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5

Towards Universal Topological Quantum Computation in the ν = 5/2 Fractional Quantum HallState

Michael Freedman,1 Chetan Nayak,1, 2 and Kevin Walker11Microsoft Research, Project Q, Kohn Hall, University of California, Santa Barbara, CA 931082Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547

The Pfaffian state, which may describe the quantized Hall plateau observed at Landau level filling fractionν = 5/2, can support topologically-protected qubits with extremely low error rates. Braiding operations alsoallow perfect implementation of certain unitary transformations of these qubits. However, in the case of thePfaffian state, this set of unitary operations is not quite sufficient for universal quantum computation (i.e. is notdense in the unitary group). If some topologically unprotected operations are also used, then the Pfaffian statesupports universal quantum computation, albeit with some operations which require error correction. On theother hand, if certain topology-changing operations can beimplemented, then fully topologically-protected uni-versal quantum computation is possible. In order to accomplish this, it is necessary to measure the interferencebetween quasiparticle trajectories which encircle other moving trajectories in a time-dependent Hall dropletgeometry.1

I. INTRODUCTION

The fractional quantum Hall regime2 contains a cornucopiaof Abelian fractional quantum Hall states, i.e. states whosequasiparticle excitations have Abelian braiding statistics3,4. Itis possible that an even more wonderful phenomenon may oc-cur there: non-Abelian quantum Hall states. The strongestcandidate is theν = 5/2 quantum Hall state. This state isquite robust in the highest-mobility samples5, and numericalstudies indicate that the Pfaffian state6,7, which has excitationsexhibiting non-Abelian braiding statistics8,9,10,11,12,13,14,15,16,has large overlap with the exact ground state for small num-bers of electrons at this filling fraction17,18. An experimenthas been proposed19 which would determine if theν = 5/2state is, indeed, in the universality class of the Pfaffian stateby observing the signature of non-Abelian statistics: a de-generate set of multi-quasiparticle states which cannot bedis-tinguished locally but can be distinguished by a non-Abeliananalogue of the Aharonov-Bohm interference measurement.In the two quasiparticle case, this is simply the observation ofa topologically-protected qubit20,21. In the presence of manypairs of quasiparticles, all kept far apart, the topologically-degenerate ground states form many qubits. These states arelocally indistinguishable. If the environment interacts onlylocally with the system, it cannot act on these qubits. Braid-ing the quasiparticles around each other, an intrinsicallynon-local operation, transforms the qubits. These gates are exactbecause small deformations of the qusiparticle trajectories donot affect their braiding topology. However, there is a sense inwhich the Pfaffian state is not quite non-Abelian enough: theset of all possible braiding operations only gives a finite set ofunitary transformations on the qubits. Thus, with these oper-ations, it is not possible to perform any desired unitary trans-formation, which would be necessary for a universal quantumcomputer.

In this paper, we suggest ways in which this apparent short-coming of the Pfaffian state (and, by implication, theν = 5/2quantum Hall state) can be circumvented. The first, morepedestrian, approach is to use some non-topological opera-tions. Consider, for instance, what happens when two quasi-

particles are brought close together. The degeneracy betweenthe two states of their qubit is broken. Since unitary evolu-tion in time will now cause a phase difference to develop be-tween the two states of the qubit, we thereby implement aphase gate. We explain how a universal quantum computercan be constructed using these ideas. The second, more inter-esting, approach relies on (1) the construction by Bravyi andKitaev24 of a universal set of gates for the Pfaffian state whichexploits topology change in an abstract context in which thereare no restrictions on the global topology of spacetime (using,for instance, overcrossings and undercrossings, which seemunlikely to be realized in a system of electrons confined to aplane ); and (2) the observation that their operations actuallycan be implemented in a way that remains entirely in the planeso long as one is able to measure the interference betweentrajectories encircling quasiparticles which are moving,merg-ing, and splitting – i.e. interference in a time-dependent back-ground Hall fluid. We note, in passing, thatν = 12/5 mayalso be a non-Abelian state, specifically one of the states pro-posed by Read and Rezayi25. It may be particularly interesting– even though it is seen more weakly – because, if it is in-deed a Read-Rezayi state25, braiding operations alone are suf-ficient to implement any unitary transformation within desiredaccuracy – i.e. it supports topologically-protected universalquantum computation26. In the event that theν = 12/5 stateproves to be simply an Abelian state or to have too small anenergy gap to permit manipulation, the protocols describedinthis paper, if they can be experimentally realized, would savethe day by boosting the computational power of theν = 5/2state so that it, too, can be regarded as a universal quantumcomputer. Furthermore, the basic architecture which we de-scribe in sections II and III is relevant to theν = 12/5 stateas well.

II. QUBITS IN THE PFAFFIAN STATE

In this paper, we will assume that theν = 5/2 plateau isin the universality class of the Pfaffian quantum Hall state.Inthis section, we list some basic facts about the Pfaffian quan-

2

tum Hall state and introduce some notation which will be use-ful in the following sections. The goal is to describe the qubitswhich arise when many quasiparticles35 are present.

The Pfaffian wavefunction6,7 takes the form:

Ψg.s.(zj) =∏

j<k

(zj − zk)2∏

j

e−|zj|2/4 · Pf(

1

zj − zk

)

.

(1)where the Pfaffian is the square root of the determinant of anantisymmetric matrix. It has Landau level filling factor1/2.(There is an obvious generalization to other even filling factorsand also to odd filling factors of bosonic particles.) it doesnotappear to be a good description of electrons at filling fraction1/2, which are in a metallic state down to the lowest observ-able temperatures (see ref. 2 and references therein). How-ever, it is a candidate for the half-filled first excited Landaulevel of the observedν = 5

2 = 2 + 12 quantum hall plateau5.

If we assume that the filled lowest Landau level of both spinsis inert and translate the Pfaffian wavefunction to the first ex-cited Landau level, it has high overlap with the exact groundstate wavefunction of small systems of electrons interactingthrough Coulomb interactions in a half-filled first excited Lan-dau level17,18. The Pfaffian state is also the exact ground stateof a certain three-body Hamiltonian7. While this three-bodyHamiltonian is unrealistic, it has the advantage that we canalso write down exact multi-quasihole wavefunctions. Sinceit appears from numerical studies of small systems that thisthree-body Hamiltonian is in the same universality class asthe actual Hamiltonian of the real system, we will assume thatthese multi-quasihole states capture the essential topologicalfeatures of the excitations of theν = 5/2 quantum Hall state.

The form of the Pfaffian factor in this wavefunction

Pf

(

1

zj − zk

)

= A(

1

z1 − z2

1

z3 − z4. . .

)

(2)

is strongly reminiscent of the real-space form of the BCSwavefunction. Indeed, the Pfaffian state may be viewed asa quantum Hall state ofp-wave paired fermions12,13,14.

The fundamental quasiparticles in this state carry half of aflux quantum and, therefore, chargee/4. A wavefunction fora two-quasihole state may be written as follows:

Ψg.s.(zj) =∏

j<k

(zj − zk)2∏

j

e−|zj|2/4×

Pf

(

(zj − η1) (zk − η2) + zj ↔ zkzj − zk

)

. (3)

When the two quasiholes atη1 andη2 are brought together atthe pointη, a single flux quantum quasiparticle results:

Ψg.s.(zj) =∏

j<k

(zj − zk)2∏

j

e−|zj|2/4×

∏

i

(zi − η) Pf

(

1

zj − zk

)

. (4)

The situation becomes more interesting when we considerstates with 4 quasiholes. A wavefunction with four quasiholes

atη1, η2, η3, η4 takes the form

Ψ(13)(24)(zj) =∏

j<k

(zj − zk)2∏

j

e−|zj|2/4 ×

Pf

(

(zj − η1)(zj − η3)(zk − η2)(zk − η4) + (j ↔ k)

zj − zk

)

(5)

However, there is another wavefunction with four quasiholesatη1, η2, η3, η4:

Ψ(14)(23)(zj) =∏

j<k

(zj − zk)2∏

j

e−|zj|2/4 ×

Pf

(

(zj − η1)(zj − η4)(zk − η2)(zk − η3) + (j ↔ k)

zj − zk

)

(6)

These two wavefunctions are linearly independent, but theyhave the same charge density profiles so long asη1, η2, η3, η4are far apart. In fact, they are indistinguishable by any localmeasurement. One might even think that there is a third four-quasihole stateΨ(12)(34), but this is not independent of theother two8,

Ψ(14)(23) −Ψ(12)(34) = x(

Ψ(13)(24) −Ψ(12)(34)

)

(7)

wherex = (η1 − η3) (η2 − η4) / (η1 − η4) (η2 − η3).It is enlightening to pick as the basis of the two dimensional

space of four-quasihole wavefunctionsΨ(13)(24) and the fol-lowing state11 (N is a normalization factor):

Ψ(13)(24) −Ψ(13)(24) = N∏

j<k

(zj − zk)2∏

j

e−|zj|2/4 ×

A(

z01z12

(z3 − η1)(z3 − η3)(z4 − η2)(z4 − η4)

z3 − z4

(z5 − η1)(z5 − η3)(z4 − η2)(z6 − η4)

z5 − z6. . .

)

(8)

The interpretation is that a Cooper pair can be broken andthe resulting neutral fermions put into zero modes11. In thiscase, the zero modes have wavefunctionsz0 andz1; in then-quasihole case, they have the formzk with 0 ≤ k ≤ n − 1.In fact, a Cooper pair can even be broken when there are onlytwo quasiparticles, but only one of the neutral fermions cango into a bulk zero mode; the other one must be at the edge.(Similarly, in the four-quasihole case, there are two additionalstates in which a neutral fermion is in one of the two zeromodes while the other one is at the edge.)

In this way, it can be seen that there are2n 2n-quasiholestates8,11 (half of them have a neutral fermion at the edge andthe other half don’t). It has been shown12 that precisely thesame degeneracy is obtained in ap + ip superconductorwhen there are2n flux hc/2e vortices present: there is onezero mode solution of the Bogoliubov-De Gennes equationsper vortex. These solutions are Majorana modes. Grouping

3

them into pairs, we haven fermionic levels, each of whichcan be occupied or unoccupied. By breaking Cooper pairs,we can change their occupancies.We interpret this degener-acy as n qubits, one qubit for each pair of quasiholes. (Ofcourse, the grouping of quasiholes into pairs is arbitrary andany two pairings are related by the a change of basis.)

Hence, we envision platform for quantum computation de-picted in Figure 1. Ann-qubit system can be created by en-dowing a Hall bar with2n antidots at which quasiholes arepinned. Each pair of quasiholes has a two-dimensional Hilbertspace spanned by|0〉 and|1〉, which correspond to the absenceor presence of a neutral fermion. In the following sections,wewill discuss how these qubits can be manipulated and mea-sured.

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FIG. 1: A system withn quasihole pairs (held at pairs of anti-dots,depicted as shaded circles) supportsn qubits. Additional antidots(hatched) can be used to move the quasiparticles, as described insection III.

These qubits will be manipulated by braiding quasiparti-cles, which causes states in this2n-dimensional Hilbert spacetransform into each other. To discuss these transformations,a different basis than (5) is useful. The effect of braidingquasiparticles is a combination of the explicit monodromy ofthe wavefunction and the Berry matrices obtained from adia-batic transport of theηis. The phase factors in (9) below havebeen chosen so that the latter are trivial and the former com-pletely encapsulate quasiparticle braiding properties8. (Wecould have worked with the basisΨ(13)(24), Ψ(14)(23), inwhich there is no explicit monodromy, but then we would haveto evaluate Berry matrix integrals9,10.)

Ψ(0,1)(zj) =(η13η24)

1

4

(1 ±√x)1/2

(

Ψ(13)(24) ± √x Ψ(14)(23)

)

(9)

whereη13 = η1 − η3, etc. andx = η14η23/η13η24. (Notethat we have taken a slightly different anharmonic ratiox thanin Ref. 8 in order to make (9) more compact than Eqs. (7.17),(7.18) of Ref. 8.) From this expression, we see, for instance,that takingη3 aroundη1 transformsΨ(0)(zj) intoΨ(1)(zj).

