Circulator function in a Josephson junction circuit and braiding of Majorana zero modes
CCirculator function in a Josephson junction circuitand braiding of Majorana zero modes
Mun Dae Kim
College of Liberal Arts, Hongik University, Sejong 30016, [email protected]
ABSTRACT
We propose a scheme for the circulator function in a superconducting circuit consisting of a three-Josephson junction loopand a trijunction. In this study we obtain the exact Lagrangian of the system by deriving the effective potential from thefundamental boundary conditions. We subsequently show that we can selectively choose the direction of current flowingthrough the branches connected at the trijunction, which performs a circulator function. Further, we use this circulator functionfor a non-Abelian braiding of Majorana zero modes (MZMs). In the branches of the system we introduce pairs of MZMs whichinteract with each other through the phases of trijunction. The circulator function determines the phases of the trijunction andthus the coupling between the MZMs to gives rise to the braiding operation. We modify the system so that MZMs might becoupled to the external ones to perform qubit operations in a scalable design.
While the ultimate goal of practical quantum computer is still far away, the noisy intermediate-scale quantum (NISQ)computing is expected to be realized in the near future due to the remarkable advancement in the qubit coherence and control.The quantum supremacy that quantum device can solve a problem that no classical computer can solve in any feasible amountof time is regarded as a notable milestone. The programmable NISQ computing for quantum supremacy requires a scalabledesign of quantum circuit, which is severely challenging. We, here, provide an approach to cope with this challenge byproposing a scheme for a circulator function which enables selective coupling between arbitrary two branches at a trijunctionby using a three-Josephson junction flux qubit as a control element in a superconducting circuit.
In this study we introduce a three-Josephson junction loop consisting of three small loops with three branches and atrijunction as shown in Fig. 1(a). Usually the Hamiltonians of the superconducting circuit with threading fluxes for quantuminformation processing have been provided phenomenologically. The effective potential in the Hamiltonian is given in anapproximate way so that the form and the coefficients have not been precisely derived from the first principle. For theunderstanding of the system we need to know the exact form of the Hamiltonian and the process by which the Hamiltonianis obtained. For the superconducting loop system in the present study we derive the Lagrangian of system exactly fromfundamental boundary conditions and obtain the effective potential of the system analytically. This Lagrangian describes thecirculating function in the ground state of the system, where we can selectively couple two branches to flow currents while theother branch does not. This kind of study will help analyzing other systems for quantum information processing.Circulator is a nonreciprocal three-port device that routes a signal to the next port. For the universal quantum computingquantum gates between different two qubits in a scalable design is required. Hence the circulator function which enablesselective coupling between arbitrary two qubits among several qubits has been studied intensively. Recently Josephson junctionbased on-chip circulators much smaller than commercial microwave circulator have been proposed for the quantum informationprocessing with superconducting devices.
The superconductor-based circulators have remarkably small photon lossescompared to the commercial nonreciprocal ferrite circulators . Moreover, the superconductor-based circulators are muchsmaller than the commercial circulators so that they can be integrated into a scalable circuit.By piercing a magnetic flux into one of three small loops we are able to make the current flow between two branchesselectively in situ , while the other is isolated, resulting in the circulator function. Usually the circulator routes a signal fromone port to the other. Present design, in contrast, performs a circulator function that routes a signal between two branches ata trijunction in a closed circuit rather than transferring the signal to outer port. In this way, we can connect arbitrary pair ofbranches to perform quantum gate operations. For the NISQ computing we need to perform the circulator function in a scalablecircuit where the trijunctions are connected with each other to form a lattice structure. We thus consider an improved designwhere the trijunction is located outside of the loop as shown in Fig. 1(b), which is topologically equivalent with the design inFig. 