IIt’s anyon’s game: the race to quantum computation
Jainendra K. Jain
Physics Department, 104 Davey Lab, Pennsylvania State University, University Park, Pennsylvania 16802, USA
In 1924, Satyendra Nath Bose dispatched a manuscript introducing the concept now known asBose statistics to Albert Einstein. Bose could hardly have imagined that the exotic statistics ofcertain emergent particles of quantum matter would one day suggest a route to fault-tolerant quan-tum computation. This non-technical Commentary on “anyons,” namely particles whose statisticsis intermediate between Bose and Fermi, aims to convey the underlying concept as well as its ex-perimental manifestations to the uninitiated.
Fractional statistics:
The quantum mechanical stateof a collection of particles is described by a complex val-ued wave function Ψ( { r j } ), with | Ψ( { r j } ) | giving theprobability that the particles are located at positions( r , r , · · · r N ). An exchange of two identical particlesdoes not produce a new state, and in particular, leavesthe probability | Ψ( { r j } ) | unchanged. This imposes cer-tain symmetry constraints on the allowed wave functions.Two possibilities are Ψ → +Ψ or Ψ → − Ψ under ex-change of two particles. Particles obeying the formerrelation are called bosons, and the latter fermions. Wespeak of Bose and Fermi statistics because the propertyof particles under exchange affects the counting of dis-tinct microscopic configurations and thus their statisticalmechanics and thermodynamics.Examples of fermions are electrons, quarks, protons,neutrons and He atoms; examples of bosons are theHiggs bosons, He atoms and photons. Even thoughparticle statistics may appear an abstract concept un-connected to our daily lives, it is central to our exis-tence. The Fermi statistics of electrons is responsible forthe structure of atoms and molecules, and thus perme-ates atomic physics, chemistry and biology; it governs, atthe most fundamental level, the properties of all matteraround and within us. The Bose statistics is responsiblefor dramatic phenomena at low temperatures, such asBose-Einstein condensation, superconductivity and Hesuperfluidity.It was recognized four decades ago that, in twospace dimensions, quantum mechanics also admits par-ticles that behave under exchange as Ψ → e iθ Ψ witharbitrary θ . To see the essential idea, let us imagineexchanging two particles along a counterclockwise pathshown in Fig. 1 (a), which produces a phase factor e iθ .(A clockwise exchange will produce e − iθ .) From the ref-erence frame of one of the particles, this is a half loop[Fig. 1 (b)]. A full loop [Fig. 1 (c)], called a winding orbraiding, represents two exchanges. The phase θ is calledthe (braiding) statistical phase. For bosons (fermions) wehave θ = mπ where m is an even (odd) integer. Particlesfor which θ is not an integer multiple of π are referredto as anyons, which stands for particles with any statis-tics. It is stressed that θ is a topological quantity, i.e. itdoes not depend on the size and shape of the exchangepath or the closed loop. Anyons can be defined only intwo space dimensions, because in higher dimensions the (a) ( b ) (c) ( d ) FIG. 1. (a) Exchange of two particles produces a phase fac-tor e iθ . (b) Particle exchange is equivalent to half a loop ofone particle around another. (c) A full loop equals two ex-changes. (d) In two dimensions the orange and green pathsare topologically distinct, but in higher dimensions they arenot. notion of a counterclockwise / clockwise exchange or ofa particle going around another is not well defined. Inthree dimensions, by lifting the paths off the page, onecan continuously deform the counterclockwise exchangeinto a clockwise one in Fig. 1 (a), or the orange loop intothe green loop in Fig. 1 (d); this implies e iθ = e − iθ andhence θ = mπ .Of course, just because anyons can be defined does notmean that they exist. Our world is three dimensional,and there certainly are no anyons in the list of parti-cles in a particle physics text book. Fortunately, thislist is incomplete. Interacting quantum matter generatesits own emergent particles, which can be rather exotic,and no principle of physics precludes the possibility thatsome of them might obey fractional braiding statistics. Aquantum state supporting such particles would have tobe a rather exotic state in two dimensions. Soon after theconcept of fractional statistics was proposed, nature pro-duced a promising candidate for its realization , namelythe fractional quantum Hall effect. Fractional quantum Hall effect:
Typically, electric cur-rent flows in the direction of an applied voltage. In the a r X i v : . [ phy s i c s . pop - ph ] A ug FIG. 2. A light-hearted depiction of how interacting electronsin a magnetic field (upper row) capture magnetic flux quantato turn into noninteracting composite fermions (lower row).Illustration by Kwon Park. presence of a magnetic field B , however, it flows at anangle; that is, there is a voltage in the direction of thecurrent flow as well as across it. The Hall resistance R H is the ratio of transverse voltage to current. The laws ofclassical electrodynamics tell us that R H is proportionalto B , as seen routinely by experiments.Dramatic quantum phenomena are revealed, however,when electrons are confined to two dimensions, cooledto near zero Kelvin, and subjected to a strong magneticfield B . The primary observation is that as B is varied,the Hall resistance exhibits a series of plateaus preciselyquantized at R H = h/ ( νe ), where h is the Planck’s con-stant, e is electron’s charge, and ν is either an integer ora fraction. The appearance of such simple and universalvalues that are utterly oblivious to all of the complexi-ties of the sample is an amazing result that has fascinatedphysicists for decades.The observation of integer and fractional values of ν are referred to as the integer quantum Hall effect (IQHE) and fractional quantum Hall effect (FQHE),respectively. The IQHE can be understood using stan-dard methods, because it occurs in a theoretical modelof noninteracting electrons, and one can make convinc-ing arguments that the interaction does not destroy it. Incontrast, the FQHE results fundamentally due to the in-teraction between electrons, which produces certain com-plex liquids of strongly correlated electrons. Close to onehundred such FQH liquids, distinguished by ν , have beenobserved so far in a variety of two-dimensional materials,such as semiconductor quantum wells and graphene. Composite fermions:
There are several ways of view-ing the FQHE. The simplest understanding of its originis achieved in terms of an emergent particle called thecomposite fermion . The composite fermion is an un-usual particle: it is often visualized as the bound stateof an electron and an even integer number (2 p ) of mag-netic flux quanta, where a flux quantum is defined as φ = h/e . (This picture of composite fermion is notto be taken literally, but is sufficient for many purposes.) FIG. 3. Extracting the statistical phase for quasiparticles, i.e.excited composite fermions, shown in blue. The compositefermions belonging to the ground state are shown in green.The statistical phase for the quasiparticles is given by thephase associated with loop on the left, which encloses anotherquasiparticle, minus the phase associated with the loop on theright, which does not.