In the 2n quasihole case, the result can be stated asfollows8,13. The2n states of the system can be grouped into arepresentation of the Clifford algebra

{γi, γj} = 2δij (10)

with i, j = 1, 2, . . . , 2n. We could, for instance, organizethe states according to their eigenvaluesψ†

iψi = ±1, where

ψj = γ2j−1 + iγ2j , j = 1, 2, . . . , n. When quasiparticlesiandj are exchanged, the states transform according to8,13:

|Ψ〉 → e−π4γiγj |Ψ〉 (11)

These braiding matrices,τij = exp(

−π4 γiγj

)

will be a set oftopologically-protected unitary transformations which we canuse to manipulate our qubits.

Several important calculational and heuristic tools followfrom field theories for the Pfaffian state. While they illumi-nate this section and section III, they are somewhat technicaland take us away from the main line of our exposition, so wehave deferred a discussion of these field theories to appendixA. For reasons which are discussed there, it is convenientto call excitations which have the same braiding properties(up to Abelian phase factors) as, respectively, (a) the vacuum,(b) charge-e/4 quasiparticles, and (c) neutral fermions either1, σ, ψ or, equivalently, isospin0, 12 , 1.

Thus far, we have assumed that the only quasiparticles inour system are the quasiparticles which we have induced onour anti-dots. There could also be thermally-excited quasipar-ticles. They are the main source of error and their density wasestimated in ref. 19 to be exponentially small at low temper-atures. However, even at zero temperature, there will alwaysbe some quasiparticles which are trapped by local variationsin the potential, such as those cause by impurities. Assumingthat they cannot move, the effect of these quasiparticles canalways be accounted for with ‘software’, i.e. quantum compu-tations must be done with some more complicated algorithmswhich compensate for the presence of these stray quasiparti-cles. As a practical matter, however, we would like to makethem as benign as possible. To the extent that we can tunethe magnetic field to the center of the plateau and use gates tomove the edge of the system to avoid impurities (as in ref. 28),we should do so. If we can remove these localized quasiparti-cles with gate or a scanning probe microscope tip (such as anatomic-force microscope (AFM) tip) we should also attemptthis. Finally, there is one simplifying feature of the Pfaffianstate in particular, noted in refs. 28,29 is that there is an even-odd effect with quasiparticles. An even number of quasiparti-cles fuse to form a quasiparticle with Abelian statistics, whilean odd number of quasiparticles fuse to form a quasiparticlewith non-Abelian statistics. Hence, we handle stray localizedquasiparticles in the following way. We should associate eachstray quasiparticle with one of the anti-dots (most naturallythe anti-dot to which it is closest). Then, we want the num-ber of stray quasiparticles associated with each anti-dot to beeven. In this way, the degrees of freedom of the anti-dot arenot modified by its associates. Finally, we need to ensure thatthe quasiparticle braiding trajectories always encircle an evennumber of stray quasiparticles. Then, as may be seem from re-peated application of (11), the braiding matrices acting onthequbit Hilbert space are unaffected by the presence of the strayquasiparticles and the Hilbert space of the stray quasiparticleswill not become entangled with it.

4

III. BRAIDING AND INTERFEROMETRY

A. Braiding

The basic process by which we will manipulate many-quasiparticles states is the counterclockwise exchange de-picted in Figure 2. We suppose that the quasiparticles arelocalized at anti-dots and that they are transferred from oneanti-dot to another by varying the voltages on the anti-dots.With three anti-dots, an exchange can be performed. Twosuccessive exchanges results in a full braid of one quasipar-ticle around the other. We may need to move one quasiparti-cle greater distances – for instance, to take it around severalothers – in which case we could use an array of anti-dots as a‘bucket brigade’ (as in CCD devices such as digital cameras).

21

FIG. 2: The exchange of two qubits through a third anti-dot.

The process depicted in Figure 2 can be used, for instance,to exchange a quasiparticle from one qubit with a quasiparticlefrom a different qubit. Such a process, which applies the gateg3 (g1 andg2 will be introduced later) has a spacetime diagramwhich is depicted in Figure 14:

g3 =1√2

1 0 0 −i0 1 −i 00 −i 1 0−i 0 0 1

. (12)

This will be one of the basic gates used below.

One can imagine another possibility, which might becomerealistic at some point in the future: one quasiparticle couldbe dragged around another with a scanning probe microscopetip, e.g. an AFM tip, which could couple to a quasiparticleelectrostatically with the required spatial resolution (which ispresumably the magnetic length, on the order of 100A), asdepicted in Figure 3.

There is one other type of braiding process which we cando, namely taking a quasiparticle from the edge of the sys-tem around one of the quasiparticles in a qubit, as depicted inFigure 4. Such an process is a NOT gate for this qubit.

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FIG. 3: Using an AFM tip to braid quasiparticles.

N

1 2

A

B Q

PM

FIG. 4: Taking a quasiparticle from the edge around one of thequasi-particles in a qubit implements a logical NOT on the qubit.

B. Interferometry

The basic process by which we will determine the state ofour system, i.e. read our qubits, is an interference measure-ment. The two states of a qubit, which differ by the absenceor presence of a neutral fermionψ, can be distinguished bytaking a chargee/4 quasiparticle around the pair. If the neu-tral fermionψ is present, an extra(−1) occurs in the ampli-tude. In ref. 19, it was shown how this minus sign could bedetected by measuring the longitudinal resistance,σxx. It isdetermined by the probability for current entering the bottomedge from the left in Figure 5 to exit along the top edge tothe left. This is given, to lowest order intMN andtPQ, bythe interference between two processes: one in which a ‘test’quasiparticle tunnels from M to N; and another in which the‘test’ quasiparticle instead continues along the bottom edge toP, tunnels to Q, and then moves along the top edge to N. (Wesubsume intotPQ the phase associated with the extra distancetravelled in the second process and the extra Aharonov-Bohmphase.)

σxx ∝ |tMN |2 + |tPQ|2 + 2Re(

tMN t∗PQ 〈ψ |B|ψ〉

)

(13)

The third term is the interference between the two possibletunneling trajectories.|ψ〉 is the state of the qubit and thetest quasiparticle, andB is the operator which takes the testquasiparticle around the qubit, i.e. the braiding matrix15. Itcan be computed by any of three equivalent ways: (1) takingη3 aroundη1 andη2 in equation (9); (2) using the expressionin (11); or (3) by evaluating the Jones polynomial atq = eπi/4

5

for the link diagrams in Figure 6 (see appendix A for more onthe meaning and evaluation of these diagrams). Either of thesemethods shows that〈ψ |B|ψ〉 = ±i for the two states of thequbit (the factor ofi comes from the Abelian sector of thetheory). Hence,

σxx ∝ |tMN ± i tPQ|2. (14)

with the+ sign corresponding to the state|0〉 and the− signcorresponding to the state|1〉.

In the many-qubit device of Figure 1, we would need tunneljunctions on either side of each qubit. By doing a sequenceof tunneling conductance measurements, we could read eachqubit in succession.

The presence or absence of a chargee/4 quasiparticle onan anti-dot can, of course, be detected simply by measuringthe charge on the anti-dot. However, we will have occasion tomeasure the topological charge contained within some com-plicated spacetime loops, so it will also be useful to detectchargee/4 quasiparticles by interferometry. This can be doneusing the experimental setup of refs. 15,27, as analyzed inrefs. 28,29. When a chargee/4 quasiparticle is present onthe anti-dot in Figure 7, the authors of refs.28,29 showed, thetwo trajectories do not interfere at all because〈ψ |B|ψ〉 = 0for equation (13) applied to this setup. This may be seen byevaluating the Jones polynomial atq = eπi/4 for the firstlink in Figure 25 (see appendix A for more details). Hence,σxx ∝ |t1|2 + |t2|2. Varying the phases oft1 andt2 will notaffect the longitudinal conductivity, which is the signature ofaσ particle.

C. Tilted Interferometry

We now consider a generalization of the interferometrymeasurements of the previous subsection. Consider the dia-gram of Figure 8, in which a quasiparticle-quasihole pair iscreated, one member of the pair winds around another quasi-particle fixed at an antidot, and then the pair is again anni-hilated. This picture has a special feature, namely that thequasiparticle-quasihole loop can be continuously deformed

N

1 2t MNt PQ

A

B Q

PM

FIG. 5: The state of a qubit can be determined from a measurementof the longitudinal conductance when inter-edge tunnelingis allowedat two interfering junctions.

=

1 10

(b)(a)

= (−1)

FIG. 6: The interference between the two trajectories in Figure 5 canbe obtained from the Jones polynomial (operator) evaluatedon thetwo diagrams in this figure. In (a) the qubit is in the state0, while in(b) it is in state1.

t21t

FIG. 7: When there is a chargee/4 qusiparticle at the anti-dot inthe middle of the device, then there is no interference between thetwo trajectories fromX to Y contributing to the longitudinal con-ductance.

into a single time slice or, for that matter, stretched out sothat it takes place over a very long time, as in the third picturein figure 8. Because the antidot is simply sitting there pas-sively, the evolution in the time direction can be chosen to ouradvantage.

FIG. 8: A quasiparticle trajectory which winds around the antidot(straight line) can be deformed into a single time slice or stretchedover a long time.

The amplitude for such a process is a measure of the totaltopological charge of the planar region bounded by this loopwhen deformed into a single time slice (as in the middle pic-ture in Figure 8) – in other words, it measures the topologicalcharge on the antidot. However, we are also free to considersuch processes even when they do not have an interpretation interms of the charge in some region of a fixed time-slice plane.This type of process can occur when the spacetime topology isnon-trivial. For instance, if the system is on a torus, then there

6

is a process in which a quasiparticle-quasihole pair is created,the quasiparticle taken around the meridian of the torus un-til it again meets the quasihole, and they are both annihilated.The corresponding loop does not enclose any region, so theusual interpretation is not available. Such loops are an impor-tant part of the Bravyi-Kitaev construction which we describein section VI. An even more exotic possibility is depicted inFigure 9. Suppose we have two antidots which we bring closetogether so that they fuse for a short period of timeT beforewe pull them apart again. The spacetime diagram for this pro-cess is depicted in figure 9. Now consider a test quasiparticlewhich travels between the two dots. We have drawn two in-terfering trajectories which the test quasiparticle can take, la-beledp andp′ in Figure 9. One of these trajectories,p, passesbetween the antidots before the merger while the other,p′,passes between the anti-dots after the merger. The curveγcaptures the matrix element for the interference between thesetwo trajectories.

anti−dot

anti−dot anti−dot

anti−dot

p

p

γt

FIG. 9: Two antidots are merged for a short while and then separatedagain. The spacetime curveγ encircles the merger region. The in-terference between the trajectoriesp andp′ measure the topologicalcharge around this curve.

Ordinarily, one thinks of the amplitude of Figure 9 as beingquite different from the middle one in Figure 8 (for instance,in Yang-Mills theory, a Wilson loop in the time direction is ameasure of the force between separated charges, and thereforeis a probe of confinement). However, in a topological phase,the curveγ in Figure 9 is put on the same footing as the loopin Figure 8 (see appendix A for more on the relation betweenthese diagrams and matrix elements in Chern-Simons theory).In a topological phase, the system does not know about anypreferred metric (at least at long distances and low energies),so the time direction is just as good as a spatial direction.The results of such interferometry measurements correspondto topological charges, even though there isn’t an interpreta-tion as the topological charge enclosed within a spatial loop.This may be familiar to some readers in the context Laughlinstates in the quantum Hall effect. Atν = 1/3, an interfer-ence experiment around the meridian of a torus can return asits answerq = 0, 13 ,

23 (modulo1). Of course, the merid-

ian of the torus does not enclose a region, so these are notcharges enclosed within a meridional loop. Rather, these re-sults correspond to the different possible quasiparticle bound-ary conditions (monodromies) around the meridian (namely

ψ → e2πim/3 ψ, m = 0, 1, 2) which are the same as thephases which a quasiparticle would acquire in going arounda region containing chargesq = 0, 13 ,

23 . Similarly, in the

Pfaffian state tilted interferometry experiments will return astheir result either1, σ, ψ, just as ordinary interferometry does.

