-ergies g iB0 are usually about 1025 times smaller com-pared to kBT, the first two terms of the Taylor ex-pansion of the exponential serve as a very goodapproximation, with the second term being about1025 times the magnitude of the first, constant term.If our measurement apparatus is a coil wrappedaround the sample detecting the classical magneti-zation signal, it turns out that the constant term isnot observable, so we ignore it and concentrate onthe remainder (called the deviation density matrix).The only time-evolution operators of interest in thissimple model of NMR (where we neglect relaxationmechanisms and higher order spin-spin interactions),will be the single-qubit rotations (u 5 g iB1 t),

Xi~u! 5 e iuIix

which corresponds to a magnetic flux density BY 1 52B1 cos(g i(B0 1 dBi)t) x, resonant with spin i, fora time t. If we think of the spin dynamics in the framerotating about the z-axis at its Larmor frequency, wecan decompose this linearly polarized field into twocounter-rotating circularly polarized fields, one ofwhich rotates with this “rotating” frame, while theother rotates oppositely at 2gi(B0 1 dBi) relative tothe nuclear magnetization. We can ignore the latterterm and concentrate on the former. Then B1 becomesa static field and B0 becomes zero, a much easier sys-tem to analyze. The BY 1 field simply rotates the spin inthe Y–Z plane in this rotating frame. We can createthis field with the same measurement coil and just con-trol the amplitude, phase, and time duration of our ra-dio-frequency (RF) field. If we time the pulse such that

giB1t 5 p/2 for the RF field, this is known in NMR lit-erature as a (p/2)x pulse. Similarly,

Yi~u! 5 e iuIiy

correspond to performing similar operations alongthe y-axis. The double-qubit rotation

ZZij~u! 5 e 2ipJtIizIjz

corresponds to evolving a pair of J-coupled spins fora time t.

It turns out the Xi , Yi , and ZZij time-evolution op-erators are computationally complete, in that to-gether they can approximate any unitary operatorwith any desired accuracy. Using our coil and an ar-bitrary waveform RF generator, we can satisfy thesecond of the quantum computing requirements.This coil actually allows us to satisfy the last require-ment as well since we use it to perform projectivemeasurement. One subtle point is that some quan-tum algorithms have been designed to give out prob-abilistic results from single quantum systems. For-tunately, it has been shown6 that they can bemodified to give deterministic results for ensemblesystems.

Decoherence is the process by which a quantum sys-tem loses quantum coherence to its environment andbecomes classical. In NMR, decoherence is measuredby how long a spin on one molecule stays in phasecoherence with another molecule. This interval iscalled the spin-spin relaxation time, T2 . The beautyof NMR for quantum computing is that T2 is muchlarger than the gate speed. This allows hundreds,even thousands of quantum gates to be implemented,allowing the third quantum computing requirementto be satisfied. One can see why this is possible: thenucleus is a perfect, frictionless trap, surrounded bya cloud of electrons that shield most external deco-herence mechanisms; we are spreading relatively fewdegrees of freedom over the 2(21023

)-sized Hilbertspace of a typical liquid. No other system to date hasshown such long coherence times and, paradoxically,the ability to control the nuclear states on fast timescales. Quantum error correction is a wonderful the-ory that has shown that one can compute indefinitelyif the decoherence rate per gate is less than a thresh-old,7–10 which has been estimated to be between2(1024) to 2(1026) per gate.11 At the cost of many moreancilla spins, NMR is the closest demonstrated technol-ogy (2(1023)) to reaching this threshold. By modify-

IBM SYSTEMS JOURNAL, VOL 39, NOS 3&4, 2000 MAGUIRE, BOYDEN, AND GERSHENFELD 825

ing the chemistry of the liquids, researchers at IBM12

are improving this threshold for NMR.

We have not satisfied the first quantum computingrequirement, the creation of a pure state. A ther-malized liquid only gives us the mixed state providedby the almost maximum entropy mixed state. Onemight think this is impossible, since to go from a highentropy state to a zero entropy state requires non-unitary operators. The key insight from a couple ofgroups is that there are degrees of freedom for re-jecting this entropy.6,13,14 Logical labeling uses extraspins to label the remaining spins as ground statespins; spatial labeling uses external gradient fieldsto erase terms in the density matrix by creating a spa-tial superposition of quantum computers and thenmixing them together; temporal labeling creates aseries of experiments that are temporally averagedto give an initial ground state and the correspond-ing evolution. For prototyping our quantum com-puter, we use an inefficient but conceptually simplemethod, exhaustive averaging. For two qubits, onemight run three experiments, for example, using pre-paratory gates to create deviation density matrices

31 0 0 00 0.6 0 00 0 20.6 00 0 0 21

4, 31 0 0 00 20.6 0 00 0 21 00 0 0 0.6

4,

31 0 0 00 21 0 00 0 0.6 00 0 0 20.6

4performing the desired computation on each, thenaveraging the three experimental results—whichgives the same result as if one had performed thecomputation on a state equal to the average of thethree initial density matrices,

31.33 0 0 0

0 0 0 00 0 0 00 0 0 0

41 3

20.33 0 0 00 20.33 0 00 0 20.33 00 0 0 20.334

The second term is unobservable and the first termis a pure state. This works because quantum mechan-ics is linear, and just as a Cartesian vector can beresolved into x, y, and z terms, we can resolve ourdensity matrix into more easily manipulated com-ponents.14

Issues for thermal nuclear resonance

In this section we discuss issues facing the scaling ofliquid nuclear resonance to nontrivial computationalcomplexity. We begin with spin polarization and itsessential role in the scalability of quantum comput-ers, then address the both quantum and classical na-ture of NMR in thermal liquids, and finally discusshow to increase the number of qubits in the system.

