Topological quantum computation away from the ground state with Majorana fermions
TTopological quantum computation away from the ground state with Majoranafermions
A. R. Akhmerov
Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands (Dated: May 2010)We relax one of the requirements for topological quantum computation with Majorana fermions.Topological quantum computation was discussed so far as manipulation of the wave function withindegenerate many body ground state. The simplest particles providing degenerate ground state,Majorana fermions, often coexist with extremely low energy excitations, so keeping the system inthe ground state may be hard. We show that the topological protection extends to the excitedstates, as long as the Majorana fermions do not interact neither directly, nor via the excited states.This protection relies on the fermion parity conservation, and so it is generic to any implementationof Majorana fermions.
PACS numbers: 03.67.Lx, 03.67.Pp, 71.10.Pm, 74.90.+n
Topological quantum computation is manipulation ofthe wave function within a degenerate many-body groundstate of many nonabelian anyons. Interchanging theanyons applies a unitary transformation to the groundstate wave function. The simplest of the nonabeliananyons useful for topological quantum computation areMajorana fermions. These are expected to exist in 5/2fractional quantum Hall effect and in certain exoticsuperconductors . In 5/2 fractional quantum Hall ef-fect the Majorana fermions are charge e/4 quasiholes,and in superconductors Majorana fermions are zero en-ergy single particle states either trapped in vortex coresor other inhomogeneities. Superconducting implementations of Majoranafermions potentially allow for a larger bulk gap of a fewKelvin as compared with 500 mK for fractional quantumHall effect. One significant difference between thesuperconductors and the fractional quantum Hall effectis that Majorana fermions in superconductors appearwhere the superconducting gap in excitation spectrumcloses. This means that Majorana fermions would notbe isolated from other excitations by the bulk gap, butcoexisting with a lot of bound fermionic states withlevel spacing of the order of the minigap ∆ /E F , where∆ ∼ E F the fermienergy. If E F ∼ This is why there is research aimed at increasing theminigap. We adopt a different strategy and show that couplingto excited states does not remove the topological protec-tion as long as different Majorana fermions stay decou-pled. The topological protection persists because cou-pling to excited states has to preserve the global fermionparity. Using only the conservation of the global fermionparity and the fact that different Majorana fermions are well separated we identify new Majorana opera-tors, which are protected even if the original Majoranafermions coexist with many excited states. We also checkthat the braiding rules for the new Majorana operatorsare the same as for original ones.We start from a brief introduction to Majoranafermions, for more information see e.g. Ref. 12. A singleMajorana fermion is described by a fermionic annihila-tion operator γ which is equal to the creation operator γ = γ † . (1)Due to this defining property of Majorana fermions theyare also called “real fermions” or “particles equal to theirown antiparticles”. Substituting Eq. 1 into the fermionanticommutation relation we get { γ, γ † } = 2 γ = 2 γ † γ = 1 . (2)The last equality is a manifestation of the fact that asingle Majorana fermion is pinned to the fermi leveland accordingly is always half-filled. Additionally it isnot possible to add a perturbation to the Hamiltonian,which would move a single Majorana level away fromfermi level, at least two Majorana fermions are required.The only possible coupling term between two Majoranafermions has the form H γ = iεγ γ . (3)The perturbation H γ hybridizes two Majorana states intoa single complex fermion state at energy ε and with cre-ation and annihilation operators a † = γ + iγ √ , a = γ − iγ √ . (4)If Majorana fermions are well separated, the couplingbetween them decays exponentially with the distancebetween them. Additionally if the superconductor isgrounded, the charging energy also vanishes, leaving theMajorana fermions completely decoupled. In the limitwhen coupling between Majorana fermions ε is negligibly a r X i v : . [ c ond - m a t . s up r- c on ] J u l small, H γ has two zero energy eigenstates which differ byfermion parity (1 − a † a ) = 2 iγ γ . (5)If the system has N decoupled Majorana fermions, theground state has 2 N/ degeneracy and it is spanned byfermionic operators with the form (4). Braiding Majo-rana fermions performs unitary rotations in the groundstate space and makes the basis for topological quantumcomputation.To understand how coupling with excited states givesnontrivial evolution to the wave function of Majoranafermions we begin from a simple example. We consider atoy model containing only two Majorana fermions γ and γ and a complex fermion a bound in the same vortex as γ . At t = 0 we turn on the coupling between γ and a with Hamiltonian H a = iε ( a + a † ) γ . (6)At t = π ¯ h/ε we turn off H a and give finite energy to thefermion by a term εa † a . We denote by | (cid:105) the state wheretwo Majorana fermions share no fermion, so an eigenstateof 2 iγ γ with eigenvalue 1, and by | (cid:105) the eigenstate of2 iγ γ with eigenvalue −