Such measurements, in which quasiparticles encircle thetime-dependent trajectories of quasiparticles on anti-dots, willbe important for the protocols proposed in this paper, so itis worth spending a little time determining the limitationsonsuch measurements, which we will call ‘tilted interferometry’because the curveγ cannot be deformed into a single timeslice. (We thank Ady Stern for pointing out that tilted in-terferometry is analogous to the less well-known experimentproposed by Aharonov and Bohm30, in which the time depen-dence ofA0 affects quasiparticle interference between trajec-tories which do not pass through regions of finite electric (ormagnetic) field.)

Unlike in the case of ‘ordinary’ interferometry, a tilted mea-surement cannot, strictly speaking, be a DC measurement. Inan ordinary interference experiment, the different interferingquasiparticle wavefunctions are plane-wave-like states,hence,even though the travel time for the two trajectories in Figure 5will be different, the wavefunctions will have spatio-temporaloverlap, so they will still be able to interfere. Consider, how-ever, the trajectories in Figure 9. Let us suppose that the twoanti-dots are merged from timet1 until time t2. Then the firsttrajectory between the anti-dots must occur beforet1 while thesecond trajectory must occur aftert2. The only way that thewavefunctions for the two quasiparticle trajectories can havean overlap is if there is a delay built into the first trajectorywhich will allow the second one to ‘catch up’. This can bedone as shown in Figure 10. We turn on and off some of thegates in order to direct the quasparticles along the specifiedtrajectories. It will also be helpful to vary the quasiparticlevelocities, as also shown in the figure.

IV. TOPOLOGICAL PROTECTION

The main advantage of using a topological state as a plat-form for quantum computation is that such states have intrin-sic fault-tolerance20. The multi-quasihole states cannot be dis-tinguished by local measurements, so long as the quasiholesare kept far apart. Hence, interactions with the environment,which are presumably local, cannot cause transitions betweendifferent topologically-degenerate states nor can it split themin energy. Suppose that we have2n quasiparticles in our sys-tem, which is the device of Figure 1. One might worry, forinstance, that there could be a local voltage fluctuation at oneof the anti-dots. This has a trivial effect, however: the voltagefluctuation changes the energy of all2n states of our Hilbertspace by precisely the same amount. Thus, it does not applya phase gate, as could happen in a non-topological quantumcomputing scheme. The only way in which errors can oc-cur is if a stray quasiparticle (created by the interaction withthe environment) moves across the system and spontaneouslyperforms a topological operation such as the braiding oper-ation of Figure 4 or the interference measurement of Figure

7

2

1 2

1 2

1

1

v2

v

FIG. 10: Two antidots are merged for a short while and then sepa-rated again. By putting delays into the possible quasiparticle trajec-tories (depicted in red and blue), we can enable them to interfere. Inorder to give the two anti-dots enough time to merge, we mightwishto make the velocitiesv1, v2 small.

5. This is closely related to the longitudinal resistance, whichis the probability of an event in which a quasiparticle travelsfrom one end of the system to another. If the spacing be-tween anti-dots is large enough that quasiparticle transport atthis scale is in the ohmic regime, then the error rate and thelongitudinal resistance will be controlled by the same kindofprocesses. (We thank Leon Balents for a discussion of thispoint.) Experimentally, the longitudinal resistance is observedto be in the thermally-activated regime. Hence, it is limited bythe density of thermally-excited quasiparticles, which isexpo-nentially smallexp(−∆/kBT ), where∆ is the quasiparticleenergy gap andT is the temperature.36 In ref. 19, the resultingerror rateΓ was estimated to be

Γ

∆∼ kBT

∆e−∆/kBT (15)

In order to minimize the error rate, we want the temperatureto be as low as possible, and we want the gap to be as largeas possible, which seems to be aided by ultra-high mobil-ity samples. The lowest temperature reached in the experi-ment of ref. 5 wasT = 5mK, while the measured gap was∆/kB = 0.5K. This leads to an error rate less than10−30.

Of course, quasiparticles cannot be kept infinitely far apart,so there will be some splitting between multi-quasihole states.This splitting can be understood as the formation of a band ofpropagating Majorana fermion modes, which mix the local-ized states. The width of this band will be proportional tothe tunneling matrix element between two quasiholes, whichshould decay asw ∼ e−R∆/c, for some constantc with di-mensions of velocity, whereR is the distance between quasi-holes and∆ is the quasiparticle gap. The condition that thequasiparticles should be kept far apart can be translated intothe statement that braiding operations should be done on timescales shorter than1/w. By keepingR large compared to theinverse of the gap we can ensure that this will always be thecase.

When quasiparticles are brought close together, however,there is no longer exponential protection. Suppose, for in-stance, that we merge two antidots into one large antidot ofradiusρ. The splitting between the states|0〉 and|1〉 is nowdetermined by processes in which a quasiparticle-quasiholepair is created at the edge of the antidot, they move in oppo-site directions around the antidot, and annihilate on the otherside. Since there is no gap for the creation of quasiparticles atthe edge of the antidot, there is no longer exponential suppres-sion of such a process. Instead, it leads to a splittingw ∼ 1/ρ.This means that the resulting phase error will be small if twoquasiparticles are merged into a large antidot for a short periodof time, even though the protection is not as good as exponen-tial. However, there is still topological protection against bitflip errors since these would require a neutral fermion to tun-nel to or from the qubit.

A second aspect of topological protection is the exactnessof braiding operations. In the case of, say, spin qubits, gatesare necessarily noisy because they depend on our ability toprecisely tune the duration of aπ pulse or the strength of anapplied magnetic field, which is necessarily imperfect. Whengates are applied by braiding quasiparticles, however, no suchtuning is necessary. The process is discrete: we either braidtwo quasiparticles or we don’t, and if we do braid them,then the corresponding unitary transformation occurs withthesame level of exactness as the vanishing of the longitudinalresistivity or the quantization of the Hall resistivity.

However, one might wonder what happens if a quasiparti-cle only goes 359 degrees around another. If our qubits werequasiparticle pairs which we created out of the vacuum andthen measured after annihilating them again in pairs at the endof the computation, then it would be clear that we would bedealing with closed braids. So long as the topological classofthe closed braid traced out by the entire history of the systemwere preserved, it would not matter whether one quasiparti-cle went 360 degrees around another or only part of the wayaround. However, we envision measuring our qubits through aquantum interference measurement of the topological charge

8

around some closed curves. Therefore, we will consider thisissue in a little more detail. If one quasiparticle were to go360 degrees around another then the initial and final statesof the system lie in the same Hilbert space and the action ofthe braiding operation is just its unitary representative on thisHilbert space. However, if a particle only goes 359 degreesaround another, then the initial and final states of the systemdo not lie in the same Hilbert space. Of course, the initial andfinal Hilbert spaces are unitarily equivalent, but the problem isthat a unitary transformation between them could be trivialorit could undo the braiding operation. So what happens? Theanswer is that it depends on how the system is now measured.If the state of the system is measured by interference, thenthe result will depend on what path the interfering test quasi-particle takes (incidentally, this is always true since we canundo the effect of a braid by choosing a convoluted interfer-ing path). If a particle goes 359 degrees around another, thenalmost all paths will give a result which is the same as if it hadgone 360 degrees. In other words, it is not necessary to havevery precise control of quasiparticle positions. Of course, if aquasiparticle were to only go 270 degrees around another, thenwe would have to exercise more care in choosing a trajectoryfor a test quasiparticle. However, even as drastic a deviationas this is not that serious a problem. The same caution holdsif, instead of measuring the system, we wanted to act on itwith yet another braiding operation. An example of such asituation is given in Figure 11. The same logic also holds forthe uncontrolled motion of a stray quasiparticle; whether ornot it has moved far enough to cause an error will depend onhow the qubit is subsequently measured.

A potential second source of error is braiding operationswhich are performed too quickly. The time scaletop overwhich a braiding operation is done must be slow comparedto the gap,top ≫ ∆. If this inequality is violated, a pair ofquasiparticles might be created. These quasiparticles mightthen execute a non-trivial braid before annihilating each other(Coulomb blockade presumably prevents them from annihilat-ing the quasiparticles on the anti-dots), thereby applyingg ·g′,for someg′, rather than our intended gateg. The amplitudesfor variousg′ depend on the ability of quasiparticles to movearound the system (semiclassically, a random walk). To avoidsuch errors, we must make sure that such quasiparticles arenot created in the first place by performing all braiding opera-tions slowly,top ≫ ∆. Again, it is advantageous to make thegap∆ as large as possible.

While the braiding operations and interference measure-ments described in this paper are similar in the sense that theyboth involve topological operations, they are actually quitedifferent in an important respect. The braiding operationsbywhich we envision manipulating our qubits areunitary opera-tions. They involve moving quasiparticles around our systemover time scales which must be long compared to the inverseof the gap and small compared to the inverse of the error rate:

1

∆≪ top ≪ T e∆/T (16)

So long as this order of time scales is respected and the quasi-particles are kept far apart (compared to the magnetic length,

(b)

(a)

(c)

FIG. 11: In (a), we have two pairs of quasiparticles. The dashed linesconnect quasiparticles which fuse to form the (topologically) trivialparticle. In (b), the second and third quasiparticles are exchanged.A measurement of the topological charge around the blue curve oran operation which takes another quasiparticle around thiscurve willgive a non-trivial result as a result of this exchange. The same re-sult is clearly obtained even if the final position of the quasiparticlesis slightly different from those shown. If the exchange is incom-plete, as in (c), then a measurement around the blue curve will givea non-trivial result but a measurement around the red curve will givea trivial result. So long as we are careful to measure the system withthe blue curves in (b) and (c), we will find the correct result.

which is the only length scale in the problem), then the systemis topologically-protected: the quantum state of the systemevolves precisely as we specify.

On the other hand, our interference measurements are dis-sipative DC measurements (since they require non-zeroσxx).As far as the multi-qubit Hilbert space is concerned, these arenot unitary operations but, rather, projection onto specifiedstates. It is worthwhile thinking a little more about how this‘wavefunction collapse’ occurs because these measurementsare potentially noisy. For instance, if a current-carryingedgequasiparticle should scatter inelastically, then the interferencebetween its two possible trajectories will be spoiled. If thisinelastic scattering rate becomes too large, then we will beunable to read the state of our quasiparticles through an inter-ference measurement because we won’t be able to resolve twodifferent values of the longitudinal conductivity. We wouldlike to know when this will occur. Also, when this occurs,there is an interesting quantum measurement theory problem:does the qubit wavefunction still ‘collapse’ even when the in-

9

elastic scattering rate for the test particles is too high toallowus to distinguish the two states of the qubit?

To answer these questions, we begin by considering Figure5. When a single test quasiparticle tunnels between the twoedges (without scattering inelastically), its wavefunction be-comes entangled with the state of the qubit: it is in one state,which we will call |a〉 when the qubit is in the state|0〉 and itis in a different state,|b〉, when the qubit is in the state|1〉. Ifthe initial state of the system wereα|0〉+ β|1〉, then it is now

α|0〉|a〉+ β|1〉|b〉 (17)

If |a〉 and |b〉 were the same, then there would be no entan-glement with the test quasiparticle at all and the coherent su-perposition of|0〉 and|1〉 is maintained. However, if|a〉 and|b〉 were orthogonal, then the entanglement between the qubitand the test quasiparticle would completely spoil the coherentsuperposition of|0〉 and|1〉 (unless they can be disentangledlater), i.e. the qubit wavefunction is ‘collapsed’.

For small tunneling amplitudestMN andtPQ, both|a〉 and|b〉 are concentrated on the bottom edge in Figure 5 and thereis very little difference between these two states of the testquasiparticle. We can write

〈b|a〉 = 1− δ (18)

with δ small. Hence, a single test quasiparticle does not do aneffective job of ‘collapsing’ the qubit wavefunction. In orderto be an effective measurement, we would like the qubit to bein the state|0〉 with probability1 when the test quasiparticleis in the state|a〉. When (18) holds, the qubit is instead in thestateα|0〉+ β(1− δ)|1〉 when test quasiparticle is in the state|a〉.

