Computing with Quantum Knots

Home , Quantum topology, Sankar Das Sarma

56 SCIENTIFIC AMERICAN A PRIL 2006 COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC. BY BRAIDING WORLD LINES (trajectories) of special particles, one can perform a quantum computation that is impossible for any ordinary (classical) computer. The particles live in a ﬂuid known as a two-dimensional electron gas.

Computing with Quantum Knots A machine based on bizarre particles called anyons that represents a calculation as a set of braids in spacetime might be a shortcut to practical quantum computation

uantum computers promise to perform calculations believed to be impossible for ordinary computers. Some of those calculations are of Qgreat real-world importance. For example, certain widely used encryption methods could be cracked given a computer capable of breaking a large number into its component factors within a reasonable length of time. Virtually all encryption methods used for highly sensitive data are vulnerable to one quantum algorithm or another. The extra power of a quantum computer comes about because it operates on information represented as qubits, or quantum bits, instead of bits. An ordinary classical bit can be either a 0 or a 1, and standard microchip architectures enforce that dichotomy rigorously. A qubit, in contrast, can be in a so-called superposition state, which entails proportions of 0 and 1 coexisting together. One can think of the possible qubit states as points on a sphere. The north pole is a classical 1, the south pole a 0, and all the points in between are all the possible superpositions of 0 and 1 [see “Rules for a Complex Quantum World,” by Michael A. Nielsen; Scien- tiﬁc American, November 2002]. The freedom that qubits have to roam across the entire sphere helps to give quantum computers their unique capabilities. Unfortunately, quantum computers seem to be extremely difﬁcult to build. The qubits are typically expressed as certain quantum properties of trapped particles,

GEORGE RETSECK By Graham P. Collins such as individual atomic ions or electrons. But their superposition states are ex-

www.sciam.com SCIENTIFIC AMERICAN 57 COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC. ceedingly fragile and can be spoiled by squashing and bending but not by cut- are instead quasiparticles—excitations the tiniest stray interactions with the ting or joining. It embraces such subjects in a two-dimensional electronic system ambient environment, which includes as knot theory. Small perturbations do that behave a lot like the particles and all the material making up the computer not change a topological property. For antiparticles of high-energy physics. itself. If qubits are not carefully isolated example, a closed loop of string with a And as a further complication, the qua- from their surroundings, such distur- knot tied in it is topologically different siparticles are of a special type called bances will introduce errors into the from a closed loop with no knot [see anyons, which have the desired mathe- computation. box on opposite page]. The only way to matical properties. Most schemes to design a quantum change the closed loop into a closed Here is a what a computation might computer therefore focus on finding loop plus knot is to cut the string, tie the look like: ﬁrst, create pairs of anyons ways to minimize the interactions of the knot and then reseal the ends of the and place them along a line [see box on qubits with the environment. Research- string together. Similarly, the only way page 60]. Each anyon pair is rather like ers know that if the error rate can be to convert a topological qubit to a dif- a particle and its corresponding antipar- reduced to around one error in every ferent state is to subject it to some such ticle, created out of pure energy. 10,000 steps, then error-correction pro- violence. Small nudges from the envi- Next, move pairs of adjacent anyons cedures can be implemented to compen- ronment will not do the trick. around one another in a carefully deter- At ﬁrst sight, a topological quantum computer does not seem much like a computer at all.

