Quantum-Mechanical Computersrobins/Quantum... · Computers by Seth Lloyd Quantum-mechanical computers, if they can be constructed, will do things no ordinary computer can E very two

Quantum-Mechanical Computers

by Seth Lloyd

Quantum-mechanical computers, if they can be constructed, will do things no ordinary computer can

E very two years for the past 50, computers have become twice as fast while theircomponents have become half as big. Circuits now contain wires and transistorsthat measure only one hundredth of a human hair in width. Because of this ex-

plosive progress, today’s machines are millions of times more powerful than their crudeancestors. But explosions do eventually dissipate, and integrated-circuit technology isrunning up against its limits.

Advanced lithographic techniques can yield parts 1/100 the size of what is currently avail-able. But at this scale—where bulk matter reveals itself as a crowd of individual atoms—

integrated circuits barely function. A tenth the size again, the individuals assert their iden-tity, and a single defect can wreak havoc. So if computers are to become much smaller inthe future, new technology must replace or supplement what we now have.

HYDROGEN ATOMS could beused to store bits of informationin a quantum computer. Anatom in its ground state, with itselectron in its lowest possible en-ergy level (blue), can represent a0; the same atom in an excitedstate, with its electron at a higherenergy level (green), can repre-sent a 1. The atom’s bit, 0 or 1,can be flipped to the oppositevalue using a pulse of laser light(yellow). If the photons in thepulse have the same amount ofenergy as the difference betweenthe electron’s ground state andits excited state, the electron willjump from one state to the other.

98 Scientific American: The Solid-State Century Copyright 1997 Scientific American, Inc.

Several decades ago pioneers such as Rolf Landauer and Charles H. Bennett, both atthe IBM Thomas J. Watson Research Center, began investigating the physics of informa-tion-processing circuits, asking questions about where miniaturization might lead: Howsmall can the components of circuits be made? How much energy must be used up in thecourse of computation? Because computers are physical devices, their basic operation isdescribed by physics. One physical fact of life is that as the components of computer cir-cuits become very small, their description must be given by quantum mechanics.

In the early 1980s Paul Benioff of Argonne National Laboratory built on Landauer andBennett’s earlier results to show that a computer could in principle function in a purely quan-tum-mechanical fashion. Soon after, David Deutsch of the Mathematical Institute at the Uni-versity of Oxford and other scientists in the U.S. and Israel began to model quantum-me-chanical computers to find out how they might differ from classical ones. In particular, theywondered whether quantum-mechanical effects might be exploited to speed computationsor to perform calculations in novel ways.

By the middle of the decade, the field languished for several reasons. First, all these re-searchers had considered quantum computers in the abstract instead of studying actualphysical systems—an approach that Landauer faulted on many counts. It also became ev-ident that a quantum-mechanical computer might be prone to errors and have trouble cor-recting them. And apart from one suggestion, made by Richard Feynman of the Califor-nia Institute of Technology, that quantum computers might be useful for simulating oth-er quantum systems (such as new or unobserved forms of matter), it was unclear thatthey could solve mathematical problems any faster than their classical cousins.

In the past few years, the picture has changed. In 1993 I described a large class of familiarphysical systems that might act as quantum computers in ways that avoid some of Lan-dauer’s objections. Peter W. Shor of AT&T Bell Laboratories has demonstrated that a quan-tum computer could be used to factor large numbers—a task that can foil the most powerfulof conventional machines. And in 1995, workshops at the Institute for Scientific Inter-change in Turin, Italy, spawned many designs for constructing quantum circuitry. Morerecently, H. Jeff Kimble’s group at Caltech and David J. Wineland’s team at the NationalInstitute of Standards and Technology have built some of these prototype parts, whereasDavid Cory of the Massachusetts Institute of Technology and Isaac Chuang of Los Ala-mos National Laboratory have demonstrated simple versions of my 1993 design. Thisarticle explains how quantum computers might be assembled and describes some of theastounding things they could do that digital computers cannot.

READING the bit an atomstores is done using a laser pulsehaving the same amount of ener-gy as the difference between theatom’s excited state, call it E1,and an even higher, less stablestate, E2. If the atom is in itsground state, representing a 0,this pulse has no effect. But if itis in E1, representing a 1, thepulse pushes it to E2. The atomwill then return to E1, emitting atelltale photon.