Polarization. It seems counterintuitive to be able tomake the observable term of the thermalized den-sity matrix behave exactly as if it were in the groundstate of the system. Delving deeper, consider the larg-est signal that can be observed by logical labeling tobe the difference between the most probable andleast probable terms of the density matrix:15,6

S 5 ~~req!max 2 ~req!min!g\~Iz!max

Solving this using the equation for the thermalizeddensity matrix r(0) given earlier in this section yields

S 5N2 N g\

sinh ~Nbg\B0 / 2!

~cosh ~bg\B0 / 2!! N

The plot of this function is shown in Figure 1A, witha baseline set as spin noise, where quantum infor-mation is lost to the environment via stimulated emis-sion with the coil. 16 It is apparent that between 20and 50 qubits, no current technique can measure thequantum computing signal. Looking at the problemin the domain of polarization, Figure 1B shows thepolarization needed to extract a specific number ofqubits with a signal-to-noise ratio of 3 decibels (dB)relative to spin noise. This plot shows the spin po-larization must approach unity to make the systemperform better than classical systems. Both plotsshow that the Boltzmann limit is not scalable forquantum computation.

It would be nice to dramatically increase the spinpolarization but keep rotational and translational de-grees of freedom thermalized. Optical pumping17 isa technique potentially able to accomplish this. Eachquantum of circularly polarized light carries one unit


of angular momentum that can put an atom into ahighly polarized electronic state. The electronic po-larization can then polarize the nuclear spins throughrelaxation mechanisms. In noble gases that do notinteract and have simple electronic orbitals, close to50 percent polarizations have been obtained. Hap-per et al.18 did pioneering work on optical pumpingof rubidium gases that transfer polarization to ei-ther xenon-129 or helium-3. Transferring this polar-ization to nuclear states efficiently is an open, dif-ficult question, but one that needs to be addressed.Another reason to have optical photons interact withnuclear spins is for quantum error correction. To im-plement any error correction algorithm, one needsa way to reject errors in the calculation in a way thatdoes not influence the other spins that carry valu-able information. A photon channel could be ameans to do this—if a spin is polarized by opticalpumping its previous state is erased. In summary,polarization is the most serious challenge facing prac-tical quantum computation using ensembles of ther-malized nuclear systems. If it can be done, remain-ing challenges should be surmountable, as will beshown below.

Is it really quantum? Even before we begin to ad-dress issues of polarization, it is paramount to de-termine if a mixed state system, in which the quan-tum system is represented by an ensemble of

quantum systems, actually has the requirements toperform quantum computing. This was a questionin the original paper by Gershenfeld and Chuang.6

The paper showed how to do state preparation, al-gorithms, and readout on an NMR system, but it wasnot clear that quantum information processing wasactually being carried out in the system. Shack andCaves19 have almost answered this question. Theyshow that for a number of qubits less than 16, at amodest spin polarization of « 5 2 3 1026, the den-sity matrix during the entire experiment is fully sep-arable. Being separable means that the statistics ofany measurement of the density matrix can be fullydescribed by classical probabilities. In the high tem-perature limit the quantum computing signal can besimplified to

S 5N2 N

\gB0

2kBT5

N2 N11 «

The density matrix of the spins in NMR can be de-scribed classically19 if

S # h ;1

1 1 2 2N21

This means that for N smaller than a certain num-ber, entanglement cannot exist. One could rephrase

Figure 1 (A) shows the intrinsic magnetization versus temperature as a function of the number of qubits. Plot (B) showsthe polarization versus the maximum number of qubits with a 3 dB signal-to-noise ratio.

10�2 10�1 100 101 102 102 103

10 0

10�5

10�10

10�15

10�20

10�25

10�30

10�35

100

10�1

10�2

10�3

10�4

N�5N�1

N�10N�20

N�50

N�100

SPIN NOISE

MA

GN

ETI

ZATI

ON

TEMPERATURE (K)P

OLA

RIZ

ATIO

NMAXIMUM NUMBER OF QUBITS

(A) (B)


the calculations and ask “What is the minimum po-larization needed so that a bulk NMR system with anumber of qubits greater than one has a density ma-trix that is not separable?” The minimum polariza-tion is 44.4 percent. Fortunately, this is not the endof the story: quantum computing is not just aboutthe density matrix, it is also about unitary evolution.Shack and Caves19 show that if a unitary transformcan be written as an outer product of matrices

V 5 V1 # V2 . . . # VN

then a classical description with positive transitionprobabilities also can fully describe the system.

