Circulator function in a Josephson junction circuit and braiding of Majorana zero modes

We propose a scheme for the circulator function in a superconducting circuit consisting of a three-Josephson junction loop and a trijunction. In this study we obtain the exact Lagrangian of the system by deriving the effective potential from the fundamental boundary conditions. We subsequently show that we can selectively choose the direction of current flowing through the branches connected at the trijunction, which performs a circulator function. Further, we use this circulator function for a non-Abelian braiding of Majorana zero modes (MZMs). In the branches of the system we introduce pairs of MZMs which interact with each other through the phases of trijunction. The circulator function determines the phases of the trijunction and thus the coupling between the MZMs to gives rise to the braiding operation. We modify the system so that MZMs might be coupled to the external ones to perform qubit operations in a scalable design.

www.nature.com/scientificreports/ with each other to form a lattice structure. We thus consider an improved design where the trijunction is located outside of the loop as shown in Fig. 1b, which is topologically equivalent with the design in Fig. 1a. Further, we can use the circulator function to realize the braiding of Majorana zero modes (MZM) 9,10 for topological quantum computing 11,12 . Topologically-protected quantum processing is expected to provide a path towards fault-tolerant quantum computing. Since quantum states are susceptible to environmental decoherence, protection from local perturbation is an emergent challenge for quantum information processing. Non-Abelian states are the building block of topological quantum computing carrying the nonlocal information. The nonlocally encoded quantum information is resilient to local noises and, if the temperature is smaller than the excitation gap, temporal excitation rate is exponentially suppressed. Majorana zero modes, γ , are predicted to exhibit non-Abelian exchange statistics, and they are self-adjoint γ † = γ in contrast to ordinary fermion operators. The theoretically proposed structures attracted a great deal of intention to realizing MZMs in condensed matter systems. MZMs are predicted to emerge in ν = 5/2 fractional quantum Hall states 11,13 , p-wave superconductors 14,15 , and one-or two-dimensional semiconductor/superconductor hybrid structures 16 . The branches in our scheme for braiding contains semiconductor/superconductor hybrid structures with p-wave-like superconductivity induced from s-wave superconductors via proximity effect.
In two-dimensional spinless p + ip topological superconductors MZMs are hosted in vortices or in the chiral edge modes as localized Andreev-bound zero-energy states at the Fermi energy. The p-wave-like superconductivity can be induced from s-wave superconductors via proximity effect in a hybrid structure 17 . Semiconductor thin film with Zeeman splitting and proximity-induced s-wave superconductivity has been expected to be a suitable platform for hosting MZMs 18 . On the other hand, the one-dimensional semiconducting nanowire has also been shown to provide MZMs at the ends of the nanowire 19 . The MZMs should be prepared, braided, and fused to implement qubit operations. In one-dimensional wire the braiding is not well defined, which can be overcome in a wire network of trijunction. However, the original scheme 17 with Josephson trijunction has not yet been explored.
Recently, an experimental evidence of MZM in a trijunction has been reported 20 . The nanowire trijunctions are manipulated by the chemical potential 21 , the charging energy 22 , and the phase 23 . In the present study a pair of MZMs can be introduced in each branch near the trijunction of Fig. 1b. Three MZMs of each pair are coupled through Josephson junctions with phase differences ϕ ′ 1 , ϕ ′ 2 , and ϕ ′ 3 in the system. The three Josephson junction loop controls the selective coupling among three MZM pairs. By applying a threading flux into one of the loops of Fig. 1b we can use the circulator function to control the phases φ ′ i and thus the couplings among MZMs in the trijunction to perform the braiding operation and, further, quantum gate operations. In contrast to the previous phase modulation scheme 23 trying to switch off the current mediated by MZMs which are inside of the loop the present proposal uses circulating function to perform braiding operations. Further, our scheme enables the interaction between MZMs outside so that we may provide a scalable design in a one or two-dimensional lattice system for coupling between MZMs which belong to different trijunctions.

Results
Three-Josephson junction loop with a trijunction. The precise fluxoid quantization condition of superconducting loop reads −� t + (m c /q c ) � v c · d � l = n� 0 with v c being the average velocity of Cooper pairs, q c = 2e the Cooper pair charge, and m c = 2m e the Cooper pair mass 24,25 . The total magnetic flux t threading the loop is the sum of the external and the induced flux t = ext + ind . With the superconducting unit flux quantum � 0 = h/2e we introduce the reduced fluxes, f t = � t /� 0 = f + f ind with f = � ext /� 0 and f ind = � ind /� 0 , expressing the fluxoid quantization condition as kl = 2π(n + f t ) with l being the circumference of the loop, k the wave vector of the Cooper pair wavefunction and n an integer.