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Quantum-Mechanical Computers100 Scientific American: The Solid-State Century

Let’s face it, quantum mechanics isweird. Niels Bohr, the Danish physicistwho helped to invent the field, said,“Anyone who can contemplate quantummechanics without getting dizzy hasn’tproperly understood it.” For better orworse, quantum mechanics predicts anumber of counterintuitive effects thathave been verified experimentally againand again. To appreciate the weirdnessof which quantum computers are capa-ble, we need accept only a single strangefact called wave-particle duality.

Wave-particle duality means thatthings we think of as solid particles,such as basketballs and atoms, behaveunder some circumstances like wavesand that things we normally describe aswaves, such as sound and light, occa-sionally behave like particles. In essence,quantum-mechanical theory sets forthwhat kind of waves are associated withwhat kind of particles, and vice versa.

The first strange implication of wave-particle duality is that small systems suchas atoms can exist only in discrete ener-gy states. So when an atom moves fromone energy state to another, it absorbsand emits energy in exact amounts, or

“chunks,” called photons, which mightbe considered the particles that makeup light waves.

A second consequence is that quan-tum-mechanical waves, like water waves,can be superposed, or added together.Taken individually, these waves offer arough description of a given particle’s po-sition. When two or more such waves arecombined, though, the particle’s positionbecomes unclear. In some weird quan-tum sense, then, an electron can some-times be both here and there at the sametime. Such an electron’s location will re-main unknown until some interaction(such as a photon bouncing off the elec-tron) reveals it to be either here or therebut not both.

When two superposed quantum wavesbehave like one wave, they are said tobe coherent; the process by which twocoherent waves regain their individualidentities is called decoherence. For anelectron in a superposition of two differ-ent energy states (or, roughly, two dif-ferent positions within an atom), deco-herence can take a long time. Days canpass before a photon, say, will collidewith an object as small as an electron, ex-

posing its true position. In principle,basketballs could be both here and thereat once as well (even in the absence ofMichael Jordan). In practice, however,the time it takes for a photon to bounceoff a ball is too brief for the eye or anyinstrument to detect. The ball is simplytoo big for its exact location to go un-detected for any perceivable amount oftime. Consequently, as a rule only small,subtle things exhibit quantum weirdness.

Quantum Information

Information comes in discrete chunks,as do atomic energy levels in quan-

tum mechanics. The quantum of infor-mation is the bit. A bit of information isa simple distinction between two alter-natives—no or yes, 0 or 1, false or true. Indigital computers, the voltage betweenthe plates in a capacitor represents a bitof information: a charged capacitor reg-isters a 1 and an uncharged capacitor, a0. A quantum computer functions bymatching the familiar discrete characterof digital information processing to thestrange discrete character of quantummechanics.

Logic gates are devices that perform elementary operations on bits of information. The Irish logician George Boole

showed in the 19th century that any complex logical or arith-

metic task could be accomplished using combinations of threesimple operations: NOT, COPY and AND. In fact, atoms, or anyother quantum system, can perform these operations. —S.L.

NOT GATE

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NOT involves nothing more than bit flipping, as thenotation above shows: if A is 0, make it a 1, and viceversa. With atoms, this can be done by applying apulse whose energy equals the difference betweenA’s ground state (its electron is in its lowest energylevel, shown as the inner ring) and its excited state(shown as the outer ring). Unlike conventional NOTgates, quantum ones can also flip bits only halfway.

Quantum Logic Gates

COPY, in the quantum world, relies on the interaction between two differ-ent atoms. Imagine one atom, A, storing either a 0 or 1, sitting next to an-other atom, B, in its ground state. The difference in energy between thestates of B will be a certain value if A is 0, and another value if A is 1. Now ap-ply a pulse of light whose photons have an energy equal to the latteramount. If the pulse is of the right intensity and duration and if A is 1, B willabsorb a photon and flip (top row); if A is 0, B cannot absorb a photon fromthe pulse and stays unchanged (bottom row). So, as in the diagram below, ifA is 1, B becomes 1; if A is 0, B remains 0.

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Quantum-Mechanical Computers Scientific American: The Solid-State Century 101

Indeed, a string of hydrogen atoms canhold bits as well as a string of capacitors.An atom in its electronic ground statecould encode a 0 and in an excited state,a 1. For any such quantum system towork as a computer, though, it must becapable of more than storing bits. An op-erator must be able to load informationonto the system, to process that informa-tion by way of simple logical manipula-tions and to unload it. That is, quantumsystems must be capable of reading, writ-ing and arithmetic.