However, when they tried to include entangling uni-taries, operators that cannot be written as in theequation above, they found a classical system coulddescribe the system with classical transition proba-bilities only at the expense of an exponential decrease« in signal-to-noise ratio for each step. They furthershowed that an exponential signal decrease per stepcan be avoided for K entangling unitaries by addingN ancillary qubits. K is the largest integer such that

« # h K11

For the classical system to be able to use a singleentangling unitary with no signal reduction at a po-larization of 2 3 1026, this corresponds to K 5 6qubits. One could pose the question, “What is theminimum polarization such that for N $ 2, no en-tangling unitaries can be simulated by the classicalsystem without exponential signal decrease?” For 2qubits, this corresponds to 3.9 3 1023, above therange of current spectrometers, but within the realmof possibility of even conventional polarization en-hancement techniques such as the Nuclear Over-hauser Effect20—dynamic nuclear polarization withelectrons. The model used by the authors for avoid-ing exponential signal decrease is only one of manypossibilities: for a system to perform genuine infor-mation processing, all possible classical models can-not account for the entangling unitary evolutionswithout an exponential cost. Most believe that al-though the initial state of thermal NMR is separable,the dynamics will not be.

The question of whether NMR is quantum or not isan academic question—unity polarization, as ad-dressed in the previous section, is paramount for scal-able quantum computation. With unity polarization

both the initial states and the dynamics will surelybe quantum mechanical.

Number of qubits. The last major barrier for ther-mal NMR quantum computing is the number ofqubits. To be able to implement error correction ona quantum computer may require somewhere be-tween 10 and 100 qubits to describe a single qubit.10

To represent thousands of fully coupled qubits withseparate resonances is impossible with current tech-niques. Fortunately, Seth Lloyd21 provided a solu-tion. If one has a polymer chain of the type D–A–B–C–A–B–C . . . , where each letter corresponds toa unique resonance of a nuclear species, then thissystem is capable of performing any quantum com-putation. This architecture suffers a slight slowdownto pass information up and down the chain, but thehuge gains that quantum algorithms offer are stillpreserved. This architecture is called a Single Instruc-tion Multiple Data (SIMD) cellular automata.Recently, it has been shown22 that only. . . A–B–A–B . . . is necessary, which will make mo-lecular synthesis considerably easier.

Hardware for the table-top quantumcomputer

We have begun the development of instrumentationto address the issues raised in the previous section.An NMR quantum computer starts with a collectionof molecules in a liquid, where each molecule is aredundant quantum computer. In the presence ofour homogeneous magnet field, a small percentageof these spins align with the magnetic field and be-gin to precess. By applying amplitude, frequency,phase and time controlled magnetic energy, we canrotate the spins around an arbitrary axis in the Hil-bert space. To do this, we wrap a coil around theliquid sample, and add reactive elements to makeit resonant at several frequencies. We use a digitallycontrolled transmitter that generates these wave-forms. With sensitive electronics and a switch, wecan detect the magnetization of the liquid and per-form a measurement of the quantum state. A gen-eral system diagram is shown in Figure 2.

Computational optimizations. Current commercialNMR spectrometers are incredible machines capa-ble of doing a range of tasks, such as imaging andquantitative chemical structure determination. Withthe instrumentation outlined below, we hope tomatch current spectrometers in performance but endup with a device that can be treated more as a com-puter. Since we are trying to make a computer, there


are optimizations that we can do to improve our sys-tem performance and size over commercial NMRspectrometers. The first has to do with the sample—commercial systems need to routinely analyze a num-ber of samples. As a result, these systems are engi-neered for cylindrical symmetry for easy access. Theends contribute an inhomogeneity in the RF field,leading to errors in the computational gates. The ma-jor constraint for this configuration is that the activeregion of a sample cannot be larger than about 1.5cm in length and 1 cm in diameter. For our com-puter, we need only a single sample and can engi-neer our system for spherical symmetry with no endeffects. A spherical probe would have higher RF ho-mogeneity and allow much smaller magnet designs.

Since we are interested in a computational resultrather than a chemical spectrum, we can use a sin-gle qubit to serially produce an answer. This posesan enormous advantage since we can make a veryhigh Q electrical,23 optical,24 or mechanical readoutsystem (decoupled from the transmitter in the lasttwo cases) that could dramatically improve our sen-sitivity. Another technique is to use an electron chan-nel to measure an answer. This is an old technique20

called ENDOR, or Electron-Nuclear Double Reso-nance. Going through the electron channel will op-timistically give a gyromagnetic ratio of improve-ment. The gyromagnetic ratio is inversely pro-portional to rest mass, and thus approximately threeorders of magnitude improvement in signal can beobtained. Species-dependent relaxation mechanismsmay make this approach insubstantial, but is worthinvestigating further for candidate quantum comput-ing molecules.