sate for decay of individual qubits. Con- At first sight, a topological quantum mined sequence. Each anyon’s world structing a functional machine that has computer does not seem much like a line forms a thread, and the movements a large number of qubits isolated well computer at all. It works its calculations of the anyons as they are swapped this enough to have such a low error rate is a on braided strings—but not physical way and that produce a braiding of all daunting task that physicists are far strings in the conventional sense. Rath- the threads. The quantum computation from achieving. er, they are what physicists refer to as is encapsulated in the particular braid so A few researchers are pursuing a very world lines, representations of particles formed. The final states of the anyons, different way to build a quantum com- as they move through time and space. which embody the result of the computa- puter. In their approach the delicate (Imagine that the length of one of these tion, are determined by the braid and quantum states depend on what are strings represents a particle’s movement not by any stray electric or magnetic in- known as topological properties of a through time and that its thickness rep- teraction. And because the braid is topo- physical system. Topology is the math- resents the particle’s physical dimen- logical—nudging the threads a little bit ematical study of properties that are un- sions.) Moreover, even the particles in- this way and that does not change the changed when an object is smoothly de- volved are unlike the electrons and pro- braiding—it is inherently protected from formed, by actions such as stretching, tons that one might first imagine. They outside disturbances. The idea of using anyons to carry out computations in this Overview/Quantum Braids fashion was proposed in 1997 by Alexei Y. Kitaev, now at Microsoft. ■ Quantum computers promise to greatly exceed the abilities of classical Michael H. Freedman, now at Mi- computers, but to function at all, they must have very low error rates. crosoft, lectured at Harvard University Achieving the required low error rates with conventional designs is far beyond in the fall of 1988 on the possibility of current technological capabilities. using quantum topology for computa- ■ An alternative design is the so-called topological quantum computer, which tion. These ideas, published in a research would use a radically different physical system to implement quantum paper in 1998, built on the discovery computation. Topological properties are unchanged by small perturbations, that certain mathematical quantities leading to a built-in resistance to errors such as those caused by stray known as knot invariants were associ- interactions with the surrounding environment. ated with the quantum physics of a two- ■ Topological quantum computing would make use of theoretically postulated dimensional surface evolving in time. If excitations called anyons, bizarre particlelike structures that are possible in an instance of the physical system could a two-dimensional world. Experiments have recently indicated that anyons be created and an appropriate measure- exist in special planar semiconductor structures cooled to near absolute zero ment carried out, the knot invariant and immersed in strong magnetic fields. would be approximately computed auto- matically instead of via an inconvenient-

Topology of a closed loop (a) is unaltered if the string is pushed by moving around the string. Instead one must cut the string, tie the around to form another shape (b) but is different from that of knot and rejoin the ends. Consequently, the topology of the loop a closed loop with a knot tied in it (c). The knot cannot be formed just is insensitive to perturbations that only push the string around.

a b c

= =

ly long calculation on a conventional it at various velocities, and so on. For when two particles are swapped. You computer. Equally diffi cult problems of example, a particle is most likely to be might say that their wave functions are more real-world importance would have found in a region where the wave func- multiplied by a factor of plus one. similar shortcuts. tion has a large amplitude. Deep mathematical reasons require Although it all sounds like wild theo- A pair of electrons is described by a that quantum particles in three dimen- rizing quite removed from reality, recent joint wave function, and when the two sions must be either fermions or bosons. experiments in a fi eld known as frac- electrons are exchanged, the resulting In two dimensions, another possibility tional quantum Hall physics have put joint wave function is minus one times arises: the factor might be a complex the anyon scheme on firmer footing. the original. That changes peaks of the phase. A complex phase can be thought Further experiments have been proposed wave into troughs, and vice versa, but it of as an angle. Zero degrees corresponds to carry out the rudiments of a topolog- has no effect on the amplitude of the to the number one; 180 degrees is minus ical quantum computation. oscillations. In fact, it does not change one. Angles in-between are complex any measurable quantity of the two numbers. For example, 90 degrees cor- Anyons electrons considered by themselves. responds to i, the square root of minus as previously mentioned, a to- What it does change is how the elec- one. As with a factor of minus one, mul- pological quantum computer braids trons might interfere with other elec- tiplying a wave function by a phase has world lines by swapping the positions of trons. Interference occurs when two absolutely no effect on the measured particles. How particles behave when waves are added together. When two properties of the individual particle, be- swapped is one of the many ways that waves interfere, the combination has a cause all that matters for those proper- quantum physics differs fundamentally high amplitude where peaks of one align ties are the amplitudes of the oscilla- from classical physics. In classical phys- with peaks of the other (“constructive tions of the wave. Nevertheless, the ics, if you have two electrons at locations interference”) and has a low amplitude phase can change how two complex A and B and you interchange their posi- where peaks align with troughs (“de- waves interfere. tions, the fi nal state is the same as the structive interference”). Multiplying Particles that pick up a complex initial state. Because the electrons are one of the waves by a phase of minus phase on being swapped are called any- indistinguishable, so, too, are the initial one interchanges its peaks and troughs ons because any complex phase might and fi nal states. Quantum mechanics is and thus changes constructive interfer- appear, not just a phase of plus or minus not so simple. ence, a bright spot, to destructive inter- one. Particles of a given species, how- The difference arises because quan- ference, a dark spot. ever, always pick up the same phase. tum mechanics describes the state of a It is not just electrons that pick up a particle with a quantity called the wave factor of minus one in this way but also Electrons in Flatland function, a wave in space that encapsu- protons, neutrons and in general any anyons exist only in a two-di- lates all the properties of the particle— particle of a class called fermions. Bo- mensional world. How can we produce the probability of fi nding it at various sons, the other chief class of particles, pairs of them for topological computing

GEORGE RETSECK locations, the probability of measuring have wave functions that are unchanged when we live in three dimensions? The

