Robust interface between flying and topological qubits

Hybrid architectures, consisting of conventional and topological qubits, have recently attracted much attention due to their capability in consolidating robustness of topological qubits and universality of conventional qubits. However, these two kinds of qubits are normally constructed in significantly different energy scales, and thus the energy mismatch is a major obstacle for their coupling, which can support the exchange of quantum information between them. Here we propose a microwave photonic quantum bus for a strong direct coupling between the topological and conventional qubits, where the energy mismatch is compensated by an external driving field. In the framework of tight-binding simulation and perturbation approach, we show that the energy splitting of Majorana fermions in a finite length nanowire, which we use to define topological qubits, is still robust against local perturbations due to the topology of the system. Therefore, the present scheme realizes a rather robust interface between the flying and topological qubits. Finally, we demonstrate that this quantum bus can also be used to generate multipartitie entangled states with the topological qubits.


Results
Interfacing topological and flying qubits. We first introduce our setup to realize strong coupling between topological and flying qubits, as schematically shown in Fig. 1. We consider a spin-orbit coupled semiconductor (InAs or InSb) nanowire deposited on a superconducting transmon qubit 53 . Topologically protected MFs, defined as γ 1 to γ 4 from left to right, can be realized when the nanowire is driven to the topological phase regime 10 . In particular, due to the presence of the MFs γ 2 and γ 3 , the single electron tunnelling across the junction will also appear besides the usual Cooper pair tunnelling. Moreover, since the difference between resonant energies of the two type of tunnelings is sufficiently large, it is reasonable to assume that only one of them can be resonantly addressed by the biased voltage.
For a finite nanowire, the coupling between the MFs leads to an energy splitting. In this case, the MFs gain a finite energy while their wave functions are still well localized at the two ends. Roughly speaking, we still have γ ≃ γ † , thus these states with nonzero energies can still be used to encode information for topological quantum computation. For the four MFs defined in Fig. 1(a), we assume their distances to be l 0 , l 1 and l 2 , respectively, which are much longer than the Cooper pair coherent length. In this case, the Hamiltonian of the MF system reads In the eigenbasis of the topological qubit, the above Hamiltonian reduces to , cosϑ = E/ω tq , and sinϑ = J 0 (θ)E M /ω tq . Obviously, since ω c − ω tq ≫ g 0 , any direct energy exchange coupling between the two type of qubits is impossible in the absence of the bias. This is expected from our analysis in the introduction.
However, when the energy mismatch between the cavity field and the topological qubit is compensated by the bias field, i.e., ω c = ω + ω tq , a parametric resonant coupling can be induced. This also means that the coupling between MFs and resonator can be switched on/off very easily by controlling the frequency of the bias field. To see this, we transform the interaction Hamiltonian in Eq. (7) into the interaction picture with respective to the qubit Hamiltonian H tq = ω tq σ z /2, the effective interaction Hamiltonian reads where g = g 0 cosϑJ 1 (θ) and g′ = g 0 J 1 (θ)sinϑ. To obtain the maximum coupling strength, one should set cosϑ = 1, which can be fulfilled when J 0 (θ) = 0 (θ ≃ 2.4). In this case, ω tq = E, g ≈ g 0 /2 and g′ = 0. Neglecting the oscillating terms using the rotating wave approximation, which is valid when ω tq ≫ g, the effective Hamiltonian reduces to H g a a 9 JC σ σ = ( + ), ( ) + − † and the neglected anti-rotating wave terms with the lowest frequency are terms oscillating with frequency of 2ω tq , that is, We therefore map the effective model in Eq. (5) to the well-known Jaynes-Cummings model. This resonant interaction -a bosonic quantum bus Hamiltonian -is readily for quantum information transfer from a topological qubit to the cavity state 55 . The first experiment may be the vacuum Rabi oscillation by preparing an initial state of 0 , the quantum information transfer can be achieved by obtaining a final state of where the excitation of the topological qubit is transferred to the cavity mode. This dynamics can be directly probed in experiments. In this way, we can consolidate the advantage of both quantum systems in a single chip.
