Extracting entangled qubits from Majorana fermions in quantum dot chains through the measurement of parity

We propose a scheme for extracting entangled charge qubits from quantum-dot chains that support zero-energy edge modes. The edge mode is composed of Majorana fermions localized at the ends of each chain. The qubit, logically encoded in double quantum dots, can be manipulated through tunneling and pairing interactions between them. The detailed form of the entangled state depends on both the parity measurement (an even or odd number) of the boundary-site electrons in each chain and the teleportation between the chains. The parity measurement is realized through the dispersive coupling of coherent-state microwave photons to the boundary sites, while the teleportation is performed via Bell measurements. Our scheme illustrates localizable entanglement in a fermionic system, which serves feasibly as a quantum repeater under realistic experimental conditions, as it allows for finite temperature effect and is robust against disorders, decoherence and quasi-particle poisoning.

where φ j,k and ψ j,k are real coefficients determined by the condition that the Hamiltonian (1) in the main text is diagonalized: H = ∑ N j=1 λ j (b † jb j − 1 2 ). We note that [b j , H] = λ jb j . If there exists a zero mode, say λ N = 0, we have [b N , H] = 0. [b N , H] can be calculated by using Eq. (1) and (1). By requiring the coefficients of c j and c † j in the calculated result to be zero, we get where the variables with subscripts equal to (N, 0) or (N, N + 1) are assumed to be 0. For a uniform chain i.e. w j = w, ∆ j = ∆ and µ j = µ in the above two equations, we notice that the transfer matrices are identical: −µ ∆+w ∆−w ∆+w 1 0 ≡ A. However, the index for φ N,i will increase upon the action of the transfer matrix, while for ψ N,i the index will decrease. So, we can set ψ N, j = φ N,N+1− j , and the Bogoliubov-de Gennes transformation (1) forb N becomes Suppose the two eigenvalues of the transfer matrix A are λ 1 and λ 2 , i.e. A|λ i = λ i |λ i , i = 1, 2, so A −1 A|λ i = |λ i = λ i A −1 |λ i , thus the two eigenvalues of A −1 are λ −1 1 and λ −1 2 (for λ i = 0). If |λ 1 | < 1 and |λ 2 | < 1 (or, |λ −1 1 | < 1 and |λ −1 2 | < 1), we will have a decaying solution for φ N,i (or φ N,N+1−i ), i = 1, 2, · · · , N, in the thermodynamic limit N → ∞. This corresponds to |µ| < 2w and ∆ = 0. For general values of w j , ∆ j and µ j , Ref. 36 proves that if w j and ∆ j are sign-ordered i.e. sign(∆ j w j )=sign(∆ j+1 w j+1 ), and |µ j | < max(|w j−1 |, |∆ j−1 |), then the chain has zero-energy Majoarana fermions. The solution (4) can be verified by substituting it into Eq. (2). Actually, there is another solution: µ 1 = µ N = 0, ∆ 1 = −w 1 , ∆ N−1 = −w N−1 , corresponding to the absence of d 2 and d 2N−2 in Eq. (3). The two MFs form a zero-energy edge mode that is similar to Eq. (5). Further analysis for this solution is also very similar to that for Eq. (4) and omitted. Next, we prove the parity equality: . The Hamiltonian (3) can be written as H = i 4 ∑ l,m A l,m d l d m . The coefficient matrix A is block diagonalized by a 2N × 2N real orthogonal matrix W : A j,k = ∑ m,n W T j,m Λ m,n W n,k , where Λ is a block diagonal matrix: Λ 2k−1,2k = −Λ 2k,2k−1 = λ j (other matrix elements are zero). Note that WW T = W T W = I so that (detW ) 2 = detW detW T = det(WW T ) = 1. When detW = 1, the parity equality holds 38 . For the parameters in Eq. (4), we have the zero-energy mode in Eq. (5). The corresponding W will be W j,1 = δ j,2N−1 , Here η = detW 0 is defined in Eq. (5). Therefore, detW can be calculated, according to the definition of determinant, as detW = η detW 0 = η 2 = 1. This concludes the proof.