However, ifN test quasiparticles tunnel, then they all be-come entangled with the qubit. The combined state of thequbit and test quasiparticles is

α|0〉|a, a, . . . , a〉+ β|1〉|b, b, . . . , b〉 (19)

Notice that we now have

〈b, b, . . . , b|a, a, . . . , a〉 = (1− δ)N (20)

For N sufficiently large, these states are nearly orthogonal.Hence, the two states of the qubit cannot be coherently super-posed unless the qubit is disentangled from the test quasipar-ticles. This cannot happen once the test quasiparticles leavethe system at the current lead and thermalize there, whichis an irreversible process. Thus, we conclude that the qubitwavefunction ‘collapses’: when the test quasiparticles are allin the state|a〉, the qubit is in the state|0〉 with probability1− (1− δ)2N , which is≈ 1 forN large.

Now, consider the effect of inelastic scattering on the testquasiparticles. Those test quasiparticles which are inelasti-cally scattered do not become entangled with the qubit. Asfar as measuring the qubit is concerned, we can forget aboutthem. However, so long as there is a large numberN of quasi-particles which coherently encircle the qubit without inelasti-cally scattering, the qubit wavefunction will collapse, accord-ing to the logic above. In principle, we can always ensure that

this happens simply by waiting long enough, but once the in-elastic scattering rate becomes of orderΓcr ∼ v/L whereL isthe device size andv the edge velocity, we would have to waitan exponentially long time.

The criterion for actually being able to read the value of thequbit is a little different, however. It depends on the resolu-tion of our ohmmeter. When the inelastic scattering rate istoo high, we won’t be able to resolve that there are actuallytwo different values of the longitudinal conductivity. How-ever, depending on how accurately we can measure the lon-gitudinal conductivity, this could occur before the boundΓcr

is reached. Thus, it is possible that the qubit wavefunctionmight ‘collapse’ by a measurement even though we would notbe able to read the result.

We should conclude this section with a comment directedto our topologically-inclined readers (perhaps 100% of thosewho have come so far). Many critical details of the interfer-ometer, such as the inelastic scattering length, travel times,quasiparticle ‘delays’, etc., are not topological in nature. Howis this to be reconciled with the fact that in a topological the-ory all information on the change of state should be encodedby the2 + 1-dimensional spacetime history of the medium?The answer is that the topology of various tunneling trajec-tories gives us operatorsh1, h2, . . . evolving the system frominitial to final states. However, experimental details determineother (dissipative) terms in the evolution equation of the den-sity matrix of the system. In the limit that these other termsare small, a pure quantum state will remain pure and differ-ent2+ 1-dimensional space-time histories will contribute co-herently to the evolution of this quantum state. When theyare large, a pure quantum state will evolve into a mixed oneand the2+1-dimensional space-time histories will effectivelycombine to form a super-operator for this mixed state densitymatrix.

V. UNIVERSAL GATE SET USING SOMETOPOLOGICALLY UNPROTECTED GATES

For all of its remarkable properties, the Pfaffian state suffersfrom one serious drawback: the transformations generated bybraiding operations are not sufficient to implement all possibleunitary transformations24,26. Hence, these operations do notpermit universal quantum computation. However, we don’tneed to supplement braiding with much in order to obtain auniversal gate set. In this section, we explain a ‘quick anddirty’ way of doing this.

First, consider single-qubit operations. If we bring thetwo quasiparticles which comprise a qubit close together, asshown in Figure 12 then their splitting will become apprecia-ble. This splitting has the form∆E(r) ∼ e−r∆/c, whereris the distance between the quasiparticles andc is some con-stant with dimensions of velocity. If we wait a timeTp beforepulling the quasiparticles apart again, then we will apply thephase gate:

UP =

(

1 00 ei∆E(r)Tp

)

(21)

10

A particularly convenient choice is∆E(r)Tp = π/4, whichwould allow us to apply the same transformation as the gateg1 described in the next section.

1 2

FIG. 12: By bringing together quasiparticles 1 and 2, which form aqubit, we can apply the gateg1. This operation is unprotected andrequires control of the distance between the quasiparticles and thelength of time that they are brought together. However, the requiredprecision might not be very stringent as a result of the availability oftopologically-protected operations such asg3.

Let us further assume that we can actually perform this op-eration on any pair of quasiparticles, not just two quasipar-ticles from the same qubit. Note that in order to do this weonly need precise control over the distance between one pairof anti-dots since we can useg3 to move any desired pairof quasiparticles to these preferred anti-dots (e.g. usingthebucket brigade of auxiliary anti-dots to move quasiparticles).If we bring together in precisely the same manner two quasi-particles from different qubits, then we will couple the twoqubits. In terms of the Majorana modes of (11), this gate isexp(

π8 γiγj

)

. (In the special case thati andj come from thesame quasiparticle, this is the same as (21) up to an overallphase.)

The other gate which we need for universal quantum com-putation is the non-destructive measurement of the total topo-logical charge of any pair of qubits. This can be done by us-ing g3 to move one qubit so that it is next to the other. Then,the interference measurement depicted in Figure 13 can deter-mine the sum of the topological charges of the two neighbor-ing qubits. This is a measurement ofγ1γ2γ3γ4. According toref. 33, this measurement is equivalent to the application ofthe gateexp

(

π4 γ1γ2γ3γ4

)

so long as we have the ability to(a) create ancilla in the state|0〉 and (b) applyexp

(

π4 γiγj

)

,which is simply the exchange of quasiparticlesi andj.

An obvious problem is that we have now given up some ofthe protection which we have worked so hard to obtain. Evenif we could calculate∆E(r) with high precision, there wouldalways be some chance of a mistake in the length of timeTpwhich the quasiparticles are close together. Thus, we wouldbe in a situation in which some gates are exact – those result-ing from braiding operations – while others are unprotected.The threshold error rate for the unprotected operations canbemuch less stringent (as high as10%), as shown by Bravyiand Kitaev34 for a specific set of perfect gates together withthe noisy creation of a one-qubit ancilla in a specified state(a‘magic state’). It is still an open problem what the threshold

is for the set of protected gates and the one unprotected gatedescribed above.

VI. BRAVYI-KITAEV CONSTRUCTION

In ref. 24, Bravyi and Kitaev constructed a universal setof gates for a system in the topological phase described bySU(2)2 Chern-Simons theory, which is the effective fieldtheory15,16 likely to describe theν = 5/2 quantum Hall state,apart from an Abelian factor which is unimportant here. Forreference, their gates{g1, g2, g3} are:

g1 =

(

1 00 eπi/4

)

, g2 =

1 0 0 00 1 0 00 0 1 00 0 0 −1

,

and g3 =1√2

1 0 0 −i0 1 −i 00 −i 1 0−i 0 0 1

. (22)

g1 is a phase gate on a single qubit.g2 andg3 are two-qubitgates.g2 is a controlled phase gate: if both qubits are in state|1〉, the state acquires a(−1). Otherwise, it is unchanged.Alternatively, if we change the basis of the second qubit to(|0〉 ± |1〉)/

√2, then g2 is simply a CNOT gate. g3 is a

two-qubit gate which, together withg1, g2 form a universalgate set. This particular gate is chosen because it can be im-plemented with the simple quasiparticle braiding process de-picted in Figure 14.

The controlled phase gateg2 is more complicated. Supposewe have two qubits composed of two pairs of quasiparticles(1,2) and (3,4). We would like to multiply the state of thesystem by(−1) only when qubits(1, 2) and(3, 4) areboth instate|1〉. The problem is that if qubit(3, 4) is in state|1〉 andwe take it around quasiparticle1, then a factor of(−1) results,regardless of whether(1, 2) is in the state|0〉 or |1〉. The trickof Bravyi and Kitaev24 is to split qubit(1, 2) in such a wayas to produce a chargee/4 quasiparticle only when(1, 2) isin the state|1〉. If this occurs, then we can take(3, 4) around

Q

MNt PQ1 2

M

N

3 4

P

t

FIG. 13: A non-demolition measurement of the total topologicalcharge of two neighboring qubits (comprised of1, 2 and3, 4) can bedone with the interference measurement shown here. Together withg3 and the operationUP shown in Figure 12, this forms a universalgate set.

11

21 43

FIG. 14: Quasiparticles 1 and 2 form a qubit; 3 and 4 form a secondqubit. Exchanging 2 and 3 applies the gateg3.

this quasiparticle, and a(−1) will occur if (3, 4) is also in thestate|1〉.

In order to do this, we perform the following steps whichwe will describe here without regard to their feasibility (whichwill be taken up in the next section). Suppose that quasipar-ticles1 and2 are at antidots, which should be understood aspunctures in the quantum Hall fluid. The state of the(1, 2)qubit is equal to the topological charge around the loopB0

in the top diagram in Figure 15. We will denote this topo-logical charge byW(B0). We create an overpass which con-nects these two punctures, as depicted in the middle part ofFigure 15. We check with an interferometry measurementthat the boundary of the antidots-plus-overpass, labeledB1

in Figure 15, has trivial topological charge,W(B1) = 1. Ifit doesn’t, we break the overpass and rebuild it again until wefindW(B1) = 1. We don’t need to repeat this very often sincethe probability forW(B1) = 1 is 1/2 and the probability forW(B1) = ψ is also1/2 (since the isospin0 and1 quasiparti-cles have the same quantum dimension – i.e. the same zero-temperature entropy per particle – inSU(2)2 Chern-Simonstheory). Each time we break the overpass, we return the qubitto its original state. This follows from the general principle(see appendix B) that adding quantum media is reversible sim-ply by deleting what was added (whereas deleting quantummedia is generally irreversible).

Once we know thatW(B1) = 1, it follows for reasonswhich we discuss below that if(1, 2) is in state|1〉 (i.e. ifW(B0) = ψ) thenW(C) = σ in figure 15. If this is the case,then taking(3, 4) around the loopC multiplies the state by(−1) if (3, 4) is in state|1〉 and leaves it unchanged if it isin state|0〉. On the other hand, if(1, 2) is in state|0〉, thenW(C) = (1 − ψ)/

√2 in Figure 15, and taking(3, 4) around

C doesn’t change the state. Therefore, this sequence of oper-ations applies the gateg2 of (22).

The Chern-Simons theory calculations which lead to thisresult are facilitated by the observation that the topologyoftwo anti-dots joined by an overpass is a torus with two punc-tures, corresponding toB0 andB1, as depicted in the bottomdiagram of Figure 15. The curveC is the meridian of thetorus. Once we have made sure thatW(B1) = 1, we knowthat we can fill in this puncture without changing anything;in other words, the system is equivalent to a torus with onepuncture,B0. Observe thatW(B0) is the state of the qubit:it is either1 or ψ. We would now like to determineW(C).This can be obtained from theS-matrix of the theory, whichrelates the topological charge around the meridian,W(M), to

C

0B

1B

0B

B1

C

0B

FIG. 15: By adding an overpass, we connect two antidots. Thedashed curve labeledB1 is the combined boundary of the antidots-plus-overpass. In the implementation of bothg1 andg2, we need tocheck thatW(B1) = 1. In the case ofg2, the controlled qubit mustbe taken aroundC in order to implement a controlled phase gate onit. To enactg1, a double Dehn twist must be performed inC. Thetwo anti-dots joined by an overpass in the middle of the figurearetopologically equivalent to a torus with two punctures correspondingto B0 andB1 as shown at the bottom. The meridian of the torus isC.

the topological charge around the longitude,W(L). If W(B0)is 1, theS-matrix is,

S1ij =

12

1√2

12

1√2

0 − 1√2

12 − 1√

212

, i, j = 1, σ, ψ. (23)

However,W(L) is simply the topological charge on each ofthe antidots, which isσ, (i.e. (0, 1, 0) in the matrix notationof (23)). Hence, theS-matrix tells us thatW(C) is the linearcombination(1 − ψ)/

√2. On the other hand, ifW(B0) =

ψ, then the vanishing of allS-matrix elementsSψij other than

Sψσσ = e−iπ/4 tells us thatW(C) = σ.We now turn tog1. The first step in the implementation of

gateg1 in (22) is the same as above: we take the two anti-dots associated with the qubit under consideration and jointhem with an overpass as in Figure 15. Again, we check thatW(B1) = 1. Now, however, we act on this qubit by perform-

12

ing a double Dehn twist on the curveC. This means that wecut along the curveC, thereby forming two boundaries. Werotate one of them by4π relative to the other (i.e. performtwo twists) and then glue them back together. Finally, we re-move the band, thereby returning the system to a state of twoσ quasiparticles, one on each of the two anti-dots.