1(a).Further, we can use the circulator function to realize the braiding of Majorana zero modes (MZM) for topological a r X i v : . [ qu a n t - ph ] J a n uantum computing. Topologically-protected quantum processing is expected to provide a path towards fault-tolerantquantum computing. Since quantum states are susceptible to environmental decoherence, protection from local perturbation isan emergent challenge for quantum information processing. Non-Abelian states are the building block of topological quantumcomputing carrying the nonlocal information. The nonlocally encoded quantum information is resilient to local noises and, ifthe 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 fermionoperators. The theoretically proposed structures attracted a great deal of intention to realizing MZMs in condensed mattersystems. MZMs are predicted to emerge in ν = / p-wave superconductors, andone- or two-dimensional semiconductor/superconductor hybrid structures. The branches in our scheme for braiding containssemiconductor/superconductor hybrid structures with p-wave-like superconductivity induced from s-wave superconductors viaproximity effect.In two-dimensional spinless p + ip topological superconductors MZMs are hosted in vortices or in the chiral edge modesas localized Andreev-bound zero-energy states at the Fermi energy. The p-wave-like superconductivity can be induced froms-wave superconductors via proximity effect in a hybrid structure. Semiconductor thin film with Zeeman splitting andproximity-induced s-wave superconductivity has been expected to be a suitable platform for hosting MZMs. On the otherhand, the one-dimensional semiconducting nanowire has also been shown to provide MZMs at the ends of the nanowire. TheMZMs should be prepared, braided, and fused to implement qubit operations. In one-dimensional wire the braiding is not welldefined, which can be overcome in a wire network of trijunction. However, the original scheme with Josephson trijunctionhas not yet been explored.Recently, an experimental evidence of MZM in a trijunction has been reported. The nanowire trijunctions are manipulatedby the chemical potential, the charging energy, and the phase. In the present study a pair of MZMs can be introduced ineach branch near the trijunction of Fig. 1(b). Three MZMs of each pair are coupled through Josephson junctions with phasedifferences ϕ (cid:48) , ϕ (cid:48) , and ϕ (cid:48) in the system. The three Josephson junction loop controls the selective coupling among three MZMpairs. By applying a threading flux into one of the loops Fig. 1(b) we can use the circulator function to control the phases φ (cid:48) 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 trying to switch off the current mediated by MZMs which are insideof the loop the present proposal uses circulating function to perform braiding operations. Further, our scheme enables theinteraction between MZMs outside so that we may provide a scalable design in a one or two-dimensional lattice system forcoupling between MZMs which belong to different trijunctions. Results
Three-Josephson junction loop with a trijunction.
The precise fluxoid quantization condition of superconducting loop reads − Φ t + ( m c / q c ) (cid:72) (cid:126) v c · d (cid:126) l = n Φ with (cid:126) v c being theaverage velocity of Cooper pairs, q c = e the Cooper pair charge, and m c = m e the Cooper pair mass. The total magneticflux Φ t threading the loop is the sum of the external and the induced flux Φ t = Φ ext + Φ ind . With the superconducting unit fluxquantum Φ = h / e we introduce the reduced fluxes, f t = Φ t / Φ = f + f ind with f = Φ ext / Φ and f ind = Φ ind / Φ , expressingthe fluxoid quantization condition as kl = π ( n + f t ) with l being the circumference of the loop, k the wave vector of theCooper pair wavefunction and n an integer.The scheme in Fig. 1(a) consists of three-Josephson junction loop and three small loops with threading fluxes f i = Φ ext , i / Φ .The fluxoid quantization conditions around three loops, including the phase differences ϕ i and ϕ (cid:48) i across the Josephson junctions,are represented as the following periodic boundary conditions, k l − k (cid:48) l (cid:48) + k (cid:48) l (cid:48) + ϕ + ϕ (cid:48) = π ( m + f + f ind , ) , (1) k l − k (cid:48) l (cid:48) + k (cid:48) l (cid:48) + ϕ + ϕ (cid:48) = π ( m + f + f ind , ) , (2) k l − k (cid:48) l (cid:48) + k (cid:48) l (cid:48) + ϕ + ϕ (cid:48) = π ( m + f + f ind , ) , (3)where k i , l , and l (cid:48) 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 junctionloop and ϕ (cid:48) i ’s phase differences of the trijunction whose positive direction, we choose, is clockwise as shown in Fig. 1(a).