Isidor Isaac Rabi, who was awardedthe Nobel Prize for Physics in 1944, firstshowed how to write information on aquantum system. Applied to hydrogenatoms, his method works as follows.Imagine a hydrogen atom in its groundstate, having an amount of energy equalto E0. To write a 0 bit on this atom, donothing. To write a 1, excite the atom toa higher energy level, E1. We can do soby bathing it in laser light made up ofphotons having an amount of energyequal to the difference between E1 andE0. If the laser beam has the proper in-tensity and is applied for the right lengthof time, the atom will gradually move

from the ground state to the excitedstate, as its electron absorbs a photon. Ifthe atom is already in the excited state,the same pulse will cause it to emit a pho-ton and go to the ground state. In termsof information storage, the pulse tellsthe atom to flip its bit.

What is meant here by gradually? Anoscillating electrical field such as laserlight drives an electron in an atom froma lower energy state to a higher one inthe same way that an adult pushes achild on a swing higher and higher. Eachtime the oscillating wave comes around,it gives the electron a little push. Whenthe photons in the field have the sameenergy as the difference between E0 andE1, these pushes coincide with the elec-tron’s “swinging” motion and graduallyconvert the wave corresponding to theelectron into a superposition of waveshaving different energies.

The amplitude of the wave associatedwith the electron’s ground state willcontinuously diminish as the amplitudeof the wave associated with the excitedstate builds. In the process, the bit regis-tered by the atom “flips” from theground state to the excited state. When

the photons have the wrong fre-quency, their pushes are out ofsync with the electron, and noth-ing happens.

If the right light is applied forhalf the time it takes to flip theatom from 0 to 1, the atom is in astate equal to a superposition ofthe wave corresponding to 0 andthe wave corresponding to 1, eachhaving the same amplitudes. Sucha quantum bit, or qubit, is thenflipped only halfway. In contrast,a classical bit will always read ei-ther 0 or 1. A half-charged capac-itor in a conventional computercauses errors, but a half-flippedqubit opens the way to new kindsof computation.

Reading bits from a quantumsystem is similar to flipping them.Push the atom to an even higher,less stable energy state, call it E2.Do so by subjecting the atom tolight having an energy equal tothe difference between E1 and E2:if the atom is in E1, it will be ex-cited to E2 but decay rapidly backto E1, emitting a photon. If theatom is already in the groundstate, nothing happens. If it is inthe “half-flipped” state, it has anequal chance of emitting a photonand revealing itself to be a 1 or of

not emitting a photon, indicating that itis a 0. From writing and reading infor-mation in a quantum system, it is onlya short step to computing.

Quantum Computation

Electronic circuits are made from lin-ear elements (such as wires, resis-

tors and capacitors) and nonlinear ele-ments (such as diodes and transistors)that manipulate bits in different ways.Linear devices alter input signals individ-ually. Nonlinear devices, on the otherhand, make the input signals passingthrough them interact. If your stereo didnot contain nonlinear transistors, for ex-ample, you could not change the bass inthe music it plays. To do so requires somecoordination of the information comingfrom your compact disc and the infor-mation coming from the adjustmentknob on the stereo.

Circuits perform computations by wayof repeating a few simple linear and non-linear tasks over and over at great speed.One such task is flipping a bit, which isequivalent to the logical operation calledNOT: true becomes false, and false be-

AND also depends on atomic interactions. Imagine three atoms, A, B and A, sitting next toone another. The difference in energy between the ground and excited states of B is a functionof the states of the two A’s. Suppose B is in its ground state. Now apply a pulse whose energyequals the difference between the two states of B only when the atom’s neighboring A’s areboth 1. If, in fact, both A’s are 1, this pulse will flip B (top row); otherwise it will leave B un-changed (all other rows).

AND GATE INITIAL STATES

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comes true. Another is COPY, whichmakes the value of a second bit the sameas the first. Both those operations are lin-ear, because in both the output reflectsthe value of a single input. Taking theAND of two bits—another useful task—

is a nonlinear operation: if two inputbits are both 1, make a third bit equal to1 as well; otherwise make the third bit a0. Here the output depends on some in-teraction between the inputs.

The devices that execute these opera-tions are called logic gates. If a digitalcomputer has linear logic gates, such asNOT and COPY gates, and nonlinearones as well, such as AND gates, it cancomplete any logical or arithmetic task.The same requirements hold for quan-tum computers. Artur Ekert, workingwith Deutsch and Adriano Barenco atOxford, and I have shown independent-ly that almost any nonlinear interactionbetween quantum bits will do. Indeed,provided a quantum computer can flipbits, any nonlinear quantum interactionenables it to perform any computation.Hence, a variety of physical phenome-na might be exploited to construct aquantum computer.