A Dehn twist onC has an equivalent effect as a2π rotationof the topological charge associated withC. Since the con-formal spins of1, σ andψ are, respectively,0, 1

16 , and 12 , the

effect of performing two successive Dehn twists is simply theidentity12 = (−1)2 if W(C) is, respectively,1 or ψ and it is(

e2πi/16)2

= eπi/4 if the topological charge isσ. SinceW(C)is perfectly correlated with the value of the qubit, the effect ofthis sequence of operations is the gateg1 in (22). Note that asingle Dehn twist would necessarily change the charge of thetwo anti-dots by transferring chargee/4 from one to the other.

VII. FROM NON-PLANAR TOPOLOGY TOTIME-DEPENDENT PLANAR TOPOLOGY

The operations described in the previous section may neverbe practical in a real quantum Hall device. Overpasses withhigh mobility are implausible, let alone gating them in and outat will. Dehn twists seem an even more remote possibility.Fortunately, there are some features of Chern-Simons theory,which is the effective field theory of our system, which can beexploited to mimic these types of operations without leavingthe plane or attempting to perform surgery on our quantumHall fluid. In this section, we will explain these features ofChern-Simons theory and how they can be used to apply thegatesg1 andg2. Once this problem in topological quantumfield theory (TQFT) has been solved, we turn in the next sec-

A

A

B

B

t

t1

t

γ

2

FIG. 16: A non-planar overpass configuration can be mimickedbybreaking the regionA and addingB. This is done by merging thetwo anti-dots between times timet1 andt2. During this merger inter-val, we allow the quantum Hall fluid the fill the regionB. This fauxpass splits the merged anti-dot in the perpendicular direction. Thesechanges are then undone to return the system to its initial configura-tion. In order to verify that removingA was harmless, we have toperform a tilted interferometry measurement to check that the topo-logical charge aroundγ is trivial. See Figure 20 for a time slicing ofthis process.

A

A

C

2

t1

tγ

t

FIG. 17: Topologically, the process depicted here is equivalent tothat depicted in Figure 16. However, as a result of the tilt ofthefaux pass, the triviality of the charge around the hole inA can bemeasured with an untilted measurement. Consequently, the curveγcan now be deformed into a single time slice. The curve labelledC inthis figure is equivalent to the curveC which goes over the overpassin Figure 15. This is the curve on which we wish to perform a Dehntwist. Measuring the topological charge around this curve allows usto effect the same transformation on the quantum state of thesystemwithout any surgery. See Figure 24 for a time slicing of this process.

tion to the new set of problems which arises when we try torealize this construction in a quantum Hall device. For thereader who is uninterested in the TQFT details and wishesto skip ahead to the next section, we summarize the results ofthis section: (1) an operation equivalent to the addition and re-moval of an overpass between two antidots can be performedby connecting the two antidots so long as a curve surround-ing the connection (which might be tilted) has trivial topo-logical charge; (2) a measurement of the topological chargearound a curveγ (with a particular framing) is equivalent toa ‘Dehn filling’ on this curve (which, in turn, is related toa Dehn twist, as explained below) modulo a few caveats de-scribed in this section. With these two observations in hand,we can replace ‘impossible’ operations with operations whichare merely very difficult. In section VIII, we give concreteillustrations of how this can be done with the device architec-ture described in this paper.

One particularly fortuitous feature of topological field the-ories, for our purposes, is the fact that when the topologicalcharge around a hole is trivial, then the part of the system out-side the hole is impervious to whether the hole is filled in ornot. For example, if the topological charge aroundγ in Figure9 is trivial, then this spacetime history is equivalent, as far asa topological field theory is concerned, with a spacetime his-tory in which there is no merger whatsoever between the twoanti-dots. This suggests a way in which we can effectivelyhave overpasses by taking advantage of the time direction sothat ‘over’ is realized as ‘at a different time’, as shown in Fig-ure 16. Suppose we want a band of materialB to form an

13

(c)(b)(a)FIG. 18: The curve in (a) can be framed, for example, as shown in(b) and (c). The self-linking number in (c) is greater by+1 than in(b).

overpass over another bandA. We instead breakA at timet1,allow B to pass, useB for whatever purpose, breakB, andthen reconstituteA at timet2. If we could measure the topo-logical charge around the resulting time-like holet1 < t < t2in A and if we found that it was trivial, it would be, as faras Chern-Simons theory were concerned, as ifA were neverbroken. The faux overpass – or, simply, faux pass –B is thenjust as good as an overpass.

Note that this figure can be continuously deformed to ourconvenience. From a topological point of view, Figure 16 isequivalent to Figure 17. However, their realizations are quitedifferent, and they have different practical advantages and dis-advantages. The tilted measurement ofγ in Figure 16 is un-tilted in Figure 17. However, Figure 17 has a moving anti-dotand island (whose velocity is the slope of the faux pass in Fig-ure 17).

A second ‘impossible’ operation which we need to performis a Dehn twist on a closed curveC. In order to make it pos-sible, we use the following two facts. (i) It is a fundamen-tal identity of the “Kirby calculus” that(−1)−framed ‘Dehnsurgery’ on a simple linking circle imparts a(+1) Dehn twistand, of course, a double Dehn twist arises if two such surg-eries are performed. By ‘frame’, we mean that the curveCis thickened into a ribbon so that a self-linking number canbe well-defined: it is the linking number of the two curvesformed by the edges of the ribbon (one edge of the ribbon isC, the other edge is traced out by the tip of the frame vec-tor). Some examples of framed curves are shown in 18. Wewould likeC to be framed so that this self-linking number is(−1). The meaning of ‘Dehn surgery’ is that a tubular neigh-borhood (in spacetime) of the loopC is deleted and then gluedback so that the meridian disk is glued to the circle defined bythe tip of the frame vector. Obviously, physical limitationsprevent us from doing this in a quantum Hall device, but wecan instead (ii) measure the particle content of a loopζ in theinterior of a2 + 1−dimensional space-time. If the result is1,we have (up to an overall normalization factor, correspondingto capping a2−sphere) accomplished Dehn surgery onC asfar as Chern-Simons theory is concerned. This Dehn surgeryon a spacetime history has the result of making it into a his-tory which interpolates between an initial state before a Dehntwist has been performed on the faux pass and a final state inwhich the faux pass has been Dehn twisted.

We would like to explain the relation between Dehn twist,Dehn surgery, and Dehn filling. Dehn twist is a method forconstructing a diffeomorphism of a surface: given a closedloop l on a surface, cut the surface alongl, twist one side by2π and reglue. This is either a+ or − Dehn twist (dependingon sign conventions). Our interest in Dehn twist is that it pro-vides a way to transform a 3-manifoldM by cutting it openalong a surface and then regluing using a Dehn twist. To dis-cuss this procedure, we will simplify matters by concentratingon an annular neighborhoodA of l. (The framing ofl definesan annulusA, as may be seen in figure 18b,c.) We split the 3-manifoldM alongA, thereby opening a toridal cavity withinM . The boundary of this cavity is a torusB which is just twocopies ofA, joined along their boundaries. This new manifoldwith the cavity is essentiallyM\neigh(l), M minus a neigh-borhood ofl. The term “Dehn filling” refers to gluing a solidtorus back into the cavity; the double process of first makingthe cavity and then refilling it is called “Dehn surgery”. Thepossible outcomes of Dehn surgery are parameterized by theslope of that curve on the cavity boundaryB which is matchedto the disk factorD2 of the reglued solid torusD2 × S1. No-tice that Dehn twist, from one copy ofA to the other, carriesa radial arc in one copy to a twisted arc in the other so that thetwo mate together to become a diagonal – either (1,1) or (1,-1)in the naturalB-coordinates (the first component counts thewinding number around the “meridian” or shortest directionand the second coordinate counts the winding number aroundthe “longitude” defined by either component of the boundaryof A within B) depending on whether the Dehn twist is+ or− respectively. Conversely, the instuction to do “−1 Dehnsurgery onl” can be expressed as: 1. open the cavity aroundl (a normal framing onl is required at this point to pick outthe surfaceA) 2. change the correspondence between the twocopies of A by a+1 Dehn twist. 3. with respect to these newcoordinates, Dehn fill the solid torus by gluing the disk to themeridian of the cavity. (Steps 2 and 3 together constitute−1Dehn filling since they tell us to match the(1,−1) curve onthe cavity boundaryB, which is taken to the meridian by step2, to the disk within the reglued solid torus.) Thus cuttingand regluing by a+/− Dehn twist is identical to doing+/−Dehn surgery. The later is simply a more three-dimensionallanguage for the former. Measuring the trivial charge alongacurveC is, from the point of view of Chern-Simons theory,equivalent to supplying a disk (containing no quasiparticles)for the curveC to bound, or an entire solid torus (with noWilson loop at its core) forB to bound. Thus, we proposeto accomplish through measurement a topological operation,namely Dehn twist, which otherwise would have no reason-able experimental realization.

An important detail to be considered is that measurementmight result in a nontrivial charge. This means, in fact, that thereglued solid torus does carry a Wilson loop of precisesly thatcharge. This Wilson loop is depicted in figure 19 as a loopβin the exterior solid torus, as shown. These states with Wilsonloops correspond to the other two ground states on the torus(up to an additional factor of2 degeneracy coming from theAbelian part of the theory). These are the states obtained byperforming the Chern-Simons functional integral over a solid

14

L−M

Mβ

1(ζ ,−1)

FIG. 19: If we measure the topological charge around(ζ1,−1) and itis not1, then we have performed a Dehn filling on(ζ1,−1) and leftbehind a Wilson loopβ, carrying this topological charge, inside thesolid torus. The state of the system is given by the functional integralin equation (24) for this manifold. The3D mapping cylinder of aDehn twist can be described as: delete a solid torus, then replaceit with a twisted Dehn filling. For convenience, we have drawnthetorus in a non-standard way so thatL−M appears simple.

torus with a Wilson loop carrying topological charge (σ or ψresp.), but expressed in the meridinal basis.

Ψ[a] =

∫

A|T2=a

DA W [A, β] e∫

MLCS[A] (24)

whereLCS[A] is the Chern-Simons Lagrangian andM =D2 × S1 is the solid torus.W [A, β] is the Wilson loop (i.e.the trace of the holonomy of the gauge fieldA), W [A, β] =

Tr(

P exp(∮

ζA))

, with the trace taken in the fundamental or

the adjoint representation (for, respectively,σ or ψ), and thefunctional integral is overSU(2) gauge fieldsA such thatA = a on T 2, the boundary of the solid torus. Since theWilson loop must be around the meridian,β is depicted as‘outside’ the torus in figure 19.

In order to perform a double Dehn twist onC, we needto measure the topological charge around two framed curveswhich run parallel toC. We will call these curves(ζ1,−1)and (ζ2,−1); the −1 denotes the framing. Note thatC isequivalent to untwisted copies of theζ ’s, which can be de-noted(ζ1, 0) and(ζ2, 0). In order to measure the topologicalcharge around(ζ1,−1) and(ζ2,−1), our measurement quasi-particles will have to go along a curve with an extra twist; away of realizing this is shown in Figure 23. If a measurementfindsW(ζ1) = 1, then we have performed the desired Dehntwist. FindingW(ζ1) = ψ isn’t the end of the world becausethe extra Wilson loop which results just gives some extra mi-nus signs. However, we want to avoidW(ζ1) = σ. One wayto do this is to take a chargeσ around the loop(ζ1, 0). Thenlet (ζ1,−1) run parallel to the charge, encircling it to yieldthe framing(−1). In Figure 23, we have depicted a time-slicing of such a framed curve: the measurement quasiparti-cle must wind around the quasiparticle which is following theloop (ζ1, 0).