Robustness of the MF wavefunction. The appearance of MFs at ends of the nanowire is ensured by the bulk topology. In this case, topological protected zero-energy edge states can be realized at the two ends when the length of the nanowire L → ∞. These localized edge states directly ensures self-hermitian, γ = γ † . The wave function of these edge states decays exponentially to zero in the bulk. For a finite system, the overlap of the two MF wave functions' tails lead to the MF energy splitting, which has been defined in Eq. (1). Here, as shown in Eq. (4), the decoherence of the topological qubit is originated from the fluctuation of hybridized energy splitting, and thus the stability of MFs energy splitting against disorder means the robustness of the defined topological qubit against disorder. It is still not quite clear how robust this splitting is in a realistic nanowire because this energy splitting is in principle not topologically protected, and thus we can not directly infer its robustness from the topological protection. Nevertheless, robustness of this splitting is crucially important for the coupling between conventional and topological qubits. Therefore, we next investigate this problem using a tight-binding numerical simulation and a perturbation approach.
(1). Tight-binding simulation. There are several sources of fluctuation in nanowires, e.g., fluctuations of order parameters (the nanowire length is much larger the Cooper pair coherence) and chemical potential (small carrier density n ~ 10 4 /cm), etc. To mimic these effects on the energy splitting, we consider the following tight-binding model where t = ħ 2 /(2m * a 2 ), λ = α/2a with a the lattice spacing, m * is the effective mass of electron, α is the spin-orbit coupling strength, chemical potential μ i↑/↓ = μ i ± h, and the Zeeman splitting h = g * μ B B z with g * being the Lande factor and B z being the external magnetic field strength along the z-direction. The topologically protected edge states appear when h h c 2 2 μ > = + Δ with Δ being the proximity induced pairing strength, see Fig. 2(a), which is protected by a finite energy around 0.1 meV for the parameters used therein. The energy splitting is an oscillation function of length L, see Fig. 2(b). This is because the localized edge states have oscillating decay function, thus the overlap may exactly disappear at some "magic points". We plot the wave function of the left edge state in Fig. 2(c) with different lengths, which shows that they are almost unchanged except their tails.
We now present our numerical results by solving the tight-binding model over different random configurations. Here we consider two typical fluctuations. In the first study, we assume on-site chemical potential fluctuation μ i = μ − 2t + δμ i , and fix Δ i = Δ . In the second study, we assume μ i = μ, and e i i i Δ = Δ δθ . These two factors are two major fluctuations in low dimensional systems with low carrier density. In both cases, we assume δμ and δθ are independent uniform random numbers distributed in a large region [− W, W] (assuming W ≥ 0). The results are presented in Fig. 3, in which we mainly focus on the lowest two non-negative eigenvalues ε 1 and ε 2 of Hamiltonian in Eq. (11). For the chemical potential fluctuation in Fig. 3(a), the averaged Hamiltonian is exactly unchanged, thus we see the mean value of ε 1 is almost unchanged. We find that the variation of ε 1 almost increase linearly with respect to W. In Fig. 3(c), we plot the overlap j j 0 ρ ψ ψ = as a function of W, where |ψ 0 〉 is the wave function without disorder. Notice that the overlap of the left and right edge states is extremely small (at the order of 10 −4 from our numerical simulation), thus ρ ≃ 1 means that the wave function of the edge state is almost unaffected in a disordered environment. In the second column, we consider robustness of MFs with respect to the phase fluctuation. Two notable differences have been observed. First, as shown in Fig. 3(b), the averaged Hamiltonian is changed because 〈 e iδθ 〉 ≠ 0, thus we find that 〈 ε 1 〉 depends strongly on W. Secondly, as shown in Fig. 3(d), we find that the overlap of the wave functions also depends strongly on the phase fluctuation magnitude. However, we have chosen extremely strong fluctuations in both cases, while these fluctuations should be much smaller in realistic experiments. As a result, we may expect the practical performance to be much better than the results presented in Fig. 3. These simulations under the extreme condition demonstrate clearly robustness of the MF wavefunctions. For this reason, we also expect that the topological qubits has a much weaker dephasing effect than conventional superconducting qubits do. It is worth to point out that we have also calculated the effect of nuclear spin polarization on the energy splitting of MFs, where we have also obtained similar results. In our simulation, we assume state, which is constructed from ψ L ~(ψ +1 + Σ † ψ +1 ) for different nanowire length. The parity of this state is + 1 because Σ † ψ L = + ψ R . The right end state with parity − 1 has similar feature thus is not shown. Other parameters from InSb nanowires are: m * = 0.015m 0 , α = 20 meV ⋅ nm, a = 10 nm, Δ = 0.5 meV and μ eff = 1.0 meV. The overlap of these two wave functions with slightly different lengths is greater than 0.99.