Magnet design. To make a table-top system, oneneeds a compact, homogeneous magnetic field. Weare investigating permanent magnet structures tosolve this problem. There are a number of constraintsfor the system: small enough to fit on a table-top;as high a magnetic flux density as possible; a homo-geneity of the flux density across the sample of aboutone part per billion. The latter two points requiresome justification. The basic signal-to-noise ratio fora coil is proportional to B0

7/4 . 25 Given NMR signalsare extremely small, it is important to maximize thissignal-to-noise ratio. If there are spatial field vari-ations, there will be spins in the sample that will res-onate slightly faster or slower than others. With thespins out of phase, this is an artificial means of in-troducing coherence that shortens the maximum pos-sible coherence time of the liquid. Empirically, it hasbeen found that for a homogeneity of one part per

billion, all experimentally induced decoherence willbe smaller than the natural decoherence processes.Due to manufacturing tolerances, this lofty goal can-not be reached in practice solely from the magnetalone. The field must be corrected in a controlledmanner with slight perturbations, a process calledshimming. With shimming, the magnet tolerance re-quired is about 1 to 10 parts per million (ppm). Thiswill be explained in more depth in the next section.

We have done finite element modeling of two basicmagnet geometries. The first is based on an arrayedmagnet geometry invented by Klaus Halbach.26 Itcan be shown27 that for a magnetization distributionwithin a cylindrical volume as shown in Figure 3, onecan obtain a perfectly homogeneous solution withinthe volume of field strength

uBu 5 Br ln~r2/r1!

Br is the remanent flux density, that is the flux den-sity output in a zero magnetic field. This is quite sur-prising since one is able to obtain flux densities in-side the volume that are much larger than thisremanence. To construct this magnet, Halbach de-veloped an approximation using segments of mag-nets that achieve a high order approximation to thisfield. The finite element model is shown in Figure4. The design goal was to achieve 1 ppm homoge-neity over a 1 cm diameter sphere in the middle ofthe magnet. This was accomplished with r1 5 2 cm,r2 5 6.5 cm, uBW u 5 1.513 T.

Figure 2 General system schematic for an NMR quantumcomputer

LOW-NOISEAMPLIFIER

MIXER

COMPUTERRFOSCILLATOR

RFCOIL

AMPLIFIER MIXER

Bo (MAGNETIC FIELD)

SAMPLE


This is very encouraging, but from a practical pointof view has a couple of limitations. Manufacturingtolerances on modern rare earth magnets, usuallymade by sintering, pressure bonding, or extruding

are about 1 to 10 thousandths of an inch and the er-ror on the magnetization angle can approach two de-grees. These two facts make the best possible ho-mogeneity on the order of one part per million. Twoinsights lead to alternatives. The first is to constructthe magnet out of many tiny magnets. With enoughmagnets, the statistics of the distribution of errorsfor the ensemble of magnets average out high spa-tial frequencies in the magnetic field. The secondmethod is more practical and simply requires judi-cious placement of high permeability materials. FromLaplace’s equation, taking the Fourier transform inspace yields

F~k! 51F~k!

~k z k!m

where F(k) is the spatial Fourier transform of ¹ zM, the magnetization. The frequency componentF(k)dk will be attenuated by a factor of (k z k)m.For ferromagnetic materials with m r approaching100000, the high permeability enhances the filter-ing of Laplace’s equation by a factor m r over usingpermanent magnet materials alone. Fermilab28 tookadvantage of this point in the design of their anti-proton recycler ring, where they required small, ho-mogeneous magnets. They iteratively shaped thepole pieces of a traditional H-loop design.

Shimming. Shimming, or the modification of the biasmagnetic field by precisely controlled electromag-nets (or, in some cases, permanent magnets, m-metal,and chunks of steel), can increase the homogeneityof the bias magnetic field by several orders of mag-nitude. This is essential for NMR due to the fact thatnonidealities in the magnetic field can greatly reducethe lifetime of quantum coherences and blur featuresin the NMR spectrum. The observed signal will be asum of the signals of many different isochromats, orcollections of spins at the same frequency. A broad-ening of a waveform in the frequency-domain is as-sociated with a shrinking of the length of the signalin the time-domain, since as the signal componentsat various frequencies dephase and become randomwith respect to one another, their sum tends towardzero. To perform a quantum computation, the sig-nal must not decay below the noise level until thecomputation is complete. Thus we require that thehomogeneity of our magnetic field be within one partin 109.

In a region without currents, a magnetic field obeysLaplace’s equation, ¹ 2BY ; ¹ 2B 5 0, which can be

Figure 3 Magnetization distribution within the magnetthat yields a perfectly homogeneous field inside.This is true for both cylinders and spheres.

