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KNOTTY CALCULATIONSA quantum version of braids could lay the groundwork

for tomorrow’s computers BY ERICA KLARREICH

When you learned to tie knots as a child, youprobably thought their main use was formaking bows on birthday presents or keep-ing your shoes on your feet. However, if asmall band of mathematicians and physi-

cists has its way, knots will form the basis for an entirelynew kind of computer, one whose power vastly out-strips that of the machines at our disposal today. In itsfirst century, the mathematical study of knots belonged squarely to the realm of pure mathematics, seemingly divorcedfrom any practical applications. In the past decade, however, math-ematicians have turned knot theory into a bridge between twoseemingly unconnected subjects: computer science and quantummechanics, the branch of physics that deals with the ultrasmall scaleof atoms and subatomic particles.

In a paper published last month, researchers propose that thisconnection between the two fields might finally enable physiciststo reach a decades-long goal: to exploit quantum physics to builda computer whose performance would far surpass that of com-puters based on the classical physics of Isaac Newton. A quantumcomputer, if it is ever built, will have the power to crack the cryp-tographic schemes that safeguard Internet transactions and tocreate incredibly detailed simulations of the behavior of the uni-verse at the tiniest scale.

The knots that mathematicians have been studying have a slightquirk: After the theoretical knot is tied, the ends of the string arejoined together so the knot can’t untie. The same knot can appearin many guises, since pulling and twisting the strings can make theknot look completely different. The basic question of knot theoryis, Given two knots that look very different, is there a way tellwhether they are knotted in the same way (SN: 12/8/01, p. 360)?

To distinguish between knots, mathematicians look for numer-ical characteristics of a knot, such as the number of times the knot’sshadow crosses itself. Some other characteristics, called knot invari-ants, don’t change when the knot is pulled and twisted about. Iftwo knots have different invariants, they must be different knots.

At the heart of the connection between computer science andquantum physics is a knot invariant called the Jones polynomial,which associates a given knot with an array of numbers. The Jonespolynomial involves a complex mathematical formula, and althoughcalculating it is easy for simple knots, it is enormously difficult formessy, tangled knots. In fact, mathematicians have found compellingevidence suggesting that as knots get more and more complicated,the difficulty of computing their Jones polynomials rises exponen-tially. Calculating the Jones polynomial for complicated knots is con-sidered beyond the reach of even the fastest computers.

That seems like bad news. A connection to quantum physics,however, has turned this apparent liability into a decided advan-tage by offering a new approach. In the late 1980s, physicist Edward

Witten, a major figure in string theory (SN: 2/27/93, p. 136),described a physical system that should calculate informationabout the Jones polynomial during the course of its regularly sched-uled activities—just as when a ball is hurled into the air, natureinstantly solves the complicated equations that govern its motion.

Now, mathematician Michael Freedman of Microsoft Researchin Redmond, Wash., and physicist Alexei Kitaev of the CaliforniaInstitute of Technology in Pasadena are pursuing a daring idea: IfWitten’s physical system somehow does calculations beyond thereach of computers, could this system be harnessed to build a com-pletely new kind of computer?

NATURE AS COMPUTER The idea of a physical system cal-culating something about knots or other loops may sound strange,but in fact examples of such systems abound, even in basic physics.In an electrical transformer, for instance, two loops of wire arecoiled around an iron core. An electric current passing through oneof the wires generates a voltage in the other wire that’s proportional

to the number of times the sec-ond wire twists around the core.Thus, even if you couldn’t see thewire, you could figure out itsnumber of twists simply by meas-uring the voltage. Witten pro-posed that in the same way, itshould be possible to obtain infor-mation about the Jones polyno-mial of a knot by taking appro-priate measurements in a morecomplicated physical system.

The connections between theJones polynomial and both com-puters and quantum physicscaught Freedman’s eye in the late

1980s. Freedman was on his home turf when it came to knots—in 1986, he was awarded a Fields Medal (the mathematical equiv-alent of the Nobel prize) for his work in topology, the mathemat-ical field to which knot theory belongs. However, he knew lessabout the challenges of building an actual physical system likeWitten’s theoretical system. Physicists “said Witten’s physics wasso abstract it wasn’t related to the real world, and that we’d neverbe able to build such a computer in our universe,” Freedman recalls.Discouraged, he put the project on the back burner.

As Freedman gradually learned more physics, however, hebecame convinced that certain extremely cold electron seas calledquantum Hall fluids might offer the right physics to do the job. Thenin 1997, working independently, Kitaev described a concrete modelfor how such a computer might work. “Kitaev’s paper was stun-ningly original,” says John Preskill, who studies quantum compu-tation at the California Institute of Technology in Pasadena. “It’sa beautiful and potentially quite significant idea.”

In the past several years, Freedman and Kitaev have joined forcesto explore the promise of their model for what they call a topologi-

At the heart of the connectionbetweencomputerscience andquantum physicsis a knot invariantcalled the Jonespolynomial.