The scheme in Fig. 1a consists of three-Josephson junction loop and three small loops with threading fluxes f i = � ext,i /� 0 . The fluxoid quantization conditions around three loops, including the phase differences ϕ i and ϕ ′ i across the Josephson junctions, are represented as the following periodic boundary conditions 26,27 , where k i , l , and l ′ are the wave vector of Cooper pairs, the length of the three-Josephson junction loop, and three branches, respectively, and m i 's are integer. Here, ϕ i 's are the phase differences of Josephson junctions in the three-Josephson junction loop and ϕ ′ i 's phase differences of the trijunction whose positive direction, we choose, is clockwise as shown in Fig. 1a. Which branches carry current, while the other not, is determined by threading a flux, f i , into a specific loop.
The induced flux f ind,1 , for example, can be written as to represent the boundary conditions as In the system of Fig. 1a three Josephson junctions with ϕ ′ i compose a trijunction which satisfies the periodic boundary condition ϕ ′ 1 + ϕ ′ 2 + ϕ ′ 3 = 2πn ′ with an integer n ′ . By using this condition and summing above three equations we can check that the boundary condition for three-Josephson junction loop can be expressed as with an integer n, which can also be derived directly from the fluxoid quantization condition. If we assume the superconducting branches in  www.nature.com/scientificreports/ where the effective inductances are defined as L eff ≡ L K + L s and L ′ eff ≡ L K + L s + 9(L ′ K + L ′ s ) . Here and after, the indices, i, are modulo 3, for example, i + 1 = i + 1 mod 3.
The dynamics of Josephson junction is described by the capacitively-shunted model, where the current relation is given by I = −I c sin φ + CV = −I c sin φ − C(� 0 /2π)φ with the critical current I c , the capacitance C of Josephson junction, and the voltage-phase relation, i with the Josephson coupling energy E J = � 0 I c /2π and the current I = −(n c Aq c /m c ) k . From the Lagrangian By using the quantum Kirchhoff relation the equation of motion, then, can be represented as which consists of the inductive energies of the loops and Josephson junction energies with E ′ J being the Josephson junction energy of trijunction. We can easily check that the effective potential U eff ({ϕ i , ϕ ′ i }) satisfy the equation of motion in Eq. (11) for φ i = ϕ i with k i 's in Eq. (9). The kinetic inductance L K is much smaller than the geometric inductance L s . For the usual parameter regime for three-Josephson junction qubit L K /L s ∼ O(10 −3 ) 30 so that we can approximate the effective inductances as L eff ≈ L s and should also satisfy the quantum Kirchhoff relation for the phase variables ϕ ′ i . In Fig. 1a we consider the currents Ĩ i across the Josephson junction with phases ϕ ′ i and I ′ i flowing in the branch, where the direction of Ĩ i is counterclockwise and Supplementary Information). Then with the current conservation relation at nodes, Limiting case. In the system of Fig. 1a we can consider the limit that the length of branches goes to zero, l ′ → 0 , and thus two nodes at the either ends of a branch collapse to a point. As a result, we have three loops with geometric inductance L s /3 which meet at the trijunction. In this limit L ′ (12) becomes which describes the inductive energies of three loops with geometric inductance L s /3 and the Josephson junction energies 25,31,32 , complying with the intuitive picture.
