2.7 Quantum Computation with Majoranas
2.7.1 Demonstration of Non-Abelian Braiding
Imagine a system of 2N well-separated vortices in a two-dimensionalp-wave super-conductor. Each vortex is assumed to bind a single Majorana zero mode. Note that we in Chapter 5 will establish a detailed understanding of how this takes place, especially how the Majorana modes are localized. In total, the set{γn}2Nn=1 contains the distinct operators at our disposal. The Majorana operators satisfy the defining properties
γn=γn† and [H, γn] = 0. (2.37) They are assumed to satisfy fundamental anticommutation relations similar to fermions, adopting a common normalization convention [11],
{γn, γm}= 2δn,m. (2.38)
Section 2.7 Quantum Computation with Majoranas 21
Note how this last property makes it difficult to speak of the occupancy of a Ma-jorana state. Trying to construct a number operator leads to the trivial result nMF = γn†γn = 1 (since γn2 = 1). In other words, a Majorana mode is in this sense always filled, and counting the occupancies is meaningless. Still, we may pair up two Majorana operators to form an ordinary fermionic state. This pairing is merely a choice of basis, and it clarifies why we need 2N Majorana modes to formN fermionic operators. We define
cj ≡ 1
2(γ2j−1+iγ2j). (2.39)
The reader may look at Figure 5.4 at this point to get a cartoon picture of the situation. These operators can readily be checked to satisfy the canonical relations {c†i, cj} = δi,j and {ci, cj} = {c†i, c†j} = 0. Occupation operators, nj = c†jcj, can conveniently be formed, and they are used to denote the ground state manifold in terms of the occupancy state|n1, n2, . . . , nNi. For the matter of this demonstration we simply state the result of adiabatically exchanging the vortices containing γj
and γj+1 in a clockwise manner. This discrete transformation is the key result of Ivanov’s derivation [6],
σi :
γi 7→γi+1
γi+1 7→ −γi
γj 7→γj for j /∈ {i, i+ 1}
. (2.40)
In section 5.6 we will suggest an argument that results in the same transformation rule. Consider the minimal (and non-trivial) example with two fermionic degrees of freedom, N = 2. Assume that the initial ground state of the system is |n1, n2i ≡
|n1i ⊗ |n2i, with n1 is the occupation number of the fermionic state formed by γ1
and γ2 according to (2.39). Similarly, n2 consists ofγ3 and γ4. Hence, |n1, n2i is an eigenstate ofn1 =c†1c1 andn2 =c†2c2 by assumption. In order to study the action of σi on|n1, n2i, we need a unitary representation of the transformation rule in (2.40), meaning an operatorUi satisfying
σi(γj) =UiγjUi†. (2.41) Up to an undetermined phase factor, the unitary operator satisfying this is given by
Ui = expπ 4γi+1γi
= 1
√2(1+γi+1γi). (2.42) One may readily check that it satisfies the Braid group properties in (2.36). The operators {Ui}3i=1 constitute a representation of the Braid group, in this case B4. The operator U2 affects γ2 and γ3 such that the state |n1, n2i is possibly left in a linear combination of the four states spanning the ground state manifold,
M ={|0,0i,|1,0i,|0,1i,|1,1i}. (2.43)
We define the fully filled state as |1,1i ≡ c†1c†2|0,0i (the order is of importance in what follows). Our system consists of two fermionic operators, c1 = 12(γ1+iγ2) and c2 = 12(γ3+iγ4), with the inverse relations
γ1 =c1+c†1, γ2 =−i(c1−c†1), γ3 =c2+c†2, γ4 =−i(c2−c†2). (2.44) Thus, acting with single Majorana operators on |n1, n2i can be summarized as
γ1|n1, n2i=|1−n1, n2i,
γ2|n1, n2i=i(−1)n1|1−n1, n2i, γ3|n1, n2i= (−1)n1|n1,1−n2i, γ4|n1, n2i=i(−1)n1+n2|n1,1−n2i.
(2.45)
Hence, we have the tools needed to see how U1, U2 and U3 act on the occupancy states,
U1|n1, n2i= 1
√2
1 +i(−1)1−n1
|n1, n2i, U2|n1, n2i= 1
√2
|n1, n2i −i|1−n1,1−n2i , U3|n1, n2i= 1
√2
1 +i(−1)2n1−n2+1
|n1, n2i.