Figure 4 Finite element model of the Halbach array witha discretized magnetization approximation thatyields a 1 ppm homogeneity inside


solved in terms of spherical harmonics. (We set thex- and y-components of the magnetic field to zero,and consider only the z-component, which we referto as B, in the following discussion.) Spherical har-monics are enumerated by indices (l, m), which arecalled the order and the degree, respectively. For m 50, the zonal harmonics, one has cylindrical symme-try (i.e., the polar coordinate f is irrelevant), whichis appropriate for many magnets, where the dom-inant inhomogeneity is due to cylindrical end effects.The equation for B at the point (r, u, f) is then ofthe form

B~r, u! 5 O Cn~r/a! nPn~cosu!

where Pn is the nth Legendre polynomial, and Cn

and a are parameters. In general, the complete so-lution for all (l, m) (including the tesseral terms, forwhich m Þ 0) is, for the point (r, u, f),

B 5 Ol50

` Om50

l

Cn~r/a! lPlm~cosu)cos~m~f 2 clm))

where Plm is called the associated Legendre polyno-mial, and the last f dependent cosine term sinusoi-dally modulates the magnetic field as one traces acircle in the xy plane. All the harmonics except forl 5 m 5 0 contribute to the inhomogeneities, sincea homogeneous field, by definition, has a magnitudewhich does not depend on any coordinates. Since thehigher-order terms rapidly decrease in amplitude formost realistic solutions of Laplace’s equation, ad-justing a few terms should suffice to remove mostinhomogeneities: this is the motivation for includ-ing between 10 and 30 shim coils with every modernNMR spectrometer.

Spherical harmonics are orthogonal, so in theory oneshould be able to tune each coil just once. In realitythey are not precisely orthogonal, since a small trans-lation of one coil with respect to another leads tohigh-order interferences between the various orders.Thus shimming with orthogonal coils is a slightly non-linear problem, and for a spectrometer with n shimcoils, one must conduct a search over an n-dimen-sional space of coil currents, trying to maximize theheight and narrowness of the peaks in the spectrum.Many cumulative weeks of human effort are usuallyrequired for the shimming operation for the lifetimeof a spectrometer, but small corrections are usuallyautomatable. We are very interested in making shim-ming an on-line process, where a classical computer

is continuously correcting the shim currents for max-imum resolution with no human intervention. Thepurpose is to begin to abstract away the spectrom-eter aspect of the machine and focus on the com-putational aspects. To do so will require modern non-linear search methods, as will be described in thesoftware section.

For a 1 tesla (T) magnetic flux density with part-per-100000 homogeneity, in order to get part-per-billionhomogeneity, the error in these small fields must beon the order of 1025 Gauss, roughly 10000 timessmaller than the Earth’s magnetic field. The phys-ical coils we developed are planar, as shown in Fig-ure 5. They are simply designed and fabricated us-ing normal techniques for printed circuit boards. Thisis an old technique, as planar, orthogonal shimmingwas developed29 in the 1950s to shim electromag-nets, but fell out of favor when superconducting mag-nets became adopted. We have built electronics toaddress any number of coils: each coil has a bipolar16-bit digital-to-analog converter (DAC) capable ofoutputting between 2100 and 100 milliampere (mA)of current, so that the least significant bit correspondsto 305 m A of current. For a 25-turn coil of diameter5 cm, this creates a magnetic field with magnitude;1 gauss (G) (1 cm away from the coil center), sothat the least significant bit corresponds to about1025 G. Thus the specified hardware is, at least intheory, capable of creating the necessary fields. Theactual performance depends on many things, includ-ing the stability of the shimming system’s power sup-ply, the susceptibility of the electronics to externalnoise, and unavoidable noise imposed by the lawsof thermodynamics.

Transmitter. To generate arbitrary time series RFsignals without prohibitive amounts of analog hard-ware, we have investigated minimal software radioapproaches. With permanent magnet systems, res-onance frequencies will reach only about 80 mega-hertz (MHz). This is well within the domain of thebest digital-to-analog converters. In making a robusttransmitter for NMR, one must be very careful withtransmitter noise, phase noise of the oscillator, andlong-term stability. To begin development, we be-gan with a VXI-based, 40 million sample per secondDAC made by Hewlett-Packard. The HPE1445 has 13bits of digital resolution and generates arbitrarywaveforms of any duration, and the clock is derivedfrom a very stable oven-controlled crystal oscillator(OCXO). Initially, we have been interested in creat-ing a system that can address two heteronuclear spe-cies such as protons and carbon-13 (many more


qubits can exist in each heteronuclear subspace). Weconstructed a small analog board that takes the DACoutput and, with filtering, creates simultaneous ar-bitrary waveforms for both species. A photographof the board is shown in Figure 6. The signal fromthe DAC is split into two paths and is filtered, thenconverted by an RF mixer to each respective Larmorfrequency. The local oscillators, also made byHewlett-Packard, are switched between transmit andreceive mode by gallium arsenide switches made byMicrowave Associates (M/A-COM). These switcheswere chosen for their unusually high isolation ofabout 70 dB between channels and fast switchingspeed. High isolation is important because any un-necessary noise leakage to the receiver will degradereceiver performance and allow high power trans-mit pulses to be demodulated in the receiver. Con-trol of the transmitter switches and other functionson this board and the receiver was accomplished witha Hitachi 32-bit processor.