In fact, all-purpose quantum logicgates have been around almost as longas the transistor! In the late 1950s, re-searchers managed to perform simpletwo-bit quantum logic operations usingparticle spins. These spins—which aresimply the orientation of a particle’s ro-tation with respect to some magneticfield—are, like energy levels, quantized.So a spin in one direction can represent a1 and in the other, a 0. The researcherstook advantage of the interaction be-tween the spin of the electron and thespin of the proton in a hydrogen atom;

they set up a system in which they flippedthe proton’s spin only if the electron’sspin represented a 1. Because theseworkers were not thinking about quan-tum logic, they called the effect doubleresonance. And yet they used double res-onance to carry out linear NOT andCOPY operations.

Since then, Barenco, David DiVincen-zo of IBM, Tycho Sleator of New YorkUniversity and Harald Weinfurter of theUniversity of Innsbruck have demon-strated how, by flipping proton andelectron spins only partway, double res-onance can be used to create an ANDgate as well. Such quantum logic gates,wired together, could make a quantumcomputer.

A number of groups have recently con-structed quantum logic gates and pro-posed schemes for wiring them together.A particularly promising developmenthas come from Caltech: by concentrat-ing photons together with a single atomin a minute volume, Kimble’s group hasenhanced the usually tiny nonlinear in-teraction between photons. The result isa quantum logic gate: one photon bit canbe flipped partway when another pho-ton is in a state signifying 1. Quantum“wires” can be constructed by havingsingle photons pass through opticalfibers or through the air, in order to fer-ry bits of information from one gate toanother.

An alternative design for a quantumlogic circuit has been proposed by J. Ig-nacio Cirac of the University of Castil-la-La Mancha in Spain and Peter Zollerof the University of Innsbruck. Theirscheme isolates qubits in an ion trap,effectively insulating them from any un-wanted external influences. Before a bitwere processed, it would be transferredto a common register, or “bus.” Specifi-cally, the information it containedwould be represented by a rocking mo-tion involving all the ions in the trap.Wineland’s group at NIST has taken thefirst step in realizing such a quantumcomputer, performing both linear andnonlinear operations on bits encoded byions and by the rocking motion.

In an exciting theoretical developmentunder experimental investigation at Cal-tech, Cirac, Zoller, Kimble and HideoMabuchi have shown how the photonand ion-trap schemes for quantum com-puting might be combined to create a“quantum Internet” in which photonsare used to shuttle qubits coherently backand forth between distant ion traps.

Although their methods can in princi-

1 0 1 0 0 0

1 1 1 1 0 0

A= 0 B = 0A=1 B =1

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0 0 1 0 1 0

0 1 1 1 1 0

LIGHT FLIPS B TO 1IF A ON ITS LEFT IS 1

FLIPS A TO 1 IF B ON LEFT IS 1

FLIPS B TO 0 IF A ON RIGHT IS 1

FLIPS A TO 0 IF B ON RIGHT IS 1

DATA HAVE MOVEDONE PLACE TO RIGHT

DATA AS WRITTEN

DATA HAVE MOVEDONE MORE PLACE TO RIGHT

SALT CRYSTAL could be made to compute by acting on pairs of neighboring ions.Flip the bit held by each B if the A on its left stores a 1; then flip each A if the B on itsright is 1. This moves the information from each A to the B on its right. Now, using thesame tactics, move the information from each B to the A on its right. The process al-lows a line of atoms to act as a quantum “wire.” Because a crystal can carry out these“double resonance” operations simultaneously in all directions with every neighboringion (bottom, right), the crystal can mimic the dynamics of any system and so serves asa general-purpose quantum analog computer.

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ple be scaled up to tens or hundreds ofquantum bits, the Caltech and NIST

groups have performed quantum logicoperations on just two bits (leading somewags to comment that a two-bit micro-processor is just a two-bit microproces-sor). In 1997, however, Neil A. Gershen-feld of M.I.T., together with Chuang ofLos Alamos, showed that my 1993method for performing quantum com-puting using the double resonance meth-ods described above could be realizedusing nuclear spins at room tempera-ture. The same result was obtained in-dependently by M.I.T.’s Cory, workingwith Amr Fahmy and Timothy F. Havelof Harvard Medical School. With con-ventional magnets of the kind used toperform magnetic resonance imaging,Chuang and Cory both succeeded inperforming quantum logic operationson three bits, with the prospect of con-structing 10-bit quantum microproces-sors in the near future.