BRAIDING Just two basic moves in a plane—a clockwise swap and a counterclockwise swap—generate all the possible braidings of the world lines (trajectories through spacetime) of a set of anyons. CLOCKWISE SWAP RESULTING BRAID COUNTERCLOCKWISE SWAP RESULTING BRAID

Anyons = =

Time Time

COMPUTING First, pairs of anyons are created and lined up in a row to represent the qubits, or quantum bits, of the computation. The anyons are moved around by swapping the positions of adjacent anyons in a particular sequence. These moves correspond to operations performed on the qubits. Finally, pairs of adjacent anyons are brought together and measured to produce the output of the computation. The output depends on the topology of the particular braiding produced by those manipulations. Small disturbances of the anyons do not change that topology, which makes the computation impervious to normal sources of errors.

Qubit

Time

BUILDING A LOGIC GATE

A logic gate known as a CNOT gate is produced by this complicated braiding—leaving one triplet in place and moving two anyons of the braiding of six anyons. A CNOT gate takes two input qubits and other triplet around its anyons—simplifi ed the calculations involved produces two output qubits. Those qubits are represented by triplets in designing the gate. This braiding produces a CNOT gate that is (green and blue) of so-called Fibonacci anyons. The particular style of accurate to about 10–3. answer lies in the fl atland realm of qua- sional electron gases, particularly when magnetic flux around with them as siparticles. Two slabs of gallium arse- the systems are immersed in high trans- though the fl ux were an integral part of nide semiconductor can be carefully en- verse magnetic fi elds at extremely low the particle. In 2005 Vladimir J. Gold- gineered to accommodate a “gas” of temperatures, because of the unusual man, Fernando E. Camino and Wei electrons at their interface. The elec- quantum properties exhibited under Zhou of Stony Brook University claimed trons move freely in the two dimensions these conditions. to have direct experimental confi rma- of the interface but are constrained from For example, in the fractional quan- tion that quasiparticles occurring in the moving in the third dimension, which tum Hall effect, excitations in the elec- fractional quantum Hall state are any- would take them off the interface. Phys- tron gas behave like particles having a ons, a crucial fi rst step in the topological icists have intensely studied such sys- fraction of the charge of the electron. approach to quantum computation.

tems of electrons, called two-dimen- Other excitations carry units of the Some researchers, however, still seek in- GEORGE RETSECK

60 SCIENTIFIC AMERICAN A PRIL 2006 COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC. dependent lines of evidence for the qua- you have three identical anyons in a quasiparticles are indeed nonabelian. siparticles’ anyonic nature because cer- row, at positions A, B and C. First swap Experiments have been proposed to set- tain nonquantum effects could conceiv- the anyons at positions A and B. Next tle that question. One was suggested by ably produce the results seen by swap the anyons now located at B and Freedman, along with Sankar Das Sar- Goldman and his colleagues. C. The result will be the original wave ma of the University of Maryland at In two dimensions, an important function modifi ed by some factor. Sup- College Park and Chetan Nayak of Mi- new issue arises in the swapping of two pose instead that the anyons at B and C crosoft, with important refinements particles: Do the particles follow clock- are swapped fi rst, followed by swap- proposed by Ady Stern of the Weizmann wise tracks or counterclockwise tracks ping those at A and B. If the result is the Institute in Israel and Bertrand Halperin as they are interchanged? The phase wave function multiplied by the same of Harvard University; the second was picked up by the wave function depends factor as before, the anyons are called presented by Kitaev, Parsa Bonderson of on that property. The two alternative abelian. If the factors differ depending the California Institute of Technology paths are topologically distinct, because on the order of the swapping, they are and Kirill Shtengel, now at the Univer- the experimenter cannot continuously nonabelian anyons. (The nonabelian sity of California, Riverside. deform the clockwise paths into coun- property arises because for these an- terclockwise paths without crossing the yons, the factor that multiplies the wave Braids and Gates paths and having the particles collide function is a matrix of numbers, and once you have nonabelian anyons, somewhere. the result of multiplying two matrices you can generate a physical representa- To build a topological quantum depends on the order in which they are tion of what is called the braid group. computer requires one additional com- multiplied.) This mathematical structure describes plication: the anyons must be what is The experiment by Goldman’s team all the ways that a given row of threads called nonabelian. This property means involved abelian anyons. Nevertheless, can be braided together. Any braid can that the order in which particles are theorists have strong reason to believe be built out of a series of elementary op- swapped is important. Imagine that that certain fractional quantum Hall erations in which two adjacent threads are moved, by either a clockwise or a PREVENTING RANDOM ERRORS counterclockwise motion. Every possible sequence of anyon manipulations Errors will occur in a topological computation if thermal fl uctuations generate a corresponds to a braid, and vice versa. stray pair of anyons that intertwine with the braid of the computation before they Also corresponding to each braid is a self-annihilate. Those strays will then corrupt (red lines) the computation. The very complicated matrix, the result of probability of this interference drops exponentially with the distance that the combining all the individual matrices of anyons travel, however. The error rate can be minimized by keeping the every anyon exchange. computation anyons suffi ciently far apart (bottom pair). Now we have all the elements in place to see how these braids correspond to a quantum computation. In a conventional computer, the state of the computer is represented by the combined state of all its bits—the particular sequence of 0s and 1s in its register. Simi- larly, a quantum computer is represented by the combined state of all its qubits. In a topological quantum computer, the qubits may be represented by groups of Strays anyons. In a quantum computer, the process of going from the initial state of all the qubits to the fi nal state is described by a matrix that multiplies the joint wave function of all the qubits. The similarity to what happens in a topological quantum computer is obvious: in that case, the matrix is the one associated with the particular braid corresponding to the Time sequence of anyon manipulations. Thus,