We can now use a similar argument to that used after (23)to show thatW(ζ1) = (1− ψ)/

√2: since(ζ1,−1) = L−M

(i.e. longitude - meridian), we claim thatW(ζ1) is given byST−1|σ〉 where n

S =

1/2√2/2 1/2√

2/2 0 −√2/2

1/2 −√2/2 1/2

andT =

1 0 00 eiπ/8 00 0 −1

in the {1, σ, ψ} basis. We check thatST−1 |σ〉 = e−iπ/8 (|1〉 − |ψ〉)/

√2. Hence, a mea-

surement ofW(ζ1) can only give 1 or ψ (with equalprobabilities). By doing this with another charge, parallel toζ2 and encircled by it, we can force a measurement ofW(ζ2)to be1 or ψ, again with equal probabilities, and completelyindependent of the result of theW(ζ1) measurement.

To see thatST−1 gives the correct transformation, notethat we wish to transform from the longitudinal basisL tothe framing= k basis,L + kM = longitude+k(meridian).To define a basisV (T 2) =

∑

particle types,a Vaa(S1 × I), we

need to select a circle, the “cuff”, to cut the torus along a dualcircle, the “seam”, to trivialize the resulting annulus asS1×I.The transformation is then done in the following two steps31:

(cuff, seam)=

(L,M)twistk−→ (L,M + kL)

S−1

−→ (M + kL, L). (25)

Hence, the composition of these transformations is given by:S−1T kS.

In the discussion above, we have shown that with two tiltedinterferometry measurements, we can accomplish the samething as a double Dehn twist, as far as Chern-Simons theoryis concerned, with the caveat that extra Wilson loops might beadded to to the final state.

One important difference between the Bravyi-Kitaevprotocol24 and our proposal to mimic this with time-dependent planar topologies is that, in the latter case,W (B1)cannot be measured until the end of the entire operation.Consequently, we must use the overpass without knowingif W (B1) = 1 or ψ (these are the only two possibilitiessince twoσs can only fuse into these two possibilities). Thismeans that we must learn to live with the possibility thatW (B1) = ψ. The saving grace is that we will at least knowafter the operation whetherW (B1) = 1 or ψ, which allowsus to compensate in the latter case. Under the assumption thatW (B1) = 1, we deduced from theS−matrix of the theorythatW(C) is perfectly correlated withW(B0), which is thevalue of the qubit.

W(B0) = 1 ⇒ W(C) = (1− ψ)/√2

W(B0) = ψ ⇒ W(C) = σ (26)

However, the same logic shows that ifW(B1) = ψ, then

W(B0) = 1 ⇒ W(C) = σ

W(B0) = ψ ⇒ W(C) = (1− ψ)/√2 (27)

In other words, the correlation betweenW(C) andW(B0) hasbeen reversed or, in other words, the roles of the states1 andψ of the qubit have been reversed.

15

In the case ofg2, this applies to the control qubit, on whichall of the operations are performed (the controlled qubit ispassive). Hence, if we find at the end of our procedure thatW(B1) = 1, then we know that we have applied the desiredphase gateg2. If , instead,W(B1) = ψ, then the gate has in-advertently interchanged the roles of1 andψ within the con-trolling qubit so that

g2 =

1 0 0 00 −1 0 00 0 1 00 0 0 1

(28)

has instead been applied. This is not too serious since repeatedapplication of the protocol gives a random walk in the groupZ2

⊕

Z2 = {1, g2, g2, g2g2}. OurW(B1) measurements tellus where we are withinZ2

⊕

Z2 as we randomly walk; wesimply halt upon reachingg2. The tails on a “long walk” de-cay exponentially so this delay is acceptable.

Now considerg1. Let us first suppose, for simplicity, thatW(ζ1) = W(ζ2) = 1. If we find at the end of our procedurethatW(B1) = 1, then we know that we have appliedg1 =(

1 00 eπi/4

)

. However, if we find thatW(B1) = ψ, then wehave applied

(

eπi/4 00 1

)

= eπi/4 g−11 . (29)

There is one added complication in the case ofg1, as com-pared tog2: we have the added uncertainty in the outcome oftheW(ζ1), W(ζ2) measurements. IfW(ζ1) = W(ζ2) = ψ,then two Wilson loops carryingψ appear within the solidtorus. However, these two Wilson loops fuse to form1, whichis again trivial, just as in the caseW(ζ1) =W(ζ2) = 1. How-ever, ifW(ζ1) ·W(ζ2) = ψ, which is as good as a single Wil-son loopζ carrying topological chargeψ parallel toC, therecan be a non-trivial effect. IfW(C) = (1 − ψ)/

√2, which

means thatW(B0) ·W(B1) = 1, the Wilson loop contributesno extra phase. However, ifW(C) = σ, which means thatW(B0) · W(B1) = ψ, then the Wilson loopζ encircles thesame topological chargeσ which is encircled byC (to whichit is parallel). When aψ encircles aσ, a (−1) results. Thismeans that ifW(ζ1) ·W(ζ2) = ψ, then instead ofg1 or g−1

1 ,we have acutally appliedσz g1 = g−3

1 or σz g−11 = g31 , where

σz =(

1 00 −1

)

.In all eight measurement outcomes forW(ζ1), W(ζ2) we

have, up to an overall phase, implemented eitherg1, g−11 , g−3

1 ,or g31 . Thus our protocol generates a type of random walk onZ8. Since we know the measurement outcomes we may iteratethe protocol until we arrive atg1, which is again efficient.

Therefore, we conclude that findingW(B1) = ψ is not acalamity for the implementation of eitherg1 or g2.

VIII. REALIZING THE BRAVYI-KITAEV GATES WITHTIME-DEPENDENT PLANAR TOPOLOGY AND TILTED

INTERFEROMETRY

In this section, we will discuss how the operations of theprevious section, which are merely difficult, rather than im-

possible, might be implemented in a quantum Hall device. Weneed to be able to do four things: (1) move our qubits at will,(2) create ‘faux passes’ – which are equivalent to overpassesin Chern-Simons theory – with the spacetimes histories de-picted in Figure 16 or Figure 17, (3) measure the topologicalcharge around a tilted trajectory such asC in Figure 17, and(4) create a quasiparticle pair and move them around as de-sired before annihilating them. We have already discussed (1)and (3) in section III. We now turn to (2) and (4).

The spacetime history shown in Figure 16 can be realizedby the sequence of steps depicted in Figure 20. In this figure,the regionA (the ‘underpass’) of the quantum Hall fluid sepa-rates the two anti-dots. At timet1, regionA is broken so thatthe two anti-dots are joined into one large oblong anti-dot.Af-ter this occurs, a strip of quantum Hall fluid is allowed to splitthe large anti-dot in the perpendicular direction (the bottomleft picture in Figure 20). This is the faux passB which playsthe role of the overpass. The spacetime region carved out bythis strip is the tube in Figure 16.

In order to check that the topological charge around thetime-like hole is trivial, we need to do a tilted interferome-try measurement similar to that depicted in figures 9, 10. Theinterference between the red and yellow curves in Figure 20measures the topological charge around the hole inA. Thereis an obvious drawback here, which is that the result of thismeasurement will not be know until after the entire procedureis complete. We will return to this issue later. For now, let usconsider the other operations which we need to perform.

During the time that the faux pass regionB is available, wemust take our qubit over it and then check whether the topo-logical charge around the curveB1 is 1 or ψ. These are bothdepicted in Figure 21. Both of these processes must occurwhile those depicted in Figure 20 are simultaneously occur-ring. We envision doing both of these with a bucket brigadeof anti-dots which are used to ferry both the measuring testquasiparticles and the controlled qubit across the faux pass,as shown in Figure 22. This is relatively straightforward forthe qubit: the two quasiparticles comprising the qubit mustbe moved across the regionB in figure 20. Of course, thisprocess must occur without the two quasiparticles of the con-trolled qubit fusing. Hence, we need to keep them far apart,either with a large faux pass or by having one follow at a dis-tance behind the other.

The test quasiparticle with which we measureB1 is trickier.Consider the red and yellow trajectories in Figure 21. Thereshould be some small amplitudetr for a quasiparticle at thebottom edge to tunnel to the top edge via the red trajectory,which takes it over the faux pass, and a small amplitudety forit to tunnel via the yellow trajectory. In the ‘bucket brigade’scenario, the red trajectory is actually composed of a seriesof hops from one anti-dot to another. Let us assume thatthe amplitude for a quasiparticle to hop from the edge ontothe first anti-dot ist1. Let us further assume that the ampli-tude for it to hop from the first anti-dot to the second anti-dot is tbb; from the second to the third, againtbb; and so onuntil the quasiparticle finally hops onto the top edge. Thentr ∼ t1 tbb tbb . . . tbb, where the number of factors oftbb de-pends on how many anti-dots are in the bucket brigade. We

16

B

1

t < t < t2

t=t2

A

A

B

A

1

t=t

FIG. 20: A non-planar overpass configuration can be mimickedby breaking the regionA and addingB. This is done by merging the twoanti-dots at timet1 and splitting them in the other direction. The faux passB is then used to either applyg1 or g2, as described in Section VII.These changes are then undone to return the system to its initial configuration. In order to verify that removingA was harmless, we have toperform a tilted interferometry measurement in which the red and yellow trajectories interfere.

need this amplitude to be small so that the topological orderof the state is not degraded, but is should be large enough tobe measurable. This might be most easily done by makingt1 small andtbb not too small. Of course, the same reason-ing holds for the yellow trajectory. Finally, we need the am-plitudes for these two processes to interfere coherently. Thismeans that the coherence time for a quasiparticle on eitherone of these trajectories must be longer than the time of flight,which might be difficult to ensure.

In order to implementg1, we need to also send an auxiliaryquasiparticle over the faux pass. This can be done as shown inFigure 23, perhaps using a ‘bucket brigade’, as in Figure 22.We must then measure the topological charge around the curve(ζ1,−1), which means that we must measure the interferencebetween the trajectories in Figure 23. We do this with(ζ2,−1)in the same way.

There is a problem with the procedure described in thesefigures, which is that the configuration shown in the third pic-ture in Figure 20 has two anti-dots merged. Consequently, thestate of the two anti-dots does not have exponential protec-

tion. The splitting between the two states of the qubit formedby this pair of anti-dots isw ∼ 1/x, wherex is the linear ex-tent of the merged antidot. This can actually be avoided in thefollowing way. Instead of merging the two anti-dots, we sendan intermediary which shuttles from one to the other. This isdone by breaking the anti-dot on the right into two anti-dots,one with electrical chargee/4 and the other with no electri-cal charge. We then move the neutral anti-dot to the left andmerge it with the left anti-dot. This would simply replace themerger by the ‘tilted merger’ shown in Figure 17. In order tohave an overpass, we need the anti-dot to be annular so thatthere is a region ofν = 5/2 quantum Hall fluid in the mid-dle of it, as shown in Figure 24. While this moving shuttleis between the two anti-dots, we check by ordinary interfer-ometry that it carries topological charge1, rather thanψ. Ifit doesn’t, then we re-merge it with the right anti-dot and re-peat the same process until we find that the topological chargearound the shuttle is1. The abortive attempts at this do not af-fect the qubit (except possibly by an irrelevant overall phase):splittinga into b ⊗ c and then re-fusing results in the original

17

B

A

A

FIG. 21: The interference between the red and yellow trajectories is ameasurement of the topological charge aroundB1. In order to applyg2, we take the controlled qubit across the faux pass

=bb

tbbtbb

t1

tbb

t

FIG. 22: The quasiparticle trajectories shown in Figures 20and 21.may actually look more like the trajectory on the right above: a seriesof hops from one small anti-dot to another.

particle type, so it is a multiple of the identity. The qubit isclearly unaffected since the phase resulting from such opera-tions is independent, by locality, of the state of the qubit.Inorder to applyg2, the controlled qubit must sit in the interiorof the shuttle as it moves from one anti-dot of the control qubitto the other. In order to applyg1, the auxiliary quasiparticlesmust do so. It must be noted that this approach again has thedifficulty that the coherence times of complicated interferingquasiparticle trajectories must be kept long.