Tremendous simplification occurs because while electronsare strongly interacting, composite fermions are, to agood approximation, noninteracting [Fig. 2]. In otherwords, the primary role of the interaction between elec-trons in the FQH regime is simply to create compositefermions. (How many flux quanta composite fermionscapture depends on B .) Many predictions of compositefermions have been confirmed. In particular, the IQHE ofcomposite fermions appears as the FQHE of electrons atfractions ν = n/ (2 pn ± n and p being integers, whichexplains almost all of the observed fractions. Compos-ite fermions have been directly observed, and provide anatural framework for understanding various propertiesof the FQH liquids.What about fractional statistics? The excitationsof the FQH liquids are nothing but excited compositefermions, referred to as quasiparticles. It turns out thatthese quasiparticles obey fractional braiding statistics.The gist of the idea is as follows. Consider a closedloop of a quasiparticle, shown in blue in Fig. 3, wind-ing around another quasiparticle. The phase associatedwith the loop can be calculated, but it is complicated: inaddition to the statistical phase 2 θ , it also involves a con-tribution from the composite fermions in the ground state FIG. 4. Experimental realization of the scheme in Fig. 3,using interference of quasiparticles moving along the sam-ple edges. Quasiparticles are injected at the upper left edge,and measured at the lower left edge. The two relevant pathsare shown in red and orange, artificially displaced in the re-gion where they coincide. The vertical dotted lines indicatequantum mechanical tunneling across a barrier. Compositefermions belonging to the ground state are not shown to avoidclutter. (shown in green) as well as a phase (called the Aharonov-Bohm phase) that a charged particle acquires when itmoves in the presence of a magnetic field. Fortunately,these other path-dependent contributions can be elimi-nated by subtracting the phase for an identical loop thatdoes not enclose any quasiparticle (see Fig. 3). A micro-scopic calculation gives 2 θ = 4 πp/ (2 pn ±
1) modulo 2 π for the quasiparticles of the ν = n/ (2 pn ±
1) FQH state,consistent with earlier results . Experimental evidence:
To measure the statisticalphase experimentally, one first needs to produce well de-fined paths. Here another property of quantum Hall ef-fect comes in handy: the current flows along the edgesof the sample, and moreover, on a given edge it flowsonly in one direction. This suggests the geometry shownin Fig. 4. A composite fermion injected into the sam-ple at the upper left edge has several choices. The mostprobable outcome is that it continues to ride the upperedge and escapes to the right. However, it can also tun-nel into the lower edge where the two edges come close.The probability of its appearing at the lower left edgedepends, according to the laws of quantum mechanics,on the phase difference between the red and the orangepaths, which is precisely the phase associated with theloop encircling the inner island. The change in this phasewhen a quasiparticle is added to the interior will give thestatistical phase 2 θ , as discussed above.Impressive progress has been made by several ex-perimental groups toward detecting fractional braidingstatistics using this geometry . One complicating fac-tor is that if the paths are too long, the composite fermionforgets its phase along the way, whereas if they aretoo short, another phenomenon called Coulomb block-ade comes into play. Overcoming these obstacles with aclever sample design, a recent experiment has reportedthe most convincing observation so far of discrete phase slips of − π/ ν = 1 /
3, precisely as expected fromfractional braiding statistics for the quasiparticles of the ν = 1 / Quantum computation with anyons?
As physicistswork further to better understand, confirm, and probefractional statistics for the excitations of various FQHstates, and search for other phenomena arising fromthem, an outsider may wonder: How will fractionalstatistics affect my life? While it would be unwise tomake predictions, because new ideas often surprise usby opening unanticipated directions, a dream applica-tion would be topological quantum computation . Thiswould actually require more complex anyons than theones discussed above, called “non-Abelian anyons.” Forsuch anyons the outcome of a sequence of windings de-pends on the order in which they are performed. A spe-cial case of such anyons, called Majorana particles, aretheoretically believed to occur in the ν = 5 / . This state has also been investigated in inter-ference experiments . Intense search for Majoranaparticles is also ongoing in contexts outside of the FQHE.How can they help with quantum computation? A funda-mental impediment to quantum computation is decoher-ence due to interaction with the environment. Here, onecan imagine a topological qubit (quantum bit) in whichinformation is stored non-locally over two distant Ma-joranas, and thus immune to decoherence by local fluc-tuations in the environment. The realization of a quan-tum computer based on non-Abelian anyons of the FQHEwould be a fitting legacy of fractional statistics.I thank Yayun Hu for discussions. Financial supportfrom the US Department of Energy, Office of Basic En-ergy Sciences, under Grant No. DE-SC0005042 is grate-fully acknowledged. Leinaas, J. M., and Myrheim, J.
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