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cal quantum computer. Such a computer would encode informationnot in the conventional zeros and ones but in the configurations ofdifferent braids, which are similar to knots but consist of several dif-ferent threads intertwined around each other. The computer wouldphysically weave braids in space-time, and then according to Wit-ten’s theory, nature would take over the hard work, carrying outcomplex Jones polynomial calculations in the blink of an eye.

POLYNOMIAL POWER A computer specially geared to calcu-late the value of some obscureknot invariant might not seemparticularly useful. However, inthe late 1980s, mathematiciansshowed that computing the Jonespolynomial belongs to the famousfamily of what they call NP-hardproblems. If someone could finda fast way to calculate the Jonespolynomial, the numerical outputof the calculation could then beused to solve a host of other diffi-cult problems. These include thetraveling salesman problem,which looks for the most efficientroute for a salesman who mustpass through many cities. “There’san impressive amount of infor-mation stored in the Jones poly-nomial,” Freedman says.

Kitaev and Freedman’s modelcomputer wouldn’t provide theexact value of the Jones polyno-mial, only indicate whether itsvalue lies within a certain range.However, even that is enough tosolve many important problems.Last year, Freedman and Kitaev,together with mathematiciansZhenghan Wang and MichaelLarsen of Indiana University inBloomington, proved that a topo-logical quantum computer wouldbe every bit as effective as anothertheoretical quantum computer,the “qubit” quantum computer,which employs quantum physicsbut uses it in a very different way(SN: 2/1/03, p. 77).

Both types of computers areexpected to have the power tocrack cryptographic schemes, suchas the RSA algorithm commonlyused in Internet security, includingcredit card transactions. “A lot ofpeople belonging to agencies withthree letters in their name will bevery interested if someone is ableto build these computers,” saysSeth Lloyd, who studies quantumcomputation at the Massachusetts Institute of Technology.

Although researchers have been thinking about how to buildqubit quantum computers for decades, so far they languish on thedrawing board. The main difficulty is that in a qubit computer,each bit of information is typically encoded in the state of a singleparticle, such as an electron or photon. This makes the informa-tion vulnerable: If a tiny disturbance in the environment changesthe state of the particle, the information is lost forever. Physicistscall this problem decoherence. “Decoherence is the number one

enemy of quantum computation,” Preskill says.By encoding information in braids instead of single particles,

a topological quantum computer neatly sidesteps this problem.A small disruption from the environment might disturb thethreads of a knot slightly but is extremely unlikely to change itsoverall knottedness — just as a breeze might make your shoelacesflutter but not untie. Thus, the topological quantum computer hasa built-in defense against decoherence. “I think it’s the right wayto go in the long run, if you want to build a quantum computer,”

Preskill says. “It’s a made-to-order solution to the problem ofdecoherence and errors.”

Topological quantum compu-tation’s stability might eventuallygive it the edge over qubit com-puters, Lloyd says. “Topologicalquantum computation is far awayfrom realization now, but it hassuch appealing features that itwouldn’t take me by surprise if itturned into the dominant para-digm,” he says. “It’s such an ele-gant idea to use what nature givesyou to build quantum computersthat are intrinsically reliable.”

COMPUTATION, ANYON?For building a topological quan-tum computer, one system thatFreedman and Kitaev are consid-ering is an exotic form of mattercalled a fractional quantum Hallfluid. It arises when electrons atthe flat interface of two semicon-ductors are subjected to a power-ful magnetic field and cooled totemperatures close to absolutezero. The electrons on the flat sur-face form a disorganized liquidsea of electrons, and if some extraelectrons are then added, strangequasi-particles called anyonsemerge. Unlike electrons, whicheach have a single negativecharge, or protons, which have asingle positive charge, anyons canhave a charge that is a fraction ofa whole number—somethingnever before seen in physics.

Anyons have a strange propertythat may make them the key totopological quantum computa-tion. If you slide a bunch of pen-nies around on a table along pathsthat return the pennies to theiroriginal spots at the end, thenafter the motion is over there isno way to tell what paths the pen-nies followed (unless you have a

very dusty table). When anyons are moved around each other,however, they remember—in a specific physical sense—the knot-tedness of the paths they followed.

Imagine several anyons on a surface, and suppose they movearound along complicated paths, ending up where they started. Onthe surface, the paths may cross each other in many spots. How-ever, in space-time—in which there is one snapshot of the surfacefor each moment in time—the paths don’t pass directly througheach other, but simply braid around each other.

SPACE-TIME BRAIDS — Particles moving about on a surface trace out paths. Here, a solid dot marks the start andend points and an open dot, the midpoint of the motion ofeach of four particles (top). Depicted in space-time, thesepaths can intertwine to form what mathematicians describeas a braid (bottom). If the particles are so-called anyons, it'spossible to recapture information about a braid by measuringphysical properties of the anyons after the motion ceases. Thisprocess may open the door to a completely new type of computer that calculates by using braids.

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Anyons come in a variety of types, and if two different anyonsbump into each other, they either annihilate each other or fuseinto a single particle. When anyons move around each other alongbraided paths, the motion changes the pattern of the anyons’ inter-actions with each other—a change physicists call a unitary trans-formation. This transformation alters what will happen if twoanyons collide, so by bumping anyons into each other it’s possibleto tell that they have moved.

