It'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 as Bose statistics to Albert Einstein. Bose could hardly have imagined that the exotic statistics of certain 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 statistics is 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 state of a collection of particles is described by a complex val- 2 ued wave function Ψ(fr jg), with jΨ(fr jg)j giving the probability that the particles are located at positions (r 1; r 2; ··· r N ). An exchange of two identical particles does not produce a new state, and in particular, leaves 2 the probability jΨ(fr jg)j unchanged. This imposes cer- (a) （b） 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 former relation are called bosons, and the latter fermions. We speak of Bose and Fermi statistics because the property of particles under exchange affects the counting of dis- tinct microscopic configurations and thus their statistical mechanics and thermodynamics. (c) （d） Examples of fermions are electrons, quarks, protons, 3 neutrons and He atoms; examples of bosons are the FIG. 1. (a) Exchange of two particles produces a phase fac- Higgs bosons, 4He atoms and photons. Even though tor eiθ. (b) Particle exchange is equivalent to half a loop of particle statistics may appear an abstract concept un- one particle around another. (c) A full loop equals two ex- connected to our daily lives, it is central to our exis- changes. (d) In two dimensions the orange and green paths tence. The Fermi statistics of electrons is responsible for are topologically distinct, but in higher dimensions they are the structure of atoms and molecules, and thus perme- not. ates atomic physics, chemistry and biology; it governs, at the most fundamental level, the properties of all matter around and within us. The Bose statistics is responsible notion of a counterclockwise / clockwise exchange or of for dramatic phenomena at low temperatures, such as a particle going around another is not well defined. In Bose-Einstein condensation, superconductivity and 4He three dimensions, by lifting the paths off the page, one superfluidity. can continuously deform the counterclockwise exchange 1,2 into a clockwise one in Fig. 1 (a), or the orange loop into It was recognized four decades ago that, in two iθ −iθ space dimensions, quantum mechanics also admits par- the green loop in Fig. 1 (d); this implies e = e and ticles that behave under exchange as Ψ ! eiθΨ with hence θ = mπ. arbitrary θ. To see the essential idea, let us imagine Of course, just because anyons can be defined does not exchanging two particles along a counterclockwise path mean that they exist. Our world is three dimensional, shown in Fig. 1 (a), which produces a phase factor eiθ. and there certainly are no anyons in the list of parti- (A clockwise exchange will produce e−iθ.) From the ref- cles in a particle physics text book. Fortunately, this arXiv:2008.06338v1 [physics.pop-ph] 8 Aug 2020 erence frame of one of the particles, this is a half loop list is incomplete. Interacting quantum matter generates [Fig. 1 (b)]. A full loop [Fig. 1 (c)], called a winding or its own emergent particles, which can be rather exotic, braiding, represents two exchanges. The phase θ is called and no principle of physics precludes the possibility that the (braiding) statistical phase. For bosons (fermions) we some of them might obey fractional braiding statistics. A have θ = mπ where m is an even (odd) integer. Particles quantum state supporting such particles would have to for which θ is not an integer multiple of π are referred be a rather exotic state in two dimensions. Soon after the to as anyons, which stands for particles with any statis- concept of fractional statistics was proposed, nature pro- 3,4 tics. It is stressed that θ is a topological quantity, i.e. it duced a promising candidate for its realization , namely does not depend on the size and shape of the exchange the fractional quantum Hall effect. path or the closed loop. Anyons can be defined only in Fractional quantum Hall effect: Typically, electric cur- two space dimensions, because in higher dimensions the rent flows in the direction of an applied voltage. In the 2 FIG. 3. Extracting the statistical phase for quasiparticles, i.e. excited composite fermions, shown in blue. The composite fermions belonging to the ground state are shown in green. The statistical phase for the quasiparticles is given by the phase associated with loop on the left, which encloses another quasiparticle, minus the phase associated with the loop on the FIG. 2. A light-hearted depiction of how interacting electrons right, which