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 , , for example, can be written as f ind , = Φ ind , / Φ = ( / Φ )( L s I / + L (cid:48) s I (cid:48) − L (cid:48) s I (cid:48) ) , where the Cooperpair current I i is given by I i = − ( n c Aq c / m c ) (cid:125) k i (4) j j j ' j ' j ' f k k k k ' k ' l ' L ' s L /3 s , l /3 f f k ' , j j j j ' j ' j ' k k k k ' k ' k ' l /3 l ' l ' l ~ L ' s L ' sLs ~ Ls , ,, l /3 l /3 f1 f2 f (a) (b) X X I ' I ' I ' Figure 1. (a)Three-Josephson junction loop with length l and geometric inductance L s has three Josephson junctions withphase differences ϕ i and three branches with length l (cid:48) and geometric inductance L (cid:48) s . k i and k (cid:48) i are the wave vectors of theCooper pairs and f i the external flux threading the loops. ϕ (cid:48) i ’s are the trijunction phase differences. (b) A scheme that threebranches and trijunction are extracted out from the three-Josephson junction loop and turned over: left and right branches havelength l (cid:48) and geometric inductance L (cid:48) s and central branch ˜ l and ˜ L s . Two schemes in (a) and (b) are topologically equivalent witheach other.with the Cooper pair density n c and the cross section A of the loop. The induced flux, Φ ind , , consists of contributionsfrom three conducting lines, L (cid:48) s I (cid:48) , L (cid:48) s I (cid:48) and L s I /
3, where L s and L (cid:48) s are the geometric inductance of the three-Josephsonjunction loop and a branch, respectively, and the inductance of one third of the loop contributes to the induced flux. Further,we introduce the kinetic inductances L K = m c l / An c q c and L (cid:48) K = m c l (cid:48) / An c q c , and then the induced fluxes become f ind , = − ( / π )[( L (cid:48) s / L (cid:48) K )( k (cid:48) − k (cid:48) ) l (cid:48) + ( L s / L K ) k l / ] , f ind , = − ( / π )[( L (cid:48) s / L (cid:48) K )( k (cid:48) − k (cid:48) ) l (cid:48) + ( L s / L K ) k l / ] , and f ind , = − ( / π )[( L (cid:48) s / L (cid:48) K )( k (cid:48) − k (cid:48) ) l (cid:48) + ( L s / L K ) k l / ] to represent the boundary conditions as (cid:18) + L s L K (cid:19) k l + (cid:18) + L (cid:48) s L (cid:48) K (cid:19) ( k (cid:48) − k (cid:48) ) l (cid:48) = π (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) (5) (cid:18) + L s L K (cid:19) k l + (cid:18) + L (cid:48) s L (cid:48) K (cid:19) ( k (cid:48) − k (cid:48) ) l (cid:48) = π (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) (6) (cid:18) + L s L K (cid:19) k l + (cid:18) + L (cid:48) s L (cid:48) K (cid:19) ( k (cid:48) − k (cid:48) ) l (cid:48) = π (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) . (7)In the system of Fig. 1(a) three Josephson junctions with ϕ (cid:48) i compose a trijunction which satisfies the periodic boundarycondition ϕ (cid:48) + ϕ (cid:48) + ϕ (cid:48) = π n (cid:48) with an integer n (cid:48) . By using this condition and summing above three equations we cancheck that the boundary condition for three-Josephson junction loop can be expressed as ( + L s / L K ) ( k + k + k )( l / ) = π [ n + f + f + f − ( ϕ + ϕ + ϕ ) / π ] with an integer n , which can also be derived directly from the fluxoid quantizationcondition. If we assume the superconducting branches in Fig. 1(a) have the same cross section A and Cooper pair density n c inEq. (29), the current conservation conditions, I = I + I (cid:48) , I = I + I (cid:48) , and I = I + I (cid:48) , at the nodes of three-Josephson junctionloop give rise to the relations, k = k + k (cid:48) , k = k + k (cid:48) , k = k + k (cid:48) . (8)From the boundary conditions in Eqs. (33)-(35) in conjunction with the relations in Eq. (8) we can readily obtain k i and k (cid:48) i in erms of ϕ i and ϕ (cid:48) i as k i = π l L K L (cid:48) eff (cid:18) m i + f i − ϕ i + ϕ (cid:48) i π (cid:19) + π l (cid:18) L K L eff − L K L (cid:48) eff (cid:19) (cid:18) n + f + f + f − ϕ + ϕ + ϕ π (cid:19) , (9) k (cid:48) i = π l L K L (cid:48) eff (cid:18) m i + − m i + + f i + − f i + − ϕ i + + ϕ (cid:48) i + π + ϕ i + + ϕ (cid:48) i + π (cid:19) , (10)where the effective inductances are defined as L eff ≡ L K + L s and L (cid:48) eff ≡ L K + L s + ( L (cid:48) K + L (cid:48) s ) . Here and after, the indices, i , aremodulo 3, for example, i + = i + I = − I c sin φ + C ˙ V = − I c sin φ − C ( Φ / π ) ¨ φ with the critical current I c , the capacitance C of Josephson junc-tion, and the voltage-phase relation, V = − ( Φ / π ) ˙ φ . The quantum Kirchhoff relation then becomes − ( Φ / π L K )( l / π ) k i = − E J sin φ i − C ( Φ / π ) ¨ φ i with the Josephson coupling energy E J = Φ I c / π and the current I = − ( n c Aq c / m c ) (cid:125) k . From theLagrangian L = ∑ i ( / ) C i ( Φ / π ) ˙ φ i − U eff ( { φ i } ) with the effective potential of the system, U eff ( { φ i } ) , the equation ofmotion, C i ( Φ / π ) ¨ φ i = − ∂ U eff / ∂ φ i , can be derived from the Lagrange equation ( d / dt ) ∂ L / ∂ ˙ φ i − ∂ L / ∂ φ i =
0. By usingthe quantum Kirchhoff relation the equation of motion then can be represented as Φ π L K l π k i − E J sin φ i = − ∂ U eff ∂ φ i . (11)We can construct the effective potential U eff ( { ϕ i , ϕ (cid:48) i } ) as follows, U eff ( { ϕ i , ϕ (cid:48) i } ) = Φ L (cid:48) eff (cid:34)(cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) + (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) + (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) (cid:35) + (cid:18) Φ L eff − Φ L (cid:48) eff (cid:19) (cid:18) n + f + f + f − ϕ + ϕ + ϕ π (cid:19) − ∑ i = ( E J cos ϕ i + E (cid:48) J cos ϕ (cid:48) i ) , (12)which consists of the inductive energies of the loops and Josephson junction energies with E (cid:48) J being the Josephson junctionenergy of trijunction. We can easily check that the effective potential U eff ( { ϕ i , ϕ (cid:48) 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 usualparameter regime for three-Josephson junction qubit L K / L s ∼ O ( − ) so that we can approximate the effective inductancesas L eff ≈ L s and L (cid:48) eff ≈ L s + L (cid:48) s .Further, the effective potential U eff ( { ϕ i , ϕ (cid:48) i } ) should also satisfy the quantum Kirchhoff relation for the phase variables ϕ (cid:48) i . InFig. 1(a) we consider the currents ˜ I i across the Josephson junction with phases ϕ (cid:48) i and I (cid:48) i flowing in the branch, where the directionof ˜ I i is counterclockwise and I (cid:48) i is opposite to k (cid:48) i (See Fig. S1(a) in the Supplementary Information). Then with the currentconservation relation at nodes, I (cid:48) i = ˜ I i + − ˜ I i + , and the current relation of Josephson junction, ˜ I i = − I (cid:48) c sin ϕ (cid:48) i − C (cid:48) ( Φ / π ) ¨ ϕ (cid:48) i , wehave I (cid:48) i = − ( I (cid:48) c sin ϕ (cid:48) i + + C (cid:48) Φ π ¨ ϕ (cid:48) i + ) + ( I (cid:48) c sin ϕ (cid:48) i + + C (cid:48) Φ π ¨ ϕ (cid:48) i + ) . Using the equation of motion, C (cid:48) i ( Φ / π ) ¨ ϕ (cid:48) i = − ∂ U eff / ∂ ϕ (cid:48) i ,obtained from the Lagrange equation, the quantum Kirchhoff relation reads − Φ π L K l π k (cid:48) i = ∂ U eff ∂ ϕ (cid:48) i + − ∂ U eff ∂ ϕ (cid:48) i + − E (cid:48) J sin ϕ (cid:48) i + + E (cid:48) J sin ϕ (cid:48) i + . (13)We can confirm that the effective potential U eff ( { ϕ i , ϕ (cid:48) i } ) in Eq. (12) also satisfies the quantum Kirchhoff relation in Eq. (13)with k (cid:48) i in Eq. (10). Limiting case.
In the system of Fig. 1(a) we can consider the limit that the length of branches goes to zero, l (cid:48) →
0, and thus two nodes at theeither ends of a branch collapse to a point. As a result, we have three loops with geometric inductance L s / L (cid:48) s → L (cid:48) eff ≈ L s + L (cid:48) s → L eff ≈ L s . Hence the effective potential U eff ( { ϕ i , ϕ (cid:48) i } ) in Eq. (12)becomes U eff ( { ϕ i , ϕ (cid:48) i } ) = Φ ( L s / ) (cid:34)(cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) + (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) + (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) (cid:35) − ∑ i = ( E J cos ϕ i + E (cid:48) J cos ϕ (cid:48) i ) , (14)which describes the inductive energies of three loops with geometric inductance L s / complying with the intuitive picture. .0 0.25 0.5-0.25-0.5 (b) e ff J U / E j /2p p m j /2p (c) e ff J U / E (d) -0.50.50.0 0.50.250.0-0.25 m j /2p m j ' / p (a) j /2p p m j / p Figure 2. (a) Contour plot for the effective potential U eff for the system in Fig. 1(b) as a function of ϕ p and ϕ m for f = f = . , and f = f =
0. (b) Profile of U eff along the dotted line in (a) for ϕ m =
0. At ϕ p / π ≈ . U eff has theminimum. (c) The profile of U eff for ϕ p / π ≈ .
124 shows ϕ m = U eff . (d) Plot of ϕ (cid:48) m as a function of ϕ m which shows ϕ (cid:48) m = U eff for ϕ p / π ≈ .