What’s more, different braids lead to different transformations,so one can figure out something about the particular braid theanyons traversed by measuring whathappens when the anyons collide.The fractional quantum Hall fluid haseffectively calculated numerical prop-erties of the braid, and measuring theanyons gives information about theresult of this calculation.

Quantum Hall fluids come in manyflavors, depending on the fractionalcharge of the anyons. Different fluidshave different unitary transformationsand so carry out different calculations.If the transformations in a particularfluid aren’t very complicated, then thecorresponding calculations aren’t veryinteresting. So, the task now is to find an anyonic system with com-plex enough transformations—called nonabelian transformations—to carry out Jones polynomial calculations. So far, the few quantumHall systems that have been studied thoroughly aren’t powerfulenough to do the job, but Freedman is optimistic that the right sys-tems exist. “Nonabelian anyons are not obscure mathematically—they’re rather simple, and there’s no overriding physical law that hasto be broken to make them,” he says. “I feel confident that they’reout there, but there’s no way of knowing right now whether they’ll

be easy or hard to put together.”Although quantum Hall fluids are the only known systems that

contain anyons, many other anyonic systems probably await dis-covery, suggests physicist Chetan Nayak of the University of Cal-ifornia, Los Angeles. “With quantum Hall fluids, we stumbled byexperiment into a new category of matter, and we’ve just scratchedthe surface in terms of new possibilities,” Nayak says. He has teamedup with Freedman to explore the possibility of creating nonabeliananyons in systems consisting of many thin layers of magnets. Kitaev,meanwhile, is investigating the possibility of building memory fora quantum computer from a system in which electrons spinningon the corners of a hexagonal grid give rise to anyons.

Freedman is reluctant to put a time frame on the constructionof a topological quantum computer, but he is confident that it willhappen. “If the physical world is the way we think it is, it’s only amatter of time,” he says.

If a quantum computer is built, its uses will go far beyond crack-ing cryptographic schemes. A quantum computer could simulatethe interactions of individual molecules and atoms, rapidly carry-ing out enormous computations that today’s computers do ago-nizingly slowly. It’s hard to guess ahead of time just how engineersand physicists will employ such a vast increase in computing capac-ity, says Daniel Gottesman, who studies quantum computation atthe Perimeter Institute in Waterloo, Ontario. “When classical com-puters were invented, no one imagined that engineers would some-day use them to test the design of airplanes or bridges,” he says.With engineers turning their attention to designing single-mole-cule drugs, robots, and machines, it would be useful to have a quan-tum computer to probe matter on this tiny scale, Gottesman says.

For Freedman, though, there’s a motivation even more com-pelling than the possibility of creating a powerful new computer.“I’m working on this because the mathematics is so beautiful,” hesays. “It’s an excuse to think about the two most interesting thingsin the world: topology and theoretical physics.” �

“With quantumHall fluids, westumbled byexperimentinto a newcategory ofmatter.”—CHETAN NAYAK

ARCHAEOLOGY

Farming sprouted inancient Ecuador

People living in the lowlands of what’snow southwestern Ecuador began togrow squash between 10,000 and 9,000years ago, about the same time that res-idents of Mexico’s southern highlandsdomesticated the vegetable (SN: 5/24/97,p. 322), according to a study in the Feb. 14Science.

The comparably ancient roots of plantcultivation in these two regions indicatethat “in South America, there was no sin-gle center of agricultural origins,” con-clude Dolores R. Piperno of the Smith-sonian Tropical Research Institute in

Balboa, Panama, and Karen E. Stothert ofthe University of Texas at San Antonio.

In the soil of two prehistoric sites inEcuador, the scientists isolated and stud-ied microscopic crystals from squashrinds that had been uncovered there.These ancient crystals were the same sizeas those in the squash’s modern domesti-cated form, but not those in its present-day wild counterpart. Piperno andStothert were able to date tiny bits of car-bon that were trapped inside the ancientcrystals as they formed. —B.B.

BIOMEDICINE

Carbon monoxidemay limit vasculardamage

Carbon monoxide in small doses can pre-vent injury to blood vessels caused by sur-gery, a study of rats suggests.

Researchers gave carbon monoxide torats before performing angioplasty, in

which a balloon-tipped catheter is usedto widen a clogged area in an artery. Theprocedure works well for people with par-tially blocked arteries, but many patientsmust undergo repeat angioplasty becausethe subtle injury caused by the ballooncan lead to new blockages later.

Carbon monoxide is poisonous, but it’s nat-urally released in low doses by cells of bloodvessels in response to surgical procedures.

In the new study, rats that inhaled car-bon monoxide–laced air for 1 hour beforeangioplasty had much less subsequentartery blockage than did rats not receiv-ing the gas, says study coauthor AugustineChoi, a pulmonologist at the Universityof Pittsburgh.

Rats that underwent a vessel transplantalso fared significantly better if given car-bon monoxide before and after the sur-gery, the researchers report in the Febru-ary Nature Medicine.

The researchers are currently testing thecarbon monoxide therapy on pigs, whoseresponses to these procedures closelyapproximate those of people. —N.S.

OFNOTE

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