Circulator function. In order to perform the NISQ computing we need to construct a scalable design with the circulator function, where the trijunctions are connected to others and the current directions can be controlled in situ in the circuit. However, in the design in Fig. 1a the trijunction is inside of the loop so it is not possible to couple the branches with others outside. Hence we consider an improved design where the trijunction is located outside of the loop as shown in Fig. 1b. In the Supplementary Information we show an archetype for a scalable design. Actually the inner branches and the trijunction are turned over, but the design is topologically equivalent with the design in Fig. 1a. Here the length l of central branch is not equal with others anymore. We then introduce more general boundary conditions for the scheme in Fig. 1b including the phase differences across the Josephson junctions as www.nature.com/scientificreports/ with integers m i . The boundary condition in Eq. (15) describes the outmost loop containing the Josephson junctions with phase differences ϕ 1 and ϕ ′ 1 and the conditions in Eqs. (16) and (17) the left and right loop in Fig. 1b. With the geometric and kinetic inductances L s and L K = m cl /An c q 2 c for the central branch, respectively, the induced fluxes become f ind, 1l − (L s /L K )k 3 l/3] to give rise to the relations similar to those in Eqs. (5), (6) and (7) where k ′ 1 l ′ 's are replaced with k ′ 1l . From these relations in conjunction with the relations in Eq. (8) we can similarly calculate k i and k ′ i with i = 1, 2, 3 in terms of ϕ i and ϕ ′ i (see the Supplementary Information). In order to induce current flowing between the branches across ϕ ′ 1 , we initially apply the flux ext,1 so that f 1 = � ext,1 /� 0 = f , but f 2 = f 3 = 0 . We then can easily check that the following effective potential satisfies the equation of motion in Eqs. (11) and (13), in Eq. (12) for the system in Fig. 1a with f 1 = f and f 2 = f 3 = 0 . Figure 2 shows the effective potential for the design in Fig. 1b, which is qualitatively similar to that for the model in Fig. 1a.
We introduce a coordinate transformation such as The effective potential in Eq. (19), then, can be expressed as  Figure 2a shows the effective potential U eff as a function of (ϕ p , ϕ m ) for m 1 = m 2 = m 3 = n = n ′ = 0 , which is minimized with respect to ϕ ′ p , ϕ ′ m and ϕ 1 . If the value of the external flux f = 0.5 , two degenerate current states, clockwise and counterclockwise, are superposed so that we cannot determine the direction of current. We thus set the value of the external flux f = 0.42 to obtain a stable minimum. The effective potential U eff (ϕ p , ϕ m ) along the dotted line in Fig. 2a is shown in Fig. 2b, where U eff (ϕ p , ϕ m ) has a minimum at ϕ p /2π ≈ 0.124 . Figure 2c shows the profile of effective potential U eff (ϕ p , ϕ m ) as a function of ϕ m for ϕ p /2π ≈ 0.124 . Here the effective potential has the minimum at ϕ m = 0, i.e., ϕ 2 = ϕ 3 . Figure 2d show that ϕ ′ m = 0, i.e., ϕ ′ 2 = ϕ ′ 3 at the minimum of the effective potential U eff (ϕ p , ϕ m ) . From Eqs. (4) and (9) we can see that k 2 = k 3 and thus I 2 = I 3 and from Eq. (10) k ′ 1 = 0 , and thus I ′ 1 = 0 , which is consistent with the current conservations, I 3 − I 2 = I ′ 1 = 0 , in Eq. (8). Hence, in Fig. 1b we can determine the direction of current such as I ′ 3 = −I ′ 2 � = 0 , and I ′ 1 = 0 . If we consider the case that Hence we can selectively determine the direction of currents flowing through a trijunction by threading a magnetic flux into a specific loop in the design of Fig. 1b, which can realize the circulator function in a scalable design.

Braiding of Majorana zero modes.
We can use the circulator function for the braiding of Majorana zero modes (MZM) for topological quantum computing. As shown in Fig. 4a we introduce three pairs of MZMs in the semiconducting nanowire with p-wave-like superconductivity induced from s-wave superconducting branch via proximity effect. For the quantum computing the scheme for quantum gate operation should be provided. Hence we consider the system of Fig. 1b because for the system of Fig. 1a the MZMs are inside of the loop so that the MZMs cannot interact with MZMs outside 23 .
In Fig. 3a we show the currents I ′ 1 = I 3 − I 2 , I ′ 2 = I 1 − I 3 , and I ′ 3 = I 2 − I 1 of the system in Fig. 1b as a function of f 1 − f 2 . If f 1 = f = 0.42 with f 2 = f 3 = 0 , the current direction is determined such that I ′ 1 = 0 , but In this case the current flows between the branch with γ 2 and the branch with γ 3 . This is the initial state of the system shown in Fig. 4b, where the three MZMs, γ ′ 1 , γ ′ 2 and γ ′ 3 , are tunnel-coupled with each other through the Hamiltonian 22,23 with Majorana Josephson energy E M and coupling energy α . Then the current carried through MZMs across trijunction is given by (21)  www.nature.com/scientificreports/ with a 4π-periodic behavior 33 . Actually we have ϕ ′ 1 /2π ≈ 0.246 and ϕ ′ 2 /2π = ϕ ′ 3 /2π ≈ −0.123 at the minimum of the effective potential U eff (ϕ p , ϕ m ) in Fig. 2a. Then the current I 1 has a larger amplitude than I 2 = I 3 as shown in Fig. 3b, which is denoted as a solid (dotted) line for I 1 (I 2 and I 3 ) in the trijunction of Fig. 4b. As shown in Eq. (22) the current mediated by MZMs I i ∝ sin ϕ ′ i /2 , while the Cooper pair current Ĩ i ∝ sin ϕ ′ i . If we consider a simplified model such that the Josephson junctions in the three-junction loop in Fig. 1a are removed (22)   www.nature.com/scientificreports/ as in the previous study 23 , the boundary condition becomes approximately ϕ ′ i − 2πf i ≈ 0 . Here, even if we set f i = 0.5 and thus ϕ ′ i ≈ π , we cannot switch off the current mediated by MZMs as I i = 0 while Ĩ i ≈ 0 . Hence, instead of switching off I i we change the current direction by using circulating function to perform the braiding operation.