GEORGE RETSECK we have veriﬁ ed that the operations car-

www.sciam.com SCIENTIFIC AMERICAN 61 COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC. TOPOLOGICAL ERRORS ried out on the anyons result in a quan- ANYON DETECTOR Boundary tum computation. Electrode current The device in this colorized micrograph was used Electron gas Another important feature must be by Vladimir J. Goldman and his co-workers to demonstrate that certain quasiparticles confi rmed: Can our topological quan- (excitations in the quantum Hall state) behave as tum computer perform any computation anyons. The device was cooled to 10 millikelvins that a conventional quantum computer and put in a strong magnetic fi eld. A two- Electron can? Freedman, working with Michael dimensional electron gas formed around the four gas electrodes, with different types of quasiparticles Larsen of Indiana University and Zheng- present in the yellow and green areas. han Wang, now at Microsoft, proved in Characteristics of the current fl owing along the 2002 that a topological quantum com- boundary confi rmed that the quasiparticles in the puter can indeed simulate any computa- yellow island were anyonic. tion of a standard quantum computer, NOT GATE but with one catch: the simulation is ap- This proposed anyonic NOT gate is based on a fractional quantum Hall state involving anyons proximate. Yet given any desired accu- having one-quarter the charge of an electron. Electrodes induce two islands on which anyons racy, such as one part in 104, a braid can can be trapped. Current fl ows along the boundary but under the right conditions can also be found that will simulate the required tunnel across the narrow isthmuses. computation to that accuracy. The fi ner 1 Initialize the gate by putting two anyons (blue) on one island and then applying voltages to the accuracy required, the greater the transfer one anyon to the other island. This pair of anyons represents the qubit in its initial number of twists in the braid. Fortu- state, which can be determined by measuring the current fl ow along the neighboring boundary. nately, the number of twists required increases very slowly, so it is not too diffi - Boundary cult to achieve very high accuracy. The Anyon current proof does not, however, indicate how to determine which actual braid corre- — Electron gas sponds to a computation that depends on the specific design of topological quantum computer, in particular the

Island ) Electrode species of anyons employed and their relation to elementary qubits. The problem of fi nding braids for do- ing specifi c computations was tackled in micrograph 2 To fl ip the qubit (the NOT operation), apply voltages to induce one anyon from the boundary (green) to tunnel across the device. 2005 by Nicholas E. Bonesteel of Florida State University, along with colleagues

there and at Lucent Technologies’s Bell VOL. 95; 2005 (

Laboratories. The team showed explic- , itly how to construct a so-called con- trolled NOT (or CNOT) gate to an accuracy of two parts in 103 by braiding six anyons. A CNOT gate takes two in- puts: a control bit and a target bit. If the control bit is 1, it changes the target bit PHYSICAL REVIEW LETTERS from 0 to 1, or vice versa. Otherwise the bits are unaltered. Acting on qubits, any 3 The passage of this anyon changes the phase relation of the two anyons so that the qubit’s computation can be built from a net- value is ﬂ ipped to the opposite state (red). work of CNOT gates and one other operation—the multiplication of individual qubits by a complex phase. This result

serves as another conﬁ rmation that to- ); FROM F. E. CAMINO ET AL. IN pological quantum computers can perform any quantum computation. illustration Quantum computers can perform ( feats believed to be impossible for classical computers. Is it possible that a topological computer is more powerful than