Scientific RepoRts | 5:12233 | DOi: 10.1038/srep12233 a random magnetic field generated by nuclear spins, B i , which introduce a Zeeman splitting smaller than 0.1 meV.
(2) Perturbational analysis. We now develop a model to understand these numerical observations. We wish to show that robustness of MFs splitting is deeply rooted in the bulk topology. To this end, we assume where  is the unperturbed model defined in Eq. (11) and V is the disordered potential, which contains all possible random fluctuations. This model has the basic particle-hole symmetry, that is, Hereafter, for convenience, we assume n > 0 and n < 0 for the eigenfunctions with position and negative eigenvalues, respectively, and thus ε +n = − ε −n , ψ −n = Σ † ψ +n . The system is protected by a fundamental gap, see Fig. 2(a), which is in the order of magnitude Δ . We attempt to understand the topological protection using the second-order perturbation theory. To this end, we assume the two localized wave functions as ψ L and ψ R , where the subscript L (left) and R (right) represent the position of the end states; see a typical example in Fig. 2(c). These two edge states have the following basic features: Σ † ψ L = ψ L and Σ † ψ R = − ψ R . The eigenfunction of ψ ±1 can be constructed using the above two edge states as is the renormalization constant, and ψ ±1 is the eigenfunction of  with eigenvalue ε ±1 .
The random potential V can affect the low-lying excitation. To this end, we assume where the first order correction is The above conclusion can be obtained using the following identity: with k = L, R, and thus 〈 ψ k |V|ψ k 〉 ≡ 0 for any weak random potential. As the wavefunction of the left and right edge states -ensured by topology -is an exponential decay function, see Fig. 2(c), their overlap, due to the random potential V, should be exponentially decay to zero with increasing the length, that is, . Generally, we find 1  . This small ration arises from the oscillation of the edge state wavefunction, in which most of the important contributions are exactly cancelled. In contrast, for conventional qubits, the first-order fluctuations play normally the leading role in the energy fluctuation (thus the decoherence) of qubits.
We next calculate the second-order correction energy, which can be written as (see the Method section for details), where l n = 〈 ψ L |V|ψ n 〉 , r n = 〈 ψ n |V|ψ R 〉 , and no correlation between them can be derived for a general random potential V. However, the most important contributions of l n and r n to 1 2 δε + ( ) are almost cancelled due to the particle-hole symmetry. Notice that the first term in the above equation is suppressed by the large energy gap, i.e., ε n − ε +1 ~ Δ , and thus the cross correction between the left and right edge states is negligible when V is not very strong in a finite length system. Meanwhile, the second term is also exponentially small when L ≫ ξ due to the exponential small overlap between the wavefunctions of the two edge states. Finally, this result is also in consistent with the bulk-edge correspondence in quantum phase transition since when L → 0, we see 0 , that is, the energy of the edge states are unaffected by V. From these results, we can conclude that the topological qubit, even with a finite coupling, is still robust against local perturbations -a basic reason relies on the topology of the bulk. Following these observations, we expect the topological qubit has much smaller dephasing rate, which can be regarded as one major advantage of it. In addition, these results are quite general, and for other topological qubits with some other symmetries, we also expect a similar conclusion.

Realization of the quantum information transfer.