In general, for f i = 0.42 with f i±1 = 0 we have ϕ ′ i /2π ≈ 0.246 and ϕ ′ i±1 /2π ≈ −0.123 . The different phases are due to the current direction, resulting in the asymmetry in the amplitude of I i at the trijunction. In next stage we adiabatically apply the flux f 3 , while decreasing f 1 (See Eq. (S27) of Supplementary Information for general f i ). In Fig. 3a, then, |I ′ 1 | increases while |I ′ 3 | decreases. In the meanwhile, |I ′ 2 | decreases to zero and then grows up to the maximum value. Finally for f 3 = 0.42 with f 1 = f 2 = 0 , we have I ′ 3 = 0 , but I ′ 1 = I ′ 2 � = 0 . Hence the current direction is changed: the current I i flows between the branch with γ 1 and the branch with γ 2 but there is no current in the branch with γ 3 as shown in Fig. 4c, and meanwhile the green MZM loses its weight in γ 3 and gains weight in γ 1 . Here the current I 3 has a larger amplitude than I 1 = I 2 , and thus the asymmetry in the amplitude of I i is changed. In this way, between t = τ and t = 2τ , the yellow MZM loses its weight in γ 2 and gains weight in γ 3 as shown in Fig. 4d. At the last stage the green MZM loses its weight in γ 1 and gains weight in γ 2 . As a result, the green and yellow MZMs are exchanged with each other as shown in Fig. 4e, completing the braiding operation.
In Fig. 5 we show an architecture for a scalable design for a superconducting circuit with MZMs. Two MZMs belong to different trijunctions (the green box in Fig. 5) can be coupled or fused to perform quantum gate operations and quantum measurements. For the green box operation, for example, we can introduce a gate voltage applied to the sector between two MZMs to control the chemical potential of the nanowire 34 . Though the system in Fig. 5 is one-dimensional, we can extend it to two-dimensional lattice straightforwardly.

Discussion
In conclusion, we proposed a scheme for the circulator function in a superconducting circuit consisting of three small loops and branches which meet at a trijunction. Usually the effective potential in the Hamiltonian for superconducting circuit is phenomenologically obtained. However in this study we obtained the boundary conditions from the fundamental fluxoid quantization condition for the superconducting loop to derive the effective potential of the system analytically, which is required for accurate and systematic study for the quantum information processing applications. We expect that this kind of study can be applied to other systems.
At the minimum of the effective potential we can see that two branches carry current while the other does not. By applying a magnetic flux into one of the loops we can determine which branches among three carry the current, achieving the circulator function. For the NISQ computing we need to perform the circulator function in a scalable design. We thus introduced an improved model where the trijunction is extracted out from the outmost loop to interact with other external branches. For the improved design we obtained the ground state of the system from the effective potential, and showed that it can perform the circulator function in the trijunction loop.
Instead of switching off the current mediated by MZMs in the previous study, in this study we selectively choose the current directions to give rise to MZM braiding. We thus use the circulator function to achieve a non-Abelian braiding operation by introducing three pairs of MZMs in the branches that meet at a trijunction in the improved model where MZMs are introduced outside of the loop. The circulator function determines the phases of the trijunction and thus the coupling between the MZMs. Initially we apply a magnetic flux into one of the three loops to selectively couple two pairs of MZMs. By applying adiabatically a flux into another loop while decreasing the previous flux we are able to gain the weight of MZM while losing in the previous branch. Consecutive executions in this way can perform the braiding operation between two MZMs. This scheme could be extended to a scalable design to implement braiding operations in one-or two-dimensional circuits.  www.nature.com/scientificreports/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.