a conventional quantum computer? An- GEORGE RETSECK

62 SCIENTIFIC AMERICAN A PRIL 2006 COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC. other theorem, proved by Freedman, relation in which they began their lives. creased. That exponential factor is the Kitaev and Wang, shows that is not the Also, it is not true that a topological essential contribution of topology, and case. They demonstrated that the opera- computer is totally immune to errors. it has no analogue in the more tradition- tion of a topological quantum computer The main source of error is thermal fluc- al approaches to quantum computing. can be simulated efficiently to arbitrary tuations in the substrate material, which The promise of extraordinarily low accuracy on a conventional quantum can generate an extra pair of anyons. error rates—many orders of magnitude computer, meaning that anything that a Both the anyons then intertwine them- lower than those achieved by any other topological quantum computer can selves with the braid of the computation, quantum computation scheme to date— compute a conventional quantum com- and finally the pair annihilates again is what makes topological quantum puter can also compute. This result sug- [see box on page 61]. Fortunately, the computing so attractive. Also, the tech- gests a general theorem: any sufficiently thermal generation process is sup- nologies involved in making fractional advanced computation system that pressed at the low temperature at which quantum Hall devices are mature, being makes use of quantum resources has ex- a topological computer would operate. precisely those of the microchip indus- actly the same computational abilities. Furthermore, the probability of the en- try; the only catch is that the devices (An analogous thesis for classical com- tire bad process occurring decreases ex- have to operate at extremely low tem- puting was proposed by Alonzo Church ponentially as the distance traveled by peratures— on the order of millikel- and Alan Turing in the 1930s.) the interlopers increases. Thus, one can vins—for the magical quasiparticles to achieve any required degree of accuracy be stable. Particles In, Answers Out by building a sufficiently large computer If nonabelian anyons actually exist, i have glossed over two process- and keeping the working anyons far topological quantum computers could es that are crucial to building a practical enough apart as they are braided. well leapfrog conventional quantum The trio estimated that the error rate for their NOT gate would be 10–30 or less.

topological quantum computer: the ini- Topological quantum computing re- computer designs in the race to scale up tialization of the qubits before the start mains in its infancy. The basic working from individual qubits and logic gates to of the computation and the readout of elements, nonabelian anyons, have not fully ﬂedged machines more deserving the answer at the end. yet been demonstrated to exist, and the of the name “computer.” Carrying out The initialization step involves gen- simplest of logic gates has yet to be built. calculations with quantum knots and erating quasiparticle pairs, and the prob- The previously mentioned experiment braids, a scheme that began as an eso- lem is knowing what species of quasi- of Freedman, Das Sarma and Nayak teric alternative, could become the stan- particle has been created. The basic pro- would achieve both those goals—if the dard way to implement practical, error- cedure is to pass test anyons around the anyons involved do turn out to be nona- free quantum computation. generated pairs and then measure how belian, as expected, the device would the test anyons have been altered by that carry out the logical NOT operation on Graham P. Collins is a staff writer process, which depends on the species of the qubit state. The trio estimated that and editor, with a Ph.D. in physics the anyons that they have passed. (If a the error rate for the process would be from Stony Brook University. He test anyon is altered, it will no longer be 10–30 or less. Such a tiny error rate oc- wishes to thank Michael H. Freedman, cleanly annihilated with its partner.) curs because the probability of errors is the director of Project Q at Microsoft, Anyon pairs not of the required type exponentially suppressed as the temper- for contributions in the preparation would be discarded. ature is lowered and the length scale in- of this article. The readout step also involves measuring anyon states. While the anyons MORE TO EXPLORE are widely separated, that measurement Topologically Protected Qubits from a Possible Non-Abelian Fractional Quantum Hall State. is impossible: the anyons must be Sankar Das Sarma, Michael Freedman and Chetan Nayak in Physical Review Letters, Vol. 94, brought together in pairs to be measured. pages 166802-1–168802-4; April 29, 2005. Roughly speaking, it is like checking to Devices Based on the Fractional Quantum Hall Effect May Fulfill the Promise of Quantum see if the pairs annihilate cleanly, like Computing. Charles Day in Physics Today, Vol. 58, pages 21–24; October 2005. true antiparticles, or if they leave behind Anyon There? David Lindley in Physical Review Focus, Vol. 16, Story 14; November 2, 2005. residues of charge and ﬂux, which re- http://focus.aps.org/story/v16/st14 veals how their states have been altered Topological Quantum Computation. John Preskill. Lecture notes available at by braiding from the exact antiparticle www.theory.caltech.edu/˜preskill/ph219/topological.pdf