Receiver. The signal magnitude produced by a coilinteracting with the spin magnetization after a (p/2)pulse is given30 by

S } Nsb~gB0!2~LVsh! 1/ 2,

where Ns is the number of spins, b is a parameterto indicate the inhomogeneity of the field B1 , L isthe coil inductance, Vs is the sample volume, and h

is the ratio of the volume of the active sample re-gion to the coil. The noise term, due to fluctuation-dissipation, is the traditional Johnson type (ignor-ing equipartition and spin-noise16):

N 5 ÎF4kBTRD f,

where F is the noise figure of the receiver, kBT isBoltzmann’s constant times the temperature, R isthe resistance of the coil, and Df is the input receiverbandwidth. For designing a receiver, we see it is im-portant to minimize both the input bandwidth andthe noise figure. Since signals in NMR are on the or-der of 2150 to 2130 dBm (dB relative to 1 mW),careful attention to these details is paramount fordetecting signals. For the receiver, we started witha VXI-based, 20-million sample per second analog-to-digital converter (ADC) made by Hewlett-Pack-ard. The HPE1437 is an amazing device capable of rep-resenting 23 bits of noise-free digital resolution. Themaximum dynamic range of an ADC without process-ing gain is 1.76 1 6.02N dB, where N is the nominalbit resolution. Thus, the theoretical dynamic rangeof this device is 140.22 dB. However, since we rou-tinely detect signals that are 2130 dBm, analog elec-tronics are still required to get the maximum digitalresolution. With this, we developed a minimal re-ceiver that performed some basic amplification andfiltering. The signals from the protons and carbon-13

Figure 5 Planar shim coils to replace complex orthogonal coils in commercial spectrometers


are amplified by an Avantek 105 amplifier, a 1.6 dBnoise figure amplifier with 27 dB of gain. After awide, bandpass filter, the signal is downconverted,filtered, and then amplified with an integrated de-vice, the RF2612 made by RF Micro Devices. The sig-nal is then buffered with an op amp and then com-bined for the HPE1437. The overall RF specificationsfor the board are a 1.7 dB NF, with 60–104 dB ofgain. A photograph of the board is shown in Figure7. The ADC digitally downsamples the signal to ob-tain the amplitude and phase of the signal and thenapplies digital filtering and downsampling to reducethe output data rate. What we hope we have shownis not that this system is superior to commercial sys-tems, but is a simple, inexpensive, and software-driven one.

FPGA version. The system that we have developedshould be capable of doing all the tasks necessaryto implement a quantum computer, but it is hardlytable-top in size or function. With so many VXI mod-ules and separate pieces, we sought a minimalimplementation of all these functions. Field Pro-grammable Gate Arrays (FPGAs) are software pro-grammable devices that give one the ability to de-velop small microprocessors, memory controllers,waveform generators, and timers usually imple-

mented in application-specific integrated circuits(ASICs). We are developing a couple of boardsaround these devices that we expect will be able toamply do all the required tasks of our current sys-tem. The design that is currently being implementedis shown in Figure 8. Control of the board is imple-mented using the Internet Protocol (IP) over Eth-ernet. An FPGA serves as a controller that interfacesto a small Ethernet physical interface and MemoryAccess Controller (MAC) using the User DatagramProtocol (UDP) over IP. A three-wire serial bus is op-toisolated and carries data over a bus to the trans-mitter and receiver cards. The transmitter card hasan FPGA controlling 512 kilobytes (kB) of static RAMand an Analog Devices AD9856, a 200 MHz comple-mentary metal oxide semiconductor (CMOS) digitalupconverter. Over Ethernet, one supplies the fre-quency, amplitude, and phase of a signal digitally andthis 3" 3 2" card creates the waveform with submi-crosecond time resolution. The same analog receiverdesign described in the previous section is imple-mented here except that the signal is mixed downto a standard intermediate frequency of 455 kHz. Athird board containing an FPGA with 512 kB of SRAMand the AD9260 samples the signal at 10 MSPS (megasamples per second) and 16-bit resolution. On theFPGA, digital filters and downsamplers are applied


and the signal is digitally downconverted with aquadrature local oscillator to recover amplitude andphase. Over the three-wire serial bus, the data aretransmitted to the FPGA Ethernet controller and canbe sent to any device on our network to be visual-ized and processed. Timing control for this wholesystem is controlled by a small oven-controlled crys-tal oscillator, which has excellent phase noise char-acteristics and temperature/time stability. To addother spins, one simply needs to add a stack of thethree aforementioned boards, a high power switch,and RF amplifier.

Software

In this section we discuss the pieces of softwareneeded to run on our quantum computer, includinglow-level programming to address our hardware andoptimization software for shimming.