1. If the system begins from astate | (cid:105) , then it evolves into an excited state a † | (cid:105) , sothe Majorana qubit flips. This seems to destroy the topo-logical protection, however there is one interesting detail:since there are two degenerate ground states | (cid:105) and | (cid:105) ,there are also two degenerate excited states: a † | (cid:105) and a † | (cid:105) . So while | (cid:105) changes into a † | (cid:105) , | (cid:105) changes into a † | (cid:105) . The two end states differ by total fermion parity,which is the actual topologically protected quantity. Inthe following we identify the degrees of freedom whichare protected by nonlocality of Majorana fermions anddo not rely on the system staying in the ground state.We consider a system with N vortices or other defectstrapping Majorana fermions with operators γ i , where i isthe number of the vortex. Additionally every vortex hasa set of m i excited complex fermion states with creationoperators a ij , with j ≤ m i the number of the excitedstate. We first consider the excitation spectrum of thesystem when the vortices are not moving and show thatit is possible to define new Majorana operators whichare protected by fermion parity conservation even whenthere are additional fermions in the vortex cores. Parityof all the Majorana fermions is given by (2 i ) n/ (cid:81) Ni =1 γ i ,so the total fermion parity of N vortices, which is a fun-damentally preserved quantity, is then equal to P = (2 i ) n/ N (cid:89) i =1 γ i × N (cid:89) i =1 m i (cid:89) j =1 [1 − a † ij a ij ]= (2 i ) n/ N (cid:89) i =1 m i (cid:89) j =1 [1 − a † ij a ij ] γ i . (7) This form of parity operator suggests to introduce newMajorana operators according toΓ i = m i (cid:89) j =1 [1 − a † ij a ij ] γ i . (8)It is easy to verify that Γ i satisfy the fermionic anti-commutation relations and the Majorana reality condi-tion (1). The total fermion parity written in terms of Γ i mimics the fermion parity without excited states in thevortices P = (2 i ) n/ N (cid:89) i =1 Γ i , (9)so the operators (2 i ) / Γ i can be identified as the localpart of the fermion parity operator belonging to a singlevortex. We now show that the operators Γ i are protectedfrom local perturbations. Let the evolution of system bedescribed by evolution operator U = U ⊗ U ⊗ · · · ⊗ U n , (10)with U i evolution operators in i -th vortex. The systemevolution must necessarily preserve the full fermion par-ity P = U † P U, (11)and hence(2 i ) n/ N (cid:89) i =1 Γ i = (2 i ) n/ N (cid:89) i =1 U † i × N (cid:89) i =1 Γ i × N (cid:89) i =1 U i = (2 i ) n/ N (cid:89) i =1 U † i Γ i U i . (12)This equation should hold for any set of allowed U i . Tak-ing U i = for all i (cid:54) = j we come to U † j Γ j U j = Γ j , (13)for any U j . In other words, the new Majorana operatorsΓ j are indeed not changed by any possible local pertur-bations.We now need to show that the protected Majorana op-erators Γ i follow the same braiding rules as the origi-nal ones. The abelian part of braiding, namely the Berryphase, is not protected from inelastic scattering invortices, so it will be completely washed out. The non-abelian part of the braiding rules is completely describedby the action of the elementary exchange of two neigh-boring vortices T on the Majorana operators. As shownin Ref. 14, exchanging Majorana fermions γ i and γ j isdescribed by γ i → γ j , γ j → − γ i . The fermion parityoperators (1 − a † ij a ij ) have trivial exchange statistics asany number operators. Applying these rules to exchangeof two vortices containing excited states givesΓ i = m i (cid:89) k =1 [1 − a † ik a ik ] γ i → m j (cid:89) k =1 [1 − a † jk a jk ] γ j = Γ j , (14a)Γ j = m j (cid:89) k =1 [1 − a † jk a jk ] γ j → m i (cid:89) k =1 [1 − a † ik a ik ]( − γ i ) = − Γ i . (14b)This finishes the proof that braiding rules are the samefor Γ i .Our proof of protection of Majorana fermions and theirbraiding properties from conservation of fermion parityonly relies on particle statistics of Majorana and complexfermions. Consequently it fully applies to the Moore-Read state of 5/2 fractional quantum Hall effect, p-wavesuperfluids of cold atoms, or any other implementationof Majorana fermions. Part of this proof can be repro-duced using topological considerations in the followingmanner. If a perturbation is added to the Hamiltonianand additional excitations are created in a vortex, the fu-sion outcome of all these excitations cannot change unlessthese excitations are braided or interchanged with thosefrom other vortices. So if a system is prepared in a cer-tain state, then excitations are created in vortices, braid-ing is performed and finally the excitations are removed,the result has to be the same as if there were no exci-tations. Our proof using parity conservation, however, allows additionally to identify which part of the Hilbertspace stays protected when excitations are present. Sinceremoving the low energy excitations does not seem fea-sible, this identification is very important. It allows amore detailed analysis of particular implementations ofthe quantum computation with Majorana fermions. Forexample we conclude that implementation of the phasegate using charging energy, as described in Ref. 18, doesnot suffer from temperature being larger than the mini-gap since it relies on fermion parity, not on the wavefunction structure.Since all the existing readout schemes of a Majoranaqubit are measuring the full fermion parity of twovortices, and not just the parity of the fermion sharedby two Majorana fermions, all these methods also workif Majorana fermions coexist with excited states. Thesignal strength however is reduced significantly when thetemperature is comparable with the minigap due to de-phasing of the internal degrees of freedom of vortices. 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