124 and ϕ m = Circulator function.
In order to perform the NISQ computing we need to construct a scalable design with the circulator function, where thetrijunctions are connected to others and the current directions can be controlled in situ in the circuit. However, in the design inFig. 1(a) the trijunction is inside of the loop so it is not possible to couple the branches with others outside. Hence we consideran improved design where the trijunction is located outside of the loop as shown in Fig. 1(b). In the Supplementary Informationwe show an archetype for a scalable design. Actually the inner branches and the trijunction are turned over, but the design istopologically equivalent with the design in Fig. 1(a). 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. 1(b) including the phase differences across theJosephson junctions as k (cid:48) l (cid:48) − k (cid:48) l (cid:48) + k l + ϕ + ϕ (cid:48) = π ( m + f − f − f + f ind , ) , (15) − k (cid:48) l (cid:48) + k (cid:48) ˜ l − k l − ϕ − ϕ (cid:48) = π ( − m − f + f ind , ) , (16) k (cid:48) l (cid:48) − k (cid:48) ˜ l − k l − ϕ − ϕ (cid:48) = π ( − m − f + f ind , ) , (17)with integers m i . The boundary condition in Eq. (23) describes the outmost loop containing the Josephson junctionswith phase differences ϕ and ϕ (cid:48) and the conditions in Eqs. (24) and (25) the left and right loop in Fig. 1(b). Withthe geometric and kinetic inductances ˜ L s and ˜ L K = m c ˜ l / An c q c for the central branch, respectively, the induced fluxes be-come f ind , = − ( / π )[( L (cid:48) s / L (cid:48) K )( k (cid:48) − k (cid:48) ) l (cid:48) + ( L s / L K ) k l / ] , f ind , = − ( / π )[ − ( L (cid:48) s / L (cid:48) K ) k (cid:48) l (cid:48) + ( ˜ L s / ˜ L K ) k (cid:48) ˜ l − ( L s / L K ) k l / ] and f ind , = − ( / π )[( L (cid:48) s / L (cid:48) K ) k (cid:48) l (cid:48) − ( ˜ L s / ˜ L K ) k (cid:48) ˜ l − ( L s / L K ) k l / ] to give rise to the relations similar to those in Eqs. (33), (34)and (35) where k (cid:48) l (cid:48) ’s are replaced with k (cid:48) ˜ l . From these relations in conjunction with the relations in Eq. (8) we can similarlycalculate k i and k (cid:48) i with i = , , ϕ i and ϕ (cid:48) i (see the Supplementary Information).In order to induce current flowing between the branches across ϕ (cid:48) , we initially apply the flux Φ ext , so that f = Φ ext , / Φ = f , but f = f =
0. We then can easily check that the following effective potential satisfies the equation of motion in Eqs. (11) nd (13), U eff ( { ϕ i , ϕ (cid:48) i } ) = Φ L eff (cid:18) − m + m + ϕ + ϕ (cid:48) π − ϕ + ϕ (cid:48) π (cid:19) + (cid:18) Φ L (cid:48) eff + Φ L eff (cid:19) (cid:18) n + f − ϕ + ϕ + ϕ π (cid:19) − Φ L (cid:48) eff (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) (cid:18) n + f − ϕ + ϕ + ϕ π (cid:19) + Φ L (cid:48) eff (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) (18) − ∑ i = ( E J cos ϕ i + E (cid:48) J cos ϕ (cid:48) i ) , where ˜ L eff ≡ L K + L s + ( L (cid:48) K + L (cid:48) s ) + ( ˜ L K + ˜ L s ) is the effective inductance of the central branch. By manipulating the thirdterm in