Pulse language and system design. We designed aprogramming language,31 q, to control the hardwarepresented in the last section. A q program is parsedby an interpreter, QOS (or Quantum Operating Sys-tem), which runs on a PC. The job of QOS is to co-ordinate the timing and configuration of the hard-ware, compile pulse sequences into waveforms forbroadcast, and process the signals that return. It is

the digital core of the software-radio architecture ofthis NMR system. A q program is essentially assem-bly microcode of the basic operations for an NMRquantum computer. A sample q program that per-forms one of the three pulse sequences required forexecuting Grover’s algorithm, which performs asearch on N unsorted items with only 2(=N) da-tabase accesses, is briefly explained in the accom-panying sidebar.

Optimization for shimming. We built a machine-learning environment for QOS. Our initial prioritywas to continuously automate the tedious shimmingprocess, but it quickly became apparent that with ap-propriate control software, the NMR spectrometercould automatically perform very complicated tasks.For example, one might then run a multidimensionalNMR experiment like COSY (COrrelated Spectros-copY), extracting the various J-couplings betweenspins, as well as any other interactions that might beapparent. This is a very difficult task, requiring ex-cellent hardware and software, since most real mol-ecules in natural environments are difficult to dif-ferentiate from their surroundings.

All nonlinear search optimization techniques requirea function to be optimized. Usually evaluating thisfunction is an expensive thing to do—in our case,

Figure 7 Photograph of the receiver board


requiring many seconds per execution. Since we wantto maximize the dephasing time (thus improving thehomogeneity of the magnetic field), a suitable func-tion is the sharpness of a peak in the 1-D NMR spec-trum. Thus a simple protocol for calculating this

sharpness is: run a p/2 pulse program with specificvalues for the parameters being optimized, and col-lect the data from the NMR spectrometer. Then com-pute the power spectrum of the signal, and fit aLorentzian (as described earlier) to the lineshape,

Declare the program type% This is a comment% A name for this program — Grover 's algorithm to discover the% u11. stateQname( 'grover u11. ')Qsample( '13CHC13 ') % Molecule — chloroformQversion( '1.0 esb ') % Program versionQdecouplingstyle( 'minimal ') % Whether decoupling is on/off

Declare the molecular parametersQnumqbs(2) % number of qubits% Proton(H) on chloroform is first qubit — give NMR specific information% for qubitQspecies(1, '1H ', 26.7510e7, 42574931.880108, 35000000, 1, 0.001, 5)% Carbon(13C) is 2nd qubitQspecies(2, '13C ', 6.7263e7, 10705040.871934, 8000000, 0.0159, 0.001, 5)

Prepare the state (for temporal averaging purposes)% A square (hard) pulse (as opposed to a soft gradient or sinc pulse)% applied about the y-axis for 90 degreesQy(1,90,square)% wait for a specified time in decimal unitsQwait(0.002325581395349, dec, 0)Qy(2, 290, square)% An x2pulseQx(2, 290, square)Qy(2, 90, square)Qy(1,290, square)Qx(1,290, square)Qy(2,90, square)Qwait(0.002325581395349, dec, 0)Qy(1,290,square)Qx(1,290,square)Qy(1,90,square)Qy(2,290,square)Qx(2,290,square)

Perform Grover 's algorithm including readoutQx(1,180,square)Qy(1,90,square)Qx(2,180,square)Qy(2,90,square)Qwait(0.002325581395349, dec, 0)Qy(2,290,square)Qx(2,90,square)Qy(1,290,square)Qx(1,90,square)Qwait(0.002325581395349, dec, 0)Qy(2,290,square)Qx(2,290,square)Qy(1,290,square)Qx(1,290,square)


extracting the sharpness v0/a. The sharpness can bemaximized using any of a number of methods—theNelder-Mead downhill simplex method, Powell’sdirection set method, or the conjugate gradientmethod (using finite differences to calculate the gra-dient).

Results

Our results are preliminary but indicative of thingsto come. The plots outlined below are relatively triv-ial on commercial spectrometers, but our resultscome from a machine that is significantly less in costand complexity. The plot in Figure 9 shows the in-

trinsic sensitivity of our spectrometer on a 0.5 cm3

50 percent glycerine in water sample. The samplewas prepared in a 5 millimeter NMR tube and thedata were collected for 3.28 milliseconds. This is shortcompared to the intrinsic coherence time of the pro-tons in the liquid, but the sample has been inhomo-geneously broadened by our low homogeneity testelectromagnet. With over 50 dB of signal to noiseover a bandwidth of approximately 1 kHz, our sys-tem is competitive with commercial systems, and withcomputational optimizations we are pursuing, wehope to develop our system beyond what is donecommercially.