Thus, as it stands, scientists can con-trol quantum logic operations on a fewbits, and in the near future, they mightwell do quantum computations using afew tens or hundreds of bits. How canthis possibly represent an improvementover classical computers that routinelyhandle billions of bits? In fact, even withone bit, a quantum computer can dothings no classical computer can. Con-sider the following. Take an atom in asuperposition of 0 and 1. Now find outwhether the bit is a 1 or a 0 by makingit fluoresce. Half of the time, the atomemits a photon, and the bit is a 1. Theother half of the time, no photon is emit-ted, and the bit is a 0. That is, the bit is

a random bit—something a classicalcomputer cannot create. The random-number programs in digital computersactually generate pseudorandom num-bers, using a function whose output isso irregular that it seems to produce bitsby chance.

Multiparticle Quantum States

Imagine what a quantum computercan do with two bits. Copying works

by putting together two bits, one with avalue to be copied and one with anoriginal value of 0; an applied pulseflips the second bit to 1 only if the firstbit is also 1. But if the value of the firstbit is a superposition of 0 and 1, thenthe applied pulse creates a superposi-tion involving both bits, such that bothare 1 or both are 0. Notice that the finalvalue of the first bit is no longer the sameas it was originally—the superpositionhas changed.

In each component of this superposi-tion, the second bit is the same as thefirst, but neither is the same as the orig-inal bit. Copying a superposition stateresults in a so-called entangled state, inwhich the original information nolonger resides in a single quantum bitbut is stored instead in the correlationsbetween qubits. Albert Einstein notedthat such states would violate all classi-cal intuition about causality. In such asuperposition, neither bit is in a definitestate, yet if you measure one bit, there-by putting it in a definite state, the otherbit also enters into a definite state. Thechange in the first bit does not cause thechange in the second. But by virtue of de-

stroying the coherence between the two,measuring the first bit also robs the sec-ond of its ambiguity. I have shown howquantum logic can be used to explorethe properties of even stranger entangledstates that involve correlations amongthree and more bits, and Chuang has usedmagnetic resonance to investigate suchstates experimentally.

Our intuition for quantum mechanicsis spoiled early on in life. A one-year-oldplaying peekaboo knows that a face isthere even when she cannot see it. Intu-ition is built up by manipulating objectsover and over again; quantum mechan-ics seems counterintuitive because wegrow up playing with classical toys. Oneof the best uses of quantum logic is toexpand our intuition by allowing us tomanipulate quantum objects and playwith quantum toys such as photons andelectrons.

The more bits one can manipulate,the more fascinating the phenomena onecan create. I have shown that with morebits, a quantum computer could be usedto simulate the behavior of any quan-tum system. When properly programmed,the computer’s dynamics would becomeexactly the same as the dynamics ofsome postulated system, including thatsystem’s interaction with its environ-ment. And the number of steps the com-puter would need to chart the evolutionof this system over time would be direct-ly proportional to the size of the system.

Even more remarkable, if a quantumcomputer had a parallel architecture,which could be realized through the ex-ploitation of the double resonance be-tween neighboring pairs of spins in theatoms of a crystal, it could mimic anyquantum system in real time, regardlessof its size. This kind of parallel quantumcomputation, if possible, would give ahuge speedup over conventional meth-ods. As Feynman noted, to simulate aquantum system on a classical comput-er generally requires a number of steps

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READOUT from a quantum computermight look like the image above. Eachcolored spot is the fluorescent light com-ing from a single mercury ion in an iontrap (left). The light indicates that eachion is in the same state, so the entire stringreads as a series of 1’s.

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Quantum-Mechanical Computers104 Scientific American: The Solid-State Century

that rises exponentially both with thesize of the system and with the amountof time over which the system’s behav-ior is tracked. In fact, a 40-bit quantumcomputer could re-create in little morethan, say, 100 steps, a quantum systemthat would take a classical computer,having a trillion bits, years to simulate.

What can a quantum computer dowith many logical operations on manyqubits? Start by putting all the input bitsin an equal superposition of 0 and 1, eachhaving the same magnitude. The com-puter then is in an equal superposition ofof all possible inputs. Run this inputthrough a logic circuit that carries out aparticular computation. The result is asuperposition of all the possible outputsof that computation. In some weirdquantum sense, the computer performsall possible computations at once.Deutsch has called this effect “quantumparallelism.”