Topological qubits embedded in an environment inevitably have the finite lifetime-this process can be modelled by two parameters: the relaxation rate Γ 1 and the dephasing rate Γ 2 . Here we investigate these two main decohenrence sources for practical experimental realizations of the current scheme. First, dissipative and incoherent quasiparticles tunneling across the Josephson junction and between nanowire and superconductors will break the parity of the qubit system and lead to decoherence. At a temperature of 20 mK 56 , the density of unpaired quasiparticles is 0.04 μm −3 , which leads the parity protection time to be in the order of magnitude ~1 ms 57 . This is sufficiently large comparing with the time for the quantum information transfer process. Secondly, we consider the influence of the superconducting phase fluctuation effect on the MF coupling, which comes from the thermal fluctuations of the bias voltage. For the free σ x term, it is a fast oscillation one, and thus low frequency modulations of the term can be negligible provided that the frequencies are much smaller comparing with ω. As for the other terms, random superconducting phase fluctuations do affect the form of MFs, thus leads to decoherence of the topological qubit. The root mean square error of the superconducting phase is δφ rms ≃ A/e ≤ 10 −3 25 with A ∈ [10 −4 , 10 −3 ]e (Ref. 58) being the amplitude of the 1/f charge noise, the error caused by which is negligible small and far below the threshold 34,35 for error correction.
We now discuss realistic parameters. In circuit QED 59 , the resonator has a wave length of λ = 25 mm and a gap of d = 5 μm between the center conductor and its ground planes, being large enough for a transmon qubit with the loop size of 4 × 4 μm 2 . The transmission probability of the junction is very small, which depends on l 1 and the magnetic field 60 , and thus can be tuneable by tuning l 1 with external electrostatic gates 12 . We modulate E M = 2π × 0.5 GHz, thus g ≈ 2π × 6 MHz for λ c /ω c ≃ 0.05 (Ref. 61). Quality factors above one million have been reported for superconducting resonators with frequencies ranged from 4 to 8 GHz 62 , i.e., the cavity decay rate κ is in the order of KHz. Here we choose κ = 2π × 6 KHz, which corresponds to κ = g/1000. Meanwhile, we choose E = E 2 = 2π × 0.2 GHz, which leads to l 2 ≈ 2.5 μm for an InSb wire with ξ = 216 nm. For l 0 = 4.5 μm, E 1 will be less than 0.01E, and thus can be safely neglected. Therefore, the total length of the wire will be less than 10 μm, which can be deposit on a transmon qubit. In addition, as ω tq = E, the resonate condition ω tq + ω = ω c is readily fulfilled with ω = 10E M = 2π × 5 GHz and ω c = 2π × 5.2 GHz. Obviously, these parameters naturally realize strong coupling between topological qubits and cavity since g ≫ {κ, Γ 1 , Γ 2 }.
We estimate the errors for the quantum information transfer process under realistic conditions. First, we consider the decay of the cavity mode (κ) with a thermal cavity photon number n, the relaxation (Γ 1 ) and dephasing (Γ 2 ) of the topological qubit. Under these decoherence effects, the dynamics of the system can be well described by the following master equation where ρ 1 is the density matrix of the combined system of the topological qubit and the cavity, n c is the number of photon in the cavity, and We simulate the quantum information transfer process using the conditional fidelity defined by F 1 = f 〈 ψ 1 |ρ a |ψ 1 〉 f , with ρ a being the reduced density matrix of the topological qubit from ρ 1 . Assuming the cavity is initially prepared in the vacuum state |0〉 c , i.e., n c = 0, we obtain a high fidelity of F 1 ≃ 99.9% for the quantum information transfer process at gt/π = 1/2 with κ = Γ 1 = Γ 2 = g/1000.
We next turn to consider the influence when the cavity is initially in a thermal state. Typically, the cavity is cooled down near its quantum mechanical ground state and the thermal occupancy related to the working temperature T of the cavity as n c = 1/[exp(ħω c /K B T) − 1]. To simplify our treatment, we assume the initial state of the thermal cavity to be n n . With the same parameters as above, as shown in Fig. 4, we plot the maximum of F 1 , with rotating wave approximation, as a function of T. We find that the infidelity is less than 0.1% when T ≤ 35 mK. For superconducting devices cooled to 20 mK inside a dilution refrigerator, the temperature effect in our scheme is negligible.