does not. in a magnetic field (upper row) capture magnetic flux quanta to turn into noninteracting composite fermions (lower row). Illustration by Kwon Park. Tremendous simplification occurs because while electrons are strongly interacting, composite fermions are, to a presence of a magnetic field B, however, it flows at an good approximation, noninteracting [Fig. 2]. In other angle; that is, there is a voltage in the direction of the words, the primary role of the interaction between elec- trons in the FQH regime is simply to create composite current flow as well as across it. The Hall resistance RH is the ratio of transverse voltage to current. The laws of fermions. (How many flux quanta composite fermions capture depends on B.) Many predictions of composite classical electrodynamics tell us that RH is proportional to B, as seen routinely by experiments. fermions have been confirmed. In particular, the IQHE of Dramatic quantum phenomena are revealed, however, composite fermions appears as the FQHE of electrons at when electrons are confined to two dimensions, cooled fractions ν = n=(2pn ± 1), n and p being integers, which to near zero Kelvin, and subjected to a strong magnetic explains almost all of the observed fractions. Compos- field B. The primary observation is that as B is varied, ite fermions have been directly observed, and provide a the Hall resistance exhibits a series of plateaus precisely natural framework for understanding various properties 2 of the FQH liquids. quantized at RH = h=(νe ), where h is the Planck's con- stant, e is electron's charge, and ν is either an integer or What about fractional statistics? The excitations a fraction. The appearance of such simple and universal of the FQH liquids are nothing but excited composite values that are utterly oblivious to all of the complexi- fermions, referred to as quasiparticles. It turns out that ties of the sample is an amazing result that has fascinated these quasiparticles obey fractional braiding statistics. physicists for decades. The gist of the idea is as follows. Consider a closed The observation of integer and fractional values of loop of a quasiparticle, shown in blue in Fig. 3, wind- ν are referred to as the integer quantum Hall effect5 ing around another quasiparticle. The phase associated (IQHE) and fractional quantum Hall effect6 (FQHE), with the loop can be calculated, but it is complicated: in respectively. The IQHE can be understood using stan- addition to the statistical phase 2θ, it also involves a con- dard methods, because it occurs in a theoretical model tribution from the composite fermions in the ground state of noninteracting electrons, and one can make convinc- ing arguments that the interaction does not destroy it. In contrast, the FQHE results fundamentally due to the in- teraction between electrons, which produces certain com- plex liquids of strongly correlated electrons. Close to one hundred such FQH liquids, distinguished by ν, have been observed so far in a variety of two-dimensional materials, such as semiconductor quantum wells and graphene. Composite fermions: There are several ways of view- FIG. 4. Experimental realization of the scheme in Fig. 3, ing the FQHE. The simplest understanding of its origin using interference of quasiparticles moving along the sam- is achieved in terms of an emergent particle called the 7 ple edges. Quasiparticles are injected at the upper left edge, composite fermion . The composite fermion is an un- and measured at the lower left edge. The two relevant paths usual particle: it is often visualized as the bound state are shown in red and orange, artificially displaced in the re- of an electron and an even integer number (2p) of mag- gion where they coincide. The vertical dotted lines indicate netic flux quanta, where a flux quantum is defined as quantum mechanical tunneling across a barrier. Composite φ0 = h=e. (This picture of composite fermion is not fermions belonging to the ground state are not shown to avoid to be taken literally, but is sufficient for many purposes.) clutter. 3 (shown in green) as well as a phase (called the Aharonov- slips of −2π=3 at ν = 1=3, precisely as expected from Bohm phase) that a charged particle acquires when it fractional braiding statistics for the quasiparticles of the moves in the presence of a magnetic field. Fortunately, ν = 1=3 FQH state. these other path-dependent contributions can be elimi- Quantum computation with anyons? As physicists nated by subtracting the phase for an identical loop that work further to better understand, confirm, and probe does not enclose any quasiparticle (see Fig.