Quantum parallelism may seem odd,but consider how waves work in gener-al. If quantum-mechanical waves weresound waves, those corresponding to 0and 1—each oscillating at a single fre-quency—would be pure tones. A wavecorresponding to a superposition of 0and 1 would then be a chord. Just as amusical chord sounds qualitatively dif-ferent from the individual tones it in-cludes, a superposition of 0 and 1 differsfrom 0 and 1 taken alone: in both cas-es, the combined waves interfere witheach other.

A quantum computer carrying out anordinary computation, in which no bitsare superposed, generates a sequence ofwaves analogous to the sound of “changeringing” from an English church tower,in which the bells are never struck simul-taneously and the sequence of sounds fol-

lows mathematical rules. A computa-tion in quantum-parallel mode is like asymphony: its “sound” is that of manywaves interfering with one another.

Shor of Bell Labs has shown that thesymphonic effect of quantum parallel-ism might be used to factor large num-bers very quickly—something classicalcomputers and even supercomputerscannot always accomplish. Shor dem-onstrated that a quantum-parallel com-putation can be orchestrated so that po-tential factors will stand out in the su-perposition the same way that a melodyplayed on violas, cellos and violins anoctave apart will stand out over thesound of the surrounding instrumentsin a symphony. Indeed, his algorithmwould make factoring an easy task fora quantum computer, if one could bebuilt. Because most public-key encryp-tion systems—such as those protectingelectronic bank accounts—rely on thefact that classical computers cannot findfactors having more than, say, 100 dig-its, quantum-computer hackers wouldgive many people reason to worry.

Whether or not quantum computers(and quantum hackers) will come aboutis a hotly debated question. Recall thatthe quantum nature of a superpositionprevails only so long as the environmentrefrains from somehow revealing the

state of the system. Because quantumcomputers might still consist of thou-sands or millions of atoms, only one ofwhich need be disturbed to damagequantum coherence, it is not clear howlong interacting quantum systems canlast in a true quantum superposition. Inaddition, the various quantum systemsthat might be used to register and processinformation are susceptible to noise,which can flip bits at random.

Shor and Andrew Steane of Oxfordhave shown that quantum logic opera-tions can be used to construct error-correcting routines that protect thequantum computation against decoher-ence and errors. Further analyses byWojciech Zurek’s group at Los Alamosand by John Preskill’s group at Caltechhave shown that quantum computerscan perform arbitrarily complex com-putations as long as only one bit in100,000 is decohered or flipped at eachtime step.

It remains to be seen whether the ex-perimental precision required to performarbitrarily long quantum computationscan be attained. To surpass the factor-ing ability of current supercomputers,quantum computers using Shor’s algo-rithm might need to follow thousandsof bits over billions of steps. Even withthe error correction, because of the tech-nical problems described by Landauer,it will most likely prove rather difficultto build a computer to perform such acomputation. To surpass classical simu-lations of quantum systems, however,would require only tens of bits followedfor tens of steps, a more attainable goal.And to use quantum logic to createstrange, multiparticle quantum statesand to explore their properties is a goalthat lies in our current grasp.

Factoring could

be an easy

task for a quantum

computer.

SETH LLOYD is the Finmeccanica Career Development Profes-sor in the mechanical engineering department at the MassachusettsInstitute of Technology. He received his first graduate degree in phi-losophy from the University of Cambridge in 1984 and his Ph.D. in

physics from the Rockefeller University in 1988. He has held post-doctoral positions at the California Institute of Technology and atLos Alamos National Laboratory, and since 1989 he has been anadjunct assistant professor at the Santa Fe Institute in New Mexico.

Further Reading

The Author

Quantum-Mechanical Models of Turing Machines ThatDissipate No Energy. Paul Benioff in Physical Review Letters,Vol. 48, No. 23, pages 1581–1585; June 7, 1982.

Quantum Theory: The Church-Turing Principle and theUniversal Quantum Computer. David Deutsch in Proceedingsof the Royal Society of London, Series A, Vol. 400, No. 1818,pages 97–117; 1985.

A Potentially Realizable Quantum Computer. Seth Lloyd in

Science, Vol. 261, pages 1569–1571; September 17, 1993.Algorithms for Quantum Computation: Discrete Loga-rithms and Factoring. Peter W. Shor in 35th Annual Sympo-sium on Foundations of Computer Science: Proceedings. Edited byShafi Goldwasser. IEEE Computer Society Press, 1994.

Quantum Computations with Cold Trapped Ions. J. I. Ciracand P. Zoller in Physical Review Letters, Vol. 74, No. 20, pages4091–4094; May 15, 1995.

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Copyright 1997 Scientific American, Inc.