FIG. 23: In order to applyg1, we must create a pair of auxiliaryquasiparticles, take one over the faux pass (not shown), andthenannihilate them again. While this is occurring, we must measurethe topological charge around a curve which follows the auxiliaryquasiparticle over the faux pass, while encircling it. By encirclingthe auxiliary quasiparticle (presumably while hopping along a bucketbrigade of anti-dots), this trajectory obtains a non-trivial framing: itis the framed curve(ξi,−1). A measurement of the interference be-tween the red and yellow curves can determine the topological chargearound this (tilted) framed curve.

IX. DISCUSSION

In this paper, we have discussed how theν = 5/2 quantizedHall plateau can be used as the basis of a quantum computer,assuming that this plateau is in the universality class of thePfaffian state. Pairs of chargee/4 quasiparticles form qubits.We propose to pin the quasiparticles at anti-dots so that bymoving the anti-dots, we move the quasiparticles. The twostates of a pair of quasiparticles,|0〉 and |1〉, can be identi-fied with the primary fields1, ψ of the conformal field theoryof the critical2D Ising model, or with the spin0 and spin1 representations ofSU(2)2 Kac-Moody algebra (or the re-lated quantum group). The value of any qubit can be readby a transport measurement which is sensitive to the inter-ference between two possible quasiparticle trajectories encir-cling the qubit. However, local measurements cannot distin-guish the two states of the qubit so long as the two quasipar-ticles are kept apart. The error rate is astronomically low,so

18

B

B

B

A

A

A

A

A

B

A

FIG. 24: By sending an annular intermediary between the two anti-dots, we can construct the spacetime history of Figure 17. The shuttlebetween the two antidots is annular and the island of quantumHall fluid in the middle of the annulus is the faux passB. A measurement of theinterference between the red and yellow trajectories can check that the tilted hole in regionA is topologically trivial as far as Chern-Simonstheory is concerned.

these qubits form an essentially perfect quantum memory19.Two simple gates can be implemented by quasiparticle braid-ing: (1) tunneling a quasiparticle from the edge between thetwo quasiparticles comprising a qubit and (2) by exchangingaquasiparticle from one qubit with a quasiparticle from anotherqubit.

In order to be able to apply any possible unitary transfor-mation to our qubits – i.e. in order to have universal quan-tum computation – we can try either of two approaches. Oneis to use the unprotected operation of bringing together thetwo quasiparticles comprising a qubit so that a phase gate

will result from the resulting energy splitting between theirtwo states. Although this operation is unprotected, the errorthreshhold may not be too strict because the other operationsare exact due to topological protection. The other approachisto use some complicated manipulations of the anti-dots whichinvolves moving, splitting, and rejoining them in order to foolthe topological field theory governing the Pfaffian state intobehaving as if we have performed non-embeddable topologychanges. Although complicated, these operations allow, inprinciple, for a universal set of exact gates. By now the readermay have decided that the former approach is more promising

19

in the short run because it does not require extremely compli-cated quasiparticle manipulations. However, if technologicaladvances make the latter approach more feasible, then it hasthe virtue of complete topological protection. Also, the beau-tiful topology involved lends it an intrinsic ‘coolness’ factor.

If the weakerν = 12/5 plateau proves to be a non-Abelianstate related toSU(2)3 Chern-Simons theory, then neither ofthese two compromises would be necessary. We would againrealize qubits as above, but, in this case, the braid group alonewould be sufficient for universal quantum computation.

Acknowledgments

We would like to thank S. Das Sarma, M. P. A. Fisher, A.Kitaev, C. Marcus, K. Shtengel, J. Slingerland, and A. Sternfor discussions. We would especially like to thank P. Bonder-son for discussions and a careful reading of earlier versions ofthis paper. We would also like to acknowledge the hospital-ity of the Kavli Institute for Theoretical Physics at UCSB andthe Aspen Center for Physics. We have been supported by theARO under Grant No. W911NF-04-1-0236. C. N. has alsobeen supported by the NSF under Grant No. DMR-0411800.

APPENDIX A: EFFECTIVE FIELD THEORY APPROACHTO THE PFAFFIAN STATE

The multi-quasihole structure which we encountered in sec-tion II is precisely the same structure as occurs in the Isingmodel. Indeed, the motivation of Moore and Read for firstwriting down the Pfaffian wavefunction was its relation toconformal blocks of the Ising model6. The first hint that theremight be such a connection comes from the observation thatthe Pfaffian factor in the wavefunction (1) is equal to a corre-lator ofψ fields:

〈ψ(z1)ψ(z2) . . . ψ(z2n)〉 = Pf

(

1

zj − zk

)

(A1)

Multi-quasihole wavefunctions are related to conformalblocks with spin fields such as6,8. Before discussing this inany more detail, however, we should first explain why thequasiparticles of the Pfaffian state have anything to do withthe primary fields of the Ising model.

The answer lies in the connection discovered by Witten22

between2 + 1-D Chern-Simons theories and1 + 1- or 2-Dconformal field theories (CFTs). According to Witten22 (seealso the explicit constructions of Elitzuret al.23), the Hilbertspace of states of a Chern-Simons theory on a3D manifoldM = X × R is equivalent to the vector space of confor-mal blocks of an associated CFT on the2D manifold X .The Hilbert space of a Chern-Simons theory with sources lo-cated atr1, r2, . . . , rn carrying representationsρ1, ρ2, . . . , ρnis equivalent to the conformal blocks of ann-point correlationfunction of primary fields associated with these representa-tions.

If we have an electron system which is described at long-wavelengths by a Chern-Simons theory, we can use the wave-functions of the Chern-Simons theory with sources as approx-imate wavefunctions of the electron system. By the aboveconnection, this means we can use conformal blocks in theassociated CFT as wavefunctions of the electron system. Ineach of these two steps, the essential features which are pre-served are the braiding properties of the quasiparticles ofthesystem. The different primary fields in the CFT correspondto the different topological charges in the Chern-Simons the-ory, which, in turn, correspond to the different quasiparticlespecies in the electron system.

As shown in ref. 15,16, the low-energy effective field the-ory associated with the Pfaffian state of bosons atν = 1 isanSU(2)2 Chern-Simons theory. The effective field theoryfor fermions atν = 1/2 (or other even-denominator fill-ing fractions and for bosons at other odd-denominator fill-ing fractions) is a deformation of this theory: a Higgs-edSU(2)2 Chern-Simons theory (independent of the filling frac-tion) coupled to aU(1) Chern-Simons theory with a filling-fraction-dependent coupling constant. The CFT correspond-ing to this Chern-Simons theory is the tensor product of anSU(2)/U(1) coset model with ac = 1 U(1) factor6. How-ever, theSU(2)/U(1) coset model is equivalent to the Isingmodel. According to this correspondence, the primary fieldswhich are associated with half-flux quantum quasiparticlesareIsing spin fields (multiplied by a field from thec = 1 sector).

TheU(1) factor, which accounts for the charge, gives thecorrect Abelian factors for quasiparticle statistics and con-tributes a factor to the ground state degeneracy. The morenon-trivial physics is contained in theSU(2) part of the the-ory. Hence, we can be a little sloppy and drop theirU(1)quantum numbers and refer to quasiparticles by theirSU(2)2quantum numbers (which we will call ‘isospin’). According tothis nomenclature, there are three primary fields, with isospins0, 12 , 1, corresponding to1, σ, ψ of the Ising model. These are,respectively, the vacuum, the half-flux quantum quasiparticle,and the neutral fermion. The topological degeneracy of multi-quasiparticle states reflects the decomposition of the productof two isospin1/2s: 1

2 ⊗ 12 = 0⊕ 1. Thus, there are two dif-

ferent reasons why it is natural to call the two states of a qubit0 and1.

Returning now to the conformal blocks which model wave-functions of electrons atν = 1/2 (in the first excited Landaulevel), we note that the corresponding CFT contains both thec = 1/2 Ising model and ac = 1 chiral boson (accounting forthe electrical charge):

S =

∫

d2z (ψ ∂zψ + (∂zφ)(∂zφ)) (A2)

We retain only the right-handed partϕ of φ = ϕ + ϕ, with∂zϕ = ∂zϕ = 0. If we assume that the operator correspond-ing to the electron is:

Ψel(z) = ψ(z) eiϕ(z)√2 (A3)

20

then we have

⟨

Ψel(z1)Ψel(z2) . . . Ψel(z2N ) e∫

d2zρ ϕ(z)√2⟩

=

∏

j<k

(zj − zk)2∏

j

e−|zj|2/4 · Pf(

1

zj − zk

)

. (A4)

The last term in the correlation function on the left-hand-sideof this equation is a neutralizing charge background withoutwhich the Coulomb gas correlation function would vanish. Tofind the primary fields of thisc = 3/2 theory, we need to puttogether the primary fields of the (right-handed chiral com-ponent of the) Ising model – the identity,1; the Ising spinfield, σ; and the Majorana fermion,ψ – with exponentials ofthe chiral bosonϕ. The primary fields must be local with re-spect to the electron operator – i.e. quasiparticles can havenon-trivial statistics with each other but they must have trivialstatistics with respect to an electron since an electron wave-function must be single-valued in electron coordinates. A fieldwhich satisfies this condition is

Φqp(η) = σ(η) eiϕ(η)/2√2 (A5)

This field corresponds to the chargee/4 quasiparticle:

⟨

Φqp(η1)Φqp(η2)×

Ψel(z1)Ψel(z2) . . .Ψel(z2N ) e∫

d2zρ ϕ(z)√2

⟩

=

Pf

(

(zj − η1) (zk − η2) + zj ↔ zkzj − zk

)

×∏

j<k

(zj − zk)2∏

j

e−|zj|2/4 (A6)

The four-quasihole wavefunctions in (9)are given by the conformal blocks of〈ΦqpΦqpΦqpΦqpΨelΨel . . .Ψel exp(

∫

d2zρϕ√2)〉. The

two different wavefunctions correspond to the two differentfusion channelsσ · σ ∼ 1 + ψ. (Furthermore, the twoconformal blocks are the wavefunctions (9) in the specialbasis in which the explicit monodromy of the wavefunctiongives the complete braiding matrices9).

The full set of primary fields is given in the table below.The columns correspond to Ising model primary fields whilethe rows correspond to thec = 1 primary fields. The entries ofthe table are the quasiparticle electrical charges. Each quasi-particle corresponds to an operator formed by multiplying theoperator at the top of its column by the operator to the left ofits row. Note that it is not a naive tensor product between thetwo theories because the quasiparticles containingσ are off-set byeiϕ/2

√2 from those containing1, ψ. (Again, these as-

signments are determined by the requirement of locality withrespect to the electron operator (A3).)

1 σ eiϕ/2√2 ψ

1 0 e/4 0

eiϕ/√2 e/2 3e/4 e/2

One of the nice features of the relationship between thePfaffian state andSU(2)2 Chern-Simons theory is that it al-lows us to use Witten’s remarkable result relating Chern-Simons theory and the Jones polynomial of knot theory22.Braiding matrix elements can be obtained by computing theJones polynomial of the corresponding link diagrams15,19,29.For instance, the qubit-flipping property of the process whichtakesη3 aroundη1 in (9) can be obtained by evaluating theJones polynomial (operator) atq = eπi/4 for the links in Fig-ure 25.

(d)(b) (c)(a)1

1

1

11 1

1

FIG. 25: By evaluating the Jones polynomial atq = exp(πi/4) forthese links, we can obtain the desired matrix elements for braidingoperations manipulating the qubit. The boxed1 is a projector on thepair of quasiparticles which puts them in the state|1〉.