Finally, we consider the influence of neglecting the counter-rotating terms in deriving the Hamiltonian in Eq. (9). Here the neglected terms with frequencies in the order of ω tq are those in the Hamiltonian of Eq. (10). This is well justified numerically for ω tq = E ≈ 33g, as shown by the insert of Fig. 4, where the blue and red dashed lines are simulated by the Hamiltonian of H JC in Eq. (9) with the absence and presence of H AJC in Eq. (10), respectively. The two results are in very good agreement, and the infidelity induced by this approximation is less than 0.1% within the three periods of Rabi oscillation.
Application to entangled states generation. When incorporating more than one qubit, we next show that our quantum bus model can be naturally used to generate entangled states of topological qubits. We consider the multi-qubit case as shown in Fig. 1(b) and modulate ν = ω c − E j − ω j > 0 for all the N qubits, which leads the total interaction Hamiltonian to be on the resonator can be obtained 59 by capacitively coupling it to a microwave source with frequency ω d , with ε being a time independent amplitude. For large amplitude driving and under a time-dependent displace- , the direct drive on the resonator can be eliminated. Under resonant driving (ω d = ω tq ), and change to a frame rotating at the frequency of ω d , the driven induced collective Rabi oscillating Hamiltonian of the topological qubits reads  Figure 4. Fidelity of the quantum information transfer between the topological qubit and the cavity. The maximum fidelity as a function of the working temperature T of the cavity with κ = Γ 1 = Γ 2 = g/1000. Insert: The fidelity dynamics as a function of gt/π at zero temperature. The blue and red dash lines are simulated with and without rotating wave approximation, respectively.
where 0 1 2 j j j ( ) ± = ± / . In the case of Ω ≫ {ν, g}, we can omit the fast oscillation terms (of frequencies Ω ± δ), then the Hamiltonian becomes H g ae a e J 17 with μ = x, y, z. In this case, the time evolution operator can be written as 64 It is obvious that B(t) is a periodical function and equals zero when t = T k = 2kπ/ν with k being a positive integer. Therefore, at these special points, Eq. (18) reduces to where A(T k ) = 2kπg 2 /ν 2 . For N qubits in an initial state of |ψ 2 〉 i = |00···0〉 , choosing A(T k ) = π/2, the final state i J exp is a GHZ state given by 66 This generation has the following distinct merits. First, the generation is fast. To be specifically, A(T k ) = π/2 can be obtained when g k 2 ν = . Then, for k = 1, one obtains ν = 2g and the entanglement generation time T = π/g, which is comparable with that of using the resonant Jaynes-Cummings interaction. This is due to a fact that the interaction used in this generation is not of the dispersive nature, and thus removes the needs of large detuning (ν ≫ g). Secondly, the generation is readily for scale up. As the operator in Eq. (19) is obtained to be independent on the number of the involved qubits, the time needed for the gate operation does not depend on the number of qubits. Therefore, this generation can be scalable provided that the qubits can be incorporated in the cavity for every wave length section of the cavity, and there can be four qubits located at the antinodes, as shown in Fig. 1. Finally, in the time evolution operator of Eq. (18), as B(t) is a periodical function, the cavity state dependent terms, i.e., the second and third terms, are removed, leading to a cavity field state insensitive operator of Eq. (19). Since the cavity will return to its original state, one can avoid cooling of the cavity to its ground state before the application of the operator in Eq. (19), which looses the limitation of the thermal effect in engineering quantum states.
