Topological qubit

In this subsection, we will introduce the topological qubit, which is composed of two far separated Majorana zero modes. Such a qubit attracts people’s attention for its robustness under local perturbations. Moreover, it is inter-esting to study if such robustness also exists in quantum decoherence process or not. It is instructive and natural to introduce the topological qubit from the framework of Kitaev’s chain model[23], which will be depicted in the following.

Kitaev’s chain model was first proposed by Kitaev in 2000 as a model to describe 1-D spinless p-wave superconductor, where the spinless fermions locate on N-site chain. The Hamiltonian of this model is setup as follows

H =−µ∑ where µ is the chemical potential, ω≥ 0 is the hopping amplitude and ∆ =

|∆|e^iθ is the induced superconducting gap. cx and c^†x are the annihilation and creation operators on site x respectively. Without loss of generality, here we set the lattice spacing to be unity.

In the momentum space, (1.55) can be expressed in terms of the quasi-particle ak as the following simple form

H = ∑

k∈BZ

E(k)a^†_ka_k, (1.56)

where BZ means the Brillouin zone and

ak = ukck+ vkc^†_−k, (1.57)

From (1.58), we know that as the chemical potential is µ = ω or µ =−ω , the excitation energies E(k) vanishes at k = ±π or k = 0 respectively. It means the system is gapless at the Fermi level as µ =±ω and it also tells us the regimes of gapped phases.

Next we pay more attention on the gapped phases: |µ| < ω and µ < −ω, where we ignore µ > ω because it is just related to µ <−ω by a particle-hole transformation; therefore they are similar in topological structure. A direct way to see the difference of these two gapped phases is to compare their corresponding ground states. As shown in [24], the ground states are closely related to the wavefunction of Cooper pair composed of fermions with momentum k and−k. The corresponding wavefunctions of these two gapped phases have very different characteristics. For µ < −ω, the wavefunction in real space is approximately as e^−|x|/ξ, which means the Cooper pair is bounded in a local range around a length ξ. On the other hand, for|µ| < ω, the wavefunction is approximately constant, which means the Cooper pair is not bounded in a local range. However the above description is still too naive to judge if these two phases are really different or not. People could argue that the unbounded case is just a special case of the bounded cases by setting ξ→ ∞; therefore, it seems that these two phases are not so different from each other.

We say two phases are different, which means these two phases can’t transform to each other by small perturbation of the coupling coefficients except by the occurrence of phase transition. This concept is closely related to topological invariants. If two phases have different values of topological invariants, we say these two phases are different. One topological invariant

⃗h(k) encodes all information of Hamiltonian. Here we don’t mention the detail of ⃗h(k) quantitatively¹, but we focus on its topological characteristic qualitatively. As k sweeps from 0 to π, ⃗h(k) maps from the Brillouin zone to a trajectory on a sphere. For the two gapped phases mentioned before, their corresponding trajectories are: The trajectory which corresponds to

1⃗h(k) is defined as

Figure 1.2: Two limiting cases of Kitaev’s chain model (a) The trivial phase.

(b) The nontrivial phase.

µ <−ω begins and ends at the same pole of the sphere. On the other hand, the trajectory which corresponds to|µ| < ω begins and ends at the different poles of the sphere. It is apparent these two trajectories are topologically different from each other; therefore, we say these two phases are different phases.

Now we decompose each spinless fermionic operator cxinto two Majorana fermions γ_A,x and γ_B,x by the following relation

c_x= e^−iθ/2

2 (γ_B,x+ iγ_A,x), (1.59)

with

γ_α,x= γ_α,x^† , {γα,x, γ_α′,x^′} = 2δαα^′δ_xx′. (1.60) Then the Hamiltonian (1.55) can be expressed in terms of Majorana fermions as follows

H =−µ 2

∑

x

(1 + iγB,xγA,x)−i 4

∑

x

[(∆ + ω)γB,xγA,x+1+ (∆− ω)γA,xγB,x+1].

(1.61) In the following, we consider two limiting cases which correspond to the two different phases mentioned before: One limiting case with µ < 0 and

ω = ∆ = 0 corresponds to the topologically trivial phase. As shown in the panel (a) of Fig.1.2, in this phase, only the Majorana modes at the same site couple with each other and no interaction exists among the Majorana modes at different sites. This phase is trivial, because a coupled pair of Majorana modes composes a spinless fermion cx as shown in (1.59) and only trivial structure occurs. Its spectrum is gapped, because it costs finite energy |µ|

to put a fermion cx on the chain.

The other limiting case with µ = 0 and ω = ∆ ̸= 0 corresponds to the topological phase. As shown in the panel (b) of Fig.1.2, in this phase, a Majorana mode always couples with its neighbor one at different site and no interaction exists between the Majorana modes at the same site. We can still define a new fermion dxcomposed of two Majorana modes with different site as

dx= 1

2(γA,x+1+ iγB,x). (1.62)

Then the Hamiltonian can be expressed as

H = ω

N∑−1 x=1

(d^†_xd_x−1

2). (1.63)

Its spectrum is gapped, because it costs finite energy ω to put a fermion dx on the chain. As shown in Fig.1.2, the main difference between the non-trivial phase from the non-trivial one is the existence of two unpaired zero energy Majorana modes γ1 := γA,1 and γ2:= γB,N on the ends of the chain. These two unpaired Majorand modes can combine into a non-local fermion called topological qubit

f = 1

2(γ₁+ iγ₂). (1.64)

Because γ₁ and γ₂ don’t appear in the Hamiltonian, it costs zero energy to produce a topological qubit, which implies a two-fold ground state degener-acy|0⟩ and |1⟩ := f^†|0⟩.

Besides these two limiting cases, next let’s talk a little bit about more general situations for topological non-trivial phase. In such general situa-tions, the corresponding Majorana zero modes are no longer the end-states as the limiting case, because the bulk contribution to the wavefunctions of

e^−L/ξ, where L is the length of the chain and ξ is the coherence length de-pendent of the coupling constants. At the same time, the two-fold ground state degeneracy is also broken by the same factor. As long as L≫ ξ, where the bulk contribution is much smaller than the boundary one, the end-states are good enough to be regarded as the Majorana zero modes. Hereafter in this thesis, when we refer to Majorana zero modes, we always mean the end-states of the limiting case.

Let’s come back to the two limiting cases. It can be said, for these two phases, their bulk structures are very similar as long as the lattice spacing is not too large. The main difference between them is the boundary structure, or naively speaking, the existence of the topological qubit. It is expected that the non-locality of a topological qubit makes many of its properties very different from those of usual qubits, e.g. its robustness against local perturbations which has been understood clearly. Besides, there are still many unsolved problems for topological qubits to be studied. What we focus on in this thesis is the decoherence behavior of the topological qubits.

The relevant topics will be discussed in the following chapters.