What do these pictures mean and how are they evaluated?They are essentially Feynman diagrams for the topological in-teractions of quasiparticles. The lines in this picture representisospin1/2 quasiparticles. Since the electric charge is not ac-counted for in these diagrams, quasiparticles and quasiholesare identical as far as these diagrams are concerned. (Theelectric charge just contributes Abelian phase factors.) If twoof these isospin1/2 quasiparticles fuse to form a spin0, thenthey can annihilate to give the vacuum. Otherwise, they fuseto form an isospin1 particle, which we denote by a boxed1on the two lines (as in Figure 25b, for example). Consider,for instance, the spacetime trajectories of the test quasipar-ticle and the quasiparticle on the antidot in Figure 7. Theywill look like Figure 26a. In this figure, we assume that aquasiparticle-quasihole pair is created atP and the quasipar-ticle goes around the antidot. The other trajectory, with whichthis interferes, is depicted in figure 26b: in this case, the testquasiparticle does not go around the antidot. The interfer-ence term between these two processes, i.e. the third termin equation 13, is given by the matrix element between thesetwo processes, depicted in Figure 26c. It is just Figure 26binverted (which corresponds to turning a ket into a bra) andthen stacked on top of 26a. This matrix element measures thetopological charge associated with the loop in Figure 26c.

21

(b)(a) (c)

FIG. 26: (a) A quasiparticle trajectory which winds around aquasi-particle at an antidot (b) a quasiparticle trajectory whichdoesn’t windaround the antidot; (c) the matrix element between the states result-ing from these two trajectories.

Hence, to summarize: we specify an initial state|ψ0〉 of nqubits by drawing2n incoming lines, which are grouped inton pairs. At the bottom of the diagram (the distant past), eachpair is either created from the vacuum (in which case this qubitis in the state|0〉) or was fused to form an isospin1 particle(in which case this qubit is in the state|1〉). The final state|ψ〉is obtained by braiding these quasiparticles, which is depictedin the figure with the vertical direction on the page being posi-tive time. Finally, the matrix element between|ψ〉 and anotherstate|χ〉 is obtained by inverting the corresponding diagramfor |χ〉 and stacking it on top of the diagram for|ψ〉, con-necting each quasiparticle line from〈χ| to the correspondingquasiparticle line from|ψ〉.

The matrix element associated with such a diagram is eval-uated recursively in the following way. Each diagram is re-placed by the sum of two diagrams both of which have onefewer crossing according to the rule:

= q1/2 + q−1/2 (A7)

The new diagrams come with corresponding coefficients asindicated in the above relation. We continue this procedureuntil all crossings are eliminated. The projector on the isospin1 state is replaced by the explicit expression:

1 = − 1

d(A8)

whered = −q − q−1. Finally, every closed loop is given thevalue

= d (A9)

APPENDIX B: THE EFFECT OF ADDING OR DELETINGBANDS OF MATERIAL TO A TOPOLOGICAL STATE

In this appendix, we discuss how aν = 5/2 fractionalquantum Hall droplet is modified by topology change, in par-ticular when bands of material are added or deleted. The basic

guideline is that adding a band of material does not change thestate of the system, so it is reversible; deleting a band of ma-terial can cause the state of the system to change irreversibly.

In the Bravyi-Kitaev construction discussed in section VI,we described a process in which a band of material (the over-pass in the middle picture in Figure 15) is added and then laterremoved. The spacetime history of such a process is depictedin Figure 27. In this figure, twoσ quasiparticles on anti-dotsfuse to form1 or ψ, which is the topological charge of theboundary. The spacetime history of the quantum Hall dropletwith two anti-dots is shown as a ‘pair of pants’ legsP × I,whereP is the disk with two punctures andI is the intervalof time [0, 1]. The spacetime history depicting the additionand subsequent removal of an overpass connecting the anti-dots is a handle connecting the two pants legs. The union ofthis1-handle and the spacetime history of the droplet with twoanti-dots isW = P × I ∪ (1 − handle).

FIG. 27: The spacetime history of a process in which an overpassconnecting two antidots is added and then removed. Warning:thishistory does not embed in2 + 1 dimensions, so thez−coordinate inthis picture cannot be literally interpreted as time. The handle mustmove into a fourth direction.

We would like to check that this procedure leaves invari-ant the qubit supported byP (rather than, say, applies aphase gate). In order to do this, we imagine evaluating theChern-Simons functional integral on the3-manifoldW . Itwill give a state in the Hilbert space of the boundary∂W .However, this boundary is divided into subsurfaces by loopsbounding specified topological charges, so the Hilbert spaceof the boundary is, in fact, the tensor product of the Hilbertspaces of the subsurfaces. Hence,W specifies a vectorψ1 in:V0,0⊗V0,σ,σ⊗V0,σ,σ⊗Vσ,σ,x⊗V ∗

σ,σ,x⊗Vx,x, where the factorscome from subsurfaces1, . . . , 6 in Figure 22. The zero labelin the first three factors is dictated by the presence of disksinW capping the boundary of the first component (a cylinder).The gluing axiom31,32 tells us that removing the1−handle de-termines a canonical isomorphism toZ(P × I) carryingψ1

toψ0 in V ∗0 ⊗ V0 ⊗ V0,σ,σ ⊗ V0,σ,σ ⊗ Vσ,σ,x ⊗ V ∗

σ,σ,x ⊗ Vx,x.

22

After supplying the canonical base vectorsβ∗0 ∈ V ∗

0 , β0,σ,σ ∈V0,σ,σ andβx,x ∈ Vx,x, ψ1 is canonically identified with id∈Hom (Vσ,σ,x) ∼= V ∗

σ,σ,x ⊗ Vσ,σ,x. Note that nox−dependentphase has entered the calculation. Thus we have proved, in theabstract language of TQFTs, that adding and then breaking aband induces the identity operator on the qubit supported inP .

The situation is rather different if, instead, we cut out a bandto join the internal punctures and then restore it (i.e. fusetheinternal punctures and then separate them). This is true re-gardless of whether we assume that we can use the electriccharge of theσs to ensure that each resulting puncture againcarries a charge-e4 σ after they are fused and split. Such anoperation is depicted in Figure 9. We show below that thisoperation acts onP by the identity ifγ has topological charge1. If it has topological chargeψ, however, it isσz . If it is alinear superposition of these two possibilities, then the appliedgate is the same linear combination of1 andσz. The moral, ingeneral, is that operations which add quantum media (in thiscase,ν = 5/2− FQHE fluid) are reversible – simply deletewhat was previously added, whereas operations which deleteare often irreversible.

This operation can be depicted in the Feynman diagram no-tation introduced in appendix A. It is either Figure 28a or 28b,according to the charge ofγ.

(a) (b)

1

FIG. 28: The spacetime history of Figure 27 is equivalent to theFeynman diagrams above in the notation introduced in appendix A.

Clearly, both operators are diagonal in the(1, ψ) basis.Furthermore, Figure 28a is simply the identity acting on theP−qubit. This may be verified by direct computation of thediagonal entries. Using the rules introduced in appendix A,we√2 · 1. The strange

√2 factor is actuallyS11 = Sψψ which

has entered because we have not rescaled the dual physicalHilbert space by1/S11 prior to gluing. Taking this factor intoaccount, we obtain the identity.

Computing the diagonal entries for the second process, weobtainσz , as claimed, after taking into account the correctiondescribed above and rescaling by1/Sψψ.

1 A related paper, cond-mat/0512072, explains the topological is-sues in greater detail. It may help the reader to look at this alter-nate presentation if confused about any point.

2 See, for instance,Perspectives in Quantum Hall Effects: NovelQuantum Liquids in Low-Dimensional Semiconductor Structures,ed. by S. Das Sarma and A. Pinczuk (Wiley, New York, 1997) andreferences therein.

3 J. Leinaas and J. Myrheim, Nuovo Cimento37B, 1 (1977); G.Goldin et al., J. Math. Phys.22, 1664 (1981); F. Wilczek, Phys.Rev. Lett.48, 1144 (1982). See also the discussion inFractionalStatistics and Anyon Superconductivity, edited by F. Wilczek(World Scientific, Singapore 1990).

4 However, direct experimental verification of this has proven dif-ficult until now, see F. E. Camino, Wei Zhou, V. J. Goldman,cond-mat/0502406.

5 J.S. Xiaet al., Phys. Rev. Lett.93, 176809 (2004); J.P. Eisensteinet al., Phys. Rev. Lett. 88, 076801 (2002); W. Panet al., Phys.Rev. Lett.83, 3530 (1999); J.P. Eisensteinet al., Surf. Sci.229,31(1990); R.L. Willettet al., Phys. Rev. Lett.59, 1776 (1987).

6 G. Moore and N. Read, Nucl. Phys. B360, 362 (1991).7 M. Greiter, X.-G. Wen, and F. Wilczek, Nucl. Phys. B374, 567

(1992).8 C. Nayak and F. Wilczek, Nucl. Phys. B479, 529 (1996).9 V. Gurarie and C. Nayak, Nucl. Phys. B506, 685 (1997).

10 Y. Tserkovnyak and S. H. Simon, Phys. Rev. Lett.90, 016802(2003)

11 N. Read and E. Rezayi, Phys. Rev. B54, 16864 (1996).12 N. Read and D. Green, Phys. Rev. B61, 10267 (2000).13 D. A. Ivanov, Phys. Rev. Lett.86, 268 (2001).14 A. Sternet al., Phys. Rev. B70, 205338 (2004).15 E. Fradkin, C. Nayak, A. M. Tsvelik, and F. Wilczek, Nucl. Phys.

B 516, 704 (1998).

16 E. Fradkin, C. Nayak, and K. Schoutens, Nucl. Phys. B546, 711(1999).

17 R. H. Morf, Phys. Rev. Lett.80, 1505 (1998).18 E. H. Rezayi, F. D. M. Haldane, Phys. Rev. Lett.84, 4685 (2000).19 S. Das Sarma, M. Freedman, and C. Nayak, Phys. Rev. Lett.94,

166802 (2005).20 A. Y. Kitaev, Ann. Phys.303, 2 (2003).21 M. H. Freedman, A. Kitaev, M. Larsen, and Z. Wang,

quant-ph/0101025.22 E. Witten, Comm. Math. Phys.121, 351 (1989).23 S. Elitzur, G. Moore, A. Schwimmer, N. Seiberg, Nucl. Phys. B

326, 108 (1989).24 S. Bravyi, A Yu. Kitaev,Quantum invariants of 3-manifolds and

quantum computation. 2001 Preprint.25 N. Read and E. Rezayi, Phys. Rev. B59, 8084 (1999).26 M. Freedman, M. Larsen, Z. Wang, Comm. Math. Phys. 227, 605

(2002).27 C. de C. Chamonet al., Phys. Rev. B55, 2331 (1997).28 A. Stern and B. I. Halperin, cond-mat/0508447.29 P. Bondersonet al., cond-mat/0508616.30 Y. Aharonov and D. Bohm, Phys. Rev.115, 485 (1959).31 K. Walker, On Witten’s 3-manifold invariants, preprint, 1991.

(Available at http://canyon23.net/math/)32 V. TuraevIntroduction to Combinatorial Torsions, Birkhauser; 1st

edition (December 15, 2000).33 S. Bravyi and A. Kitaev, Ann. of Phys.298, 210 (2002).34 S. Bravyi and A. Kitaev, Phys. Rev. A71, 022316 (2005)35 It is easier to write down explicit wavefunctions for quasiholes

than for qusiparticles, so we restrict attention in this section toquasiholes. For the particular device configurations whichwe dis-cuss in this paper, it is also easier to work with quasiholes.How-ever, the underlying topological properties are the same, so we

23

will use the terms ‘quasiparticle’ and ‘quasihole’ almost inter-changeably.

36 Presumably it is controlled by variable-range hopping at stilllower temperatures. In this regime, a quasiparticle which is in onelocalized state at zero temperature is thermally excited into an-

other localized state. Through a sequence of such hops, withanaverage range which varies with temperature, it travels across thesystem.