However, the time evolution does involve the excitation of the cavity during the generation, so that we need to include its influence as well as others. Then, we estimate the fidelity for the generation process by the Lindblad master equation. For the N = 2 case, we can obtain a high fidelity of F 2 ≃ 99.3% at t = π/g for the generation with κ = Γ 1 = Γ 2 = g/1000 at T = 0. For N > 2 cases, the maximum of F 2 will decrease gradually due to the decoherence of the increased number of qubits. Nevertheless, we can still obtain fidelities of 98.5% and 96.8% for the entanglement generation with N = 4 and N = 8, respectively. As it is well known, the fidelity of the generation drops with the increase of the decohenrence rates. For the cases of N = 4 and N = 8, as shown in Fig. 5, we also plot the maximum of F 2 with decohenrence rates in the range of Γ 1,2 ∈ [1, 10]κ. It should be emphasized that the dephasing term Γ 2 has a leading effect in F 2 for the multipartitie entangled state. In the previous sections, we demonstrate that the topological qubits is much more stable than the conventional qubits in environment, and thus we expect Γ 2 to be much smaller than that in conventional qubits, namely, F 2 for topological qubits can be much higher than that for conventional qubits.
In summary, we have proposed a microwave photonic quantum bus for a direct coupling between flying and topological qubits, in which the energy mismatch is compensated by the external driving field. Strong coupling between these two qubits can be realized. It has also been shown that from the realistic tight-binding simulation and perturbation theory that the energy splitting of the MF wavefunctions in a finite length nanowire is still robust against local perturbations, which is ensured by the topology. Thus our scheme is rather promising for implementing a robust interface between the flying and topological qubits. Finally, we have demonstrated that this quantum bus can be used to generate multipartitie entangled states with the topological qubits.

Method
Derivation of Eq. (6). We begin with the Hamiltonian in Eq. (5) To obtain a time-independent effective Hamiltonian for Eq. (23), we first need to deal with the time-dependent driven terms of S(t). This time-dependency can be safely neglected when J n (θ)E M / (nω) ≪ 1. To see this, we perform n transformations with frequencies nω, which are defined by U(t) = U n U n−1 …U 2 U 1 with . First, we can expect that the second-order correction energy to ψ ±1 is exactly equals to zero when L → ∞. It is easy to understand from the following identity (k = L, R), where we have assumed that the left and right edge states ψ L,R have well-defined chirality. Note that ψ L,R are not necessary to be the eigenvectors of the Hamiltonian. Physically, it means that the second-order correction energy from the particle and hole sectors exactly cancels with each other, and thus 0 2 δε = ± ( ) when L → ∞.
In the following, we wish to show that in the finite length case, contributions from the particle and hole sectors will also almost be canceled, and thus the net second-order correction energy is also very small. To this end, we need to calculate The correlation energy to ψ −1 can be calculated using a similar manner, and we can prove exactly that , which ensures that the perturbation method also respects the particle-hole symmetry. Using the identity Eq. (32), we obtain the correction energy as in Eq. (13), the matrix elements have the following general properties: Notice that ψ n may contain an arbitrary phase, thus both 〈 ψ L |V|ψ n 〉 and 〈 ψ R |V|ψ n 〉 are generally complex numbers. In other words, the first term in Eq. (13) is in general non zero. In fact, the second-order correction is exactly equal to zero only when the left and right wavefunctions are well separated. In this case, ψ L and ψ R are also the eigenvectors of the Hamiltonian, and thus V 0 L R ψ ψ = and V 0 n L ψ ψ = . The numerical results show that the contribution of the first term in Eq. (13) is much smaller than the second-term in a finite length system. This can be understood as follows. First, the system protected by a large energy gap, so the second-order contribution is greatly suppressed. Secondly, the edge states are fast oscillating function in real space, while the extended states ψ n are well-extended in the real space. Thus the overlap between the localized state and extended state mediated by the random potential is very small. Therefore, the major contribution to the second-order correction energy comes from the second term in Eq. (13). Note that V L exp [ 2 ] L R ψ ψ ξ ∼ − /( ) and ε +1 ~ exp[− L/(2ξ)], thus it is reasonable to expect that the second term is also very small. Obviously, lim L→∞ ε +1 (L) = 0, which is in consistent with the well-known bulk-edge correspondence in topological phase transitions.