(2.46)
The two-qubit state |n1, n2i is an eigenstate of U1 and U3 as anticipated. However, the operatorU2 is seen to rotate the state into a linear combination of two states. In other words, the representation is non-Abelian, and the order of which one succes-sively applies U operators do matter. Alternatively, we may represent the operators as matrices, with the consequence that U2 is non-diagonal. We do this with the basis states in the same order as in (2.43):
U1 = 1
√2
1−i 0 0 0 0 1+i 0 0 0 0 1−i 0 0 0 0 1+i
, U2 = 1
√2
1 0 0 −i 0 1 −i 0 0 −i 1 0
−i 0 0 1
, U3 = 1
√2
1−i 0 0 0 0 1−i 0 0 0 0 1+i 0
0 0 0 1+i
. (2.47) Unfortunately, the two-qubit braiding operators U1, U2 and U3 alone do not allow for universal quantum computation; they do not span all unitary transformations [12]. However, there are schemes, even though we will not pursue them, on how to approximate any quantum computation to arbitrary accuracy with only the braid operations. Proposals have also been put forward on how to extend the set of braiding operations with the gates needed to obtain universal computation.
Chapter 3
The Closed Kitaev Chain
We open the content of this thesis with a detailed study of a toy model of his-torical impact. The motivation is to fill in details, and to gain a more complete understanding than obtained with the presentation by Jason Alicea in [11]. By con-sidering closed boundary conditions, we will be able to analytically derive the energy spectrum and characterize the ground state. We explore how a certain Berry phase distinguishes the two topological phases of the model. With this classification in mind, it is of fundamental interest to study if fermionic correlation functions show signs of the topological phase transition. This is done numerically.
In 2001 Alexei Kitaev proposed a toy model for spinless fermions on a one-dimensional superconducting chain [10]. The model captures concisely the physics of topological phenomena, and it has proven to be central in the field of topological superconduc-tivity. The Kitaev Hamiltonian for a closed system is
H =−µ XN x=1
c†xcx− 1 2
XN x=1
tc†xcx+1+ ∆eiφcxcx+1+ h.c.
. (3.1)
Above, cx denotes the annihilation operator of a spinless fermion at site x. We define cN+1 ≡c1 so that the N-site chain obeys periodic boundary conditions. The chemical potential is denoted by µ, t is the real and positive neighbouring site hopping strength, ∆ is the superconducting energy gap (the order parameter) and φa superconducting phase. The fermion operators obey canonical anticommutation relations,
{cx, c†x0}=δx,x0 and {cx, cx0}={c†x, c†x0}= 0. (3.2) The discrete Kitaev chain has a notable similarity with the Ising model. Despite the affinity with onsite terms and nearest neighbour couplings along the chain, the Ising model is semi-classical – classical in the sense of involving commuting variables and quantum mechanical in the sense of describing quantized spins – while the Kitaev chain plays the role of being a truly quantum analogue.
23
3.1 Bogoliubov-de-Gennes Hamiltonian
Discrete Fourier transforms ck of the operators cx are introduced. The context should reveal clearly when we refer to momentum space operators (subscriptk orl) and when we refer to real space operators (subscript x ory),
ck = 1
√N XN x=1
eikx2πNcx and cx = 1
√N XN
k=1
e−ikx2πNck. (3.3)
The Kronecker delta function δk,k0 of period N in this discrete formulation may be expressed in terms of a sum of complex exponentials,
δk,k0 = 1 N
XN x=1
e−i(k−k0)x2πN, (3.4)
and will appear in the deduction below. This formula may be proven by using the summation of a geometric series. The Fourier transformed operators are defined for k-values 1 ≤ k ≤ N, but we can by periodicity extend the allowed range. In particular, when for example−kappears in the following deduction, it is understood to be equivalent toN−k. The transformed operators obey similar anticommutation relations as in equation (3.2). This follows from (3.2), (3.3) and (3.4),
{c†k, ck0}= 1 N
XN x=1
e−ikx2πN XN x0=1
eik0x02πN{c†x, cx0}=δk,k0. (3.5)
The relations{c†k, c†k0}={ck, ck0}= 0 follow analogously.