Figure 8 Overall schematic for the transmitter and receiver electronics for an FPGA-based software radio

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The next plot is much more important. It is a cor-relation spectroscopy plot of the correlation of onespin to another. In this case, we are showing the cor-relation of a carbon atom and a hydrogen atom, proofthat we can in principle create logical gates with oursystem. Richard Ernst32 invented this technique asa means to show coupling patterns between spins,that is to identify pJ ijI ziI zj terms of the Hamiltonianfrom vI i , terms that were ambiguous in previous ef-forts. Without delving into the details of the rich sub-ject of 2-D correlation spectroscopy, we have dem-onstrated an experiment that isolates the couplingterm to show that the nonlinear term of the Ham-iltonian is present. The experiment goes as follows:we apply a p/2 pulse on the hydrogen spin that cre-ates a (1/=2)(u1& 1 u0&) state, then we let the systemevolve for a variable time t1 that is changed fromexperiment to experiment. During this time, the u0&state of the hydrogen spin will interact with the u0&state of the carbon spin to make this state resonateslightly faster during this time. The other u1& stateof the hydrogen spin will interact with the u0& stateof the carbon spins to resonate slightly slower. Thedifference in resonance frequency is the J couplingfrequency. Our sample was 1 milliliter (mL) of C 13

labeled chloroform with a coupling frequency of 215Hz. A plot is shown in Figure 10, highlighting a split-ting of the hydrogen resonance by 215 Hz. This is

direct evidence of the coupling between spins, whichwith shimming will be used to implement gates. Wecould not go further because of the inhomogeneityof our magnet. With our shimming system, we shouldbe able to implement a simple quantum algorithm.

Figure 10 Three-dimensional plot showing the resolvementof J-coupling in chloroform, required toimplement gates. This proton spectrum showsa 215 Hz splitting via the interaction with C13.

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Conclusions

We have demonstrated the feasibility of a table-topNMR apparatus performing quantum computations,approaching the relevant performance of commer-cial spectrometers, and potentially scaling to non-trivial sizes through optimizations for computationrather than spectroscopy. Success beyond engineer-ing diligence almost wholly depends on the successof unity polarization enhancement for thermal en-sembles of nuclear spins. Optical pumping tech-niques show promise but still fall short of what is re-quired for this technique to be useful. However, ifthis major obstacle can be overcome, we will havea machine that will have unprecedented computa-tional degrees of freedom. This could potentially bevery important not only for implementing quantumalgorithms but also for controlling and manipulat-ing large coherent quantum systems. Such a systemcould one day lend itself to simulating problems thatphysicists and semiconductor engineers only dreamof. The important realization is that nature itself isa very powerful computational architecture. We justneeded to figure out how to talk to it.

Acknowledgments

We gratefully acknowledge the MIT Media Lab’sThings That Think consortium and the support ofthe Defense Advanced Research Projects Agency(DARPA) under the Nuclear Magnetic ResonanceQuantum Computer (NMRQC) initiative. We thankour collaborators at Stanford, Berkeley, MIT, andIBM. Yael Maguire acknowledges financial supportfrom IBM and the National Science Research Coun-cil of Canada (NSERC).

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18. W. Happer, E. Miron, S. Schaefer, D. Schreiber, W. A. V.Wijngaarden, and X. Zeng, “Polarization of the Nuclear Spinsof Noble-Gas Atoms by Spin Exchange with OpticallyPumped Alkali-Metal Atoms,” Physical Review A 29, 3092–3110 (1984).

19. R. Shack and C. M. Caves, “Explicit Product Ensembles forSeparable Quantum States,” Journal of Modern Optics 47, No.2–3, 387–399 (2000).

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Accepted for publication July 5, 2000.

Yael Maguire MIT Media Laboratory, 20 Ames Street, Cambridge,Massachusetts 02139-4307 (electronic mail: [email protected]).Mr. Maguire is a Ph.D. candidate at the MIT Media Laboratory.He is an IBM Student Fellow holding a master’s degree in mediaarts and sciences from MIT. Mr. Maguire received a degree inengineering physics with Highest Honors from Queen’s Univer-sity in Canada. He is currently working on building a table-topdevice for quantum computing, which holds the promise of com-puting faster than traditional, semiconductor-based computers.He currently holds an NSERC fellowship from Canada and isgenerally interested in the fundamental ties between informationprocessing and physics.

Edward S. Boyden III Stanford Neurosciences Program, Beck-man Center B103, Stanford, California 94305 (electronic mail:[email protected]). Mr. Boyden is a doctoral student in neu-rosciences at Stanford University, where he works on molecularanalyses of small neural networks using genetic, electrophysiologi-cal, and optical techniques. He received his master’s degree inelectrical engineering and computer science, and his bachelor’sdegree in physics, both from MIT. He worked on many of theprojects in the MIT Physics and Media Group throughout his un-dergraduate years.

Neil Gershenfeld MIT Media Laboratory, 20 Ames Street, Cam-bridge, Massachusetts 02139-4307. Dr. Gershenfeld leads the Phys-ics and Media Group at the MIT Media Laboratory and directsthe Things That Think research consortium. His laboratory in-vestigates the relationship between the content of informationand its physical representation, from developing molecular quan-tum computers, to smart furniture, to virtuosic musical instru-ments. Author of the books When Things Start to Think, The Na-ture of Mathematical Modeling, and The Physics of InformationTechnology, Dr. Gershenfeld has a B.A. degree in physics withHigh Honors from Swarthmore College, a Ph.D. from CornellUniversity, was a Junior Fellow of the Harvard University So-ciety of Fellows, and a member of the research staff at AT&TBell Laboratories.


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Toward a table-top quantum computer