can be calculated by using these expressions and Eq. (6), and the result is 1 4 . Thus, x 0 can be chosen to be √ 2 2 , and x 1 is determined by normalization condition of the state (8): For general values of ∆ j , w j , µ j , we only have Eq. (2) and (5), butb j ( j = 1, 2, · · · , N − 1) is unknown, which is a function of c k , c † k (k = 1, 2, · · · , N) and can be obtained by diagonalizing the coefficient matrix in the Hamiltonian (1). Then, we can solve for c 1 and c N to determine x 0 , x 1 , η in the state (8). The state |G 2 =b † N |G 1 by using Hermitian conjugate of Eq. (5). For the situation N = 2, the calculation is similar, where the two ground states are those before |S e and |S o in Eq. (5).

Estimation of J
The coupling between the microwave and the spin of the quantum dots in our proposal is analogous to the electron spin resonance which is usually weak as compared with the coupling through the electric dipole moment and thus ignored. However, in our experiment, this coupling is significant due to the presence of strong spin-orbit coupling in the quantum dots. We shall estimate the coupling strength J below. Note that the microwave is applied to the two boundary quantum dots which are decoupled from the inner part of the chain. Therefore, the superconducting proximity effect are not present in the measurement process.
With the measurement setup in Fig. 5, the Hamiltonian for a single quantum dot and the microwave changes tohω m a † a + 1 2m * (p + eA) 2 − eφ m + αẑ · [(p + eA) × σ ] + V (r), where a † (a) is the creation (annihilation) operator of the microwave photons with frequency ω m , p is the momentum of the electron, A and φ m are the vector and scalar potentials of the microwave, σ = (σ x , σ y , σ z ) is a vector of Pauli matrices, V (r) is the confining potential. The term −eφ m ∼ er · (E + ∂ t A) describes the interaction of the electromagnetic filed with the dipole moment of the electron. This term is considered for coupled double quantum dots, 62 but it can be ignored here for a single quantum dot (or two decoupled quantum dots) because the only possible transition is that between the two energy levels in Eq. (6) which is electric-dipole forbidden ( ψ + |r|ψ − = 0). An exception is to use a third energy level to induce Raman transition, 63 but this method will not be considered here due to its incapability of realizing the dispersive coupling. One could also consider this third level and |ψ − forming a two-level system which allows transition through the electric dipole (usually with optical frequency). This method is feasible for achieving the dispersive coupling, but transitions involving other levels (e.g. from the aforementioned third level to |ψ + ) must be suppressed, otherwise spin-photon entanglement will be generated. 64 Instead, when restricted to the subspace spanned by |ψ ± , the relevant part describing the electron-microwave interaction in the Hamiltonian is H r = e m * (p x A x + p y A y + p z A z ) + eα(σ y A x − σ x A y ). We assume that ψ 0 |p|ψ 0 = 0, and notice that ψ + |p x |ψ − = im * α. The electromagnetic gauge is chosen to be 62,65 where k is the wave vector of the microwave, B 0 is the rms vacuum fluctuations of the magnetic field of the microwave, c tot is the total capacitance of the transmission line resonator (TLR), a (a † ) is the annihilation (creation) operator of the microwave photons, l is the coordinate along the electric field lines, and l 0 is the distance between the two planes facing the quantum dots (see Fig. 7). The second quantization of H r with the rotating-wave approximation is where c η = − ψ 0 | cos( 2πx l so + 2η)|ψ 0 . Experimentally, α = π/(m * l so ) with m * = 0.023m e for the InAs quantum dot, l so ∼ 100 nm, and |B 0 | ∼ |∇ lφm |/c = 1 l 0 c hωm c tot (c the speed of light), ω m /2π ∼ 10 11 Hz, c tot ∼ 1 pF, l 0 ∼ 5 µm, k = 2π/λ ∼ 2 × 10 3 m −1 and assume x|ψ 0 = 2 L sin( πx L ) with L ∼ 120 nm the width of the quantum dot in x direction, J/2π can be tuned to 200 MHz. The phase φ in Eq. (11) can be generated through the time evolution e itδ f † j f j f † j e −itδ f † j f j = f † j e itδ with t = (φ + 2nπ)/δ , n an integer. If we ignore the spin-orbit coupling, but instead consider the direct interaction of the microwave's magnetic field with the electron's spin via the Zeeman effect: H z = −µ · B = e 2m * σ · B 0 (a + a † ). The corresponding coupling strength after the second quantization and the rotating-wave approximation are performed will be J z = e|B 0 |/(2m * ). The ratio J/J z = 2αc η m * /k = c η λ /l so ∼ 10 4 , indicating that the spin-microwave interaction will be considerably enhanced when the spin-orbit coupling is significant. We notice that this enhancement has already been pointed out in Ref. 47

Teleportation
The state (9) will be useful for performing quantum teleportation if its parity is determined through the microwave measurement. Suppose the boundary sites are in the state: 1 √ 2 (|00 1,N + |11 1,N ). We have an extra fermionic site with the index 0. The quantum state of this site may be correlated with its environment. Denote the state of the site 0 and its environment as |χ 0 = y 0 |0 0 |φ 0 + y 1 |1 0 |φ 1 , where |0 0 (|1 0 ) means that there is no (one) fermion in the site 0, y 0 and y 1 are the amplitudes, and |φ 0 and |φ 1 are some states of the environment. The task is to teleport the state in the site 0 to the site N. As we know, one need to perform Bell measurement. This seems not possible, but we can realize it indirectly through basis transformation in the Hilbert space of the site 0 and 1. The initial state is Then the interaction (hopping and pairing) between the sites 0 and 1 is switched on, and their chemical potential is finely tuned to be zero. The Hamiltonian is Choose the hopping amplitude w 0 and the superconducting gap ∆ 0 to satisfy w 0 = ∆ 0 > 0. The time evolution operator is exp(−itH 01 ). Setting t = t 0 ≡ π 4∆ 0 , we have where we have written the states of the sites 0 and 1 first, and then the states of the site N and the environment of the site 0. It can be seen that we have four results when measuring the site 0 and 1 in the number basis: | jk , j, k ∈ {0, 1} (see Ref. 28 for the experimental realization of charge measurement). For each result, the corresponding state for the site N and the environment of the site 0 is equivalent to the original state |χ 0 up to a local unitary transformation (gate) on the site N. The results 00 and 11 involve phase gates which are fulfilled by applying electric voltage, while the results 01 and 10 involve bit-flip gates which are realized by coupling the site N to a chain supporting zero-energy edge modeb † mbm . The chain has a Hamiltonian similar to Eq. (1) with parameters in Eq. (4), but N there is all replaced by another length m. The inner sites of a sub-chain in Fig. 1 can be chosen to serve as this chain. The coupling between the site N and the chain is 46 : where κ denotes the coupling strength. We have exp(−itH f )|0 N |G 1 = −i|1 N |G 2 and exp(−itH f )|1 N |G 1 = −i|0 N |G 2 when t = π/(2κ), realizing the bit-flip gate. Here |G 1 and |G 2 are the ground states of the chain (see Eq. (6) with N replaced by m), and their positions can be interchanged to obtain the other two Eqs. of time evolution.
For the other maximally entangled state (|10 1,N + |01 1,N )/ √ 2, the discussion is very similar. We only need to flip the state of the site N in Eq. (10) to obtain the result. To suppress the charge noise, it is preferred to use the basis of |10 and |01 rather than |00 and |11 of the double quantum dots for encoding the qubit. In this situation, the state (|00 1,N + |11 1,N )/ √ 2 and the pairing interaction in (9) should be avoided. When ∆ 0 is set to be zero in (9), it can be verified that the teleportation succeeds with the probability 1/2.