Eq. (18) (see the Supplementary Information) we can obtain the effective potential of the system in Fig. 1(b) as U eff ( { ϕ i , ϕ (cid:48) i } ) = Φ L (cid:48) eff (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) + (cid:18) Φ L (cid:48) eff + Φ L eff (cid:19)(cid:34)(cid:18) m − ϕ + ϕ (cid:48) π (cid:19) + (cid:18) m − ϕ + ϕ (cid:48) π (cid:19) (cid:35) (19) + (cid:18) Φ L (cid:48) eff − Φ L eff (cid:19)(cid:18) m − ϕ + ϕ (cid:48) π (cid:19)(cid:18) m − ϕ + ϕ (cid:48) π (cid:19) + (cid:18) Φ L eff − Φ L (cid:48) eff (cid:19) (cid:18) n + f − ϕ + ϕ + ϕ π (cid:19) − ∑ i ( E Ji cos ϕ i + E (cid:48) Ji cos ϕ (cid:48) i ) . If we consider that the inductances of left, right and central branches are all equal, ˜ l = l (cid:48) , ˜ L s = L (cid:48) s , ˜ L K = L (cid:48) K , and thus ˜ L eff = L (cid:48) eff ,the effective potential U eff ( { ϕ i , ϕ (cid:48) i } ) in Eq. (19) can be reduced to U eff ( { ϕ i , ϕ (cid:48) i } ) in Eq. (12) for the system in Fig. 1(a) with f = f and f = f =
0. Figure 2 shows the effective potential for the design in Fig. 1(b), which is qualitatively similar to thatfor the model in Fig. 1(a).We introduce a coordinate transformation such as ϕ p = ( ϕ + ϕ ) / , ϕ m = ( ϕ − ϕ ) / , ϕ (cid:48) p = ( ϕ (cid:48) + ϕ (cid:48) ) / , and ϕ (cid:48) m =( ϕ (cid:48) − ϕ (cid:48) ) /
2. The effective potential in Eq. (19), then, can be expressed as U eff ( ϕ p , ϕ m , ϕ (cid:48) p , ϕ (cid:48) m , ϕ ) = Φ L (cid:48) eff (cid:18) m − n (cid:48) + f − ϕ − ϕ (cid:48) p π (cid:19) + (cid:18) Φ L eff − Φ L (cid:48) eff (cid:19) (cid:18) n + f − ϕ + ϕ p π (cid:19) + (cid:18) Φ L (cid:48) eff + Φ L eff (cid:19) (cid:34)(cid:18) m − ϕ p + ϕ m + ϕ (cid:48) p + ϕ (cid:48) m π (cid:19) + (cid:18) m − ϕ p − ϕ m + ϕ (cid:48) p − ϕ (cid:48) m π (cid:19) (cid:35) + (cid:18) Φ L (cid:48) eff − Φ L eff (cid:19) (cid:18) m − ϕ p + ϕ m + ϕ (cid:48) p + ϕ (cid:48) m π (cid:19) (cid:18) m − ϕ p − ϕ m + ϕ (cid:48) p − ϕ (cid:48) m π (cid:19) (20) − E J cos ϕ − E J cos ϕ p cos ϕ m − E (cid:48) J cos 2 ϕ (cid:48) p − E (cid:48) J cos ϕ (cid:48) p cos ϕ (cid:48) m , where we use ϕ (cid:48) = π n (cid:48) − ( ϕ (cid:48) + ϕ (cid:48) ) = π n (cid:48) − ϕ (cid:48) p . Figure 2(a) shows the effective potential U eff as a function of ( ϕ p , ϕ m ) for m = m = m = n = n (cid:48) =
0, which is minimized with respect to ϕ (cid:48) p , ϕ (cid:48) m and ϕ . If the value of the external flux f = . f = .
42 to obtain a stable minimum. The effective potential U eff ( ϕ p , ϕ m ) along the dotted line in Fig. 2(a) is shown in Fig. 2(b), where U eff ( ϕ p , ϕ m ) has a minimum at ϕ p / π ≈ . U eff ( ϕ p , ϕ m ) as a function of ϕ m for ϕ p / π ≈ . ϕ m = , i . e ., ϕ = ϕ . Figure 2(d) show that ϕ (cid:48) m = , i . e ., ϕ (cid:48) = ϕ (cid:48) at the minimum of the effective potential U eff ( ϕ p , ϕ m ) . From Eqs. (29) and (9) we can see that k = k and thus I = I and from Eq. (10) k (cid:48) =
0, and thus I (cid:48) =
0, whichis consistent with the current conservations, I − I = I (cid:48) =
0, in Eq. (8). Hence, in Fig. 1(b) we can determine the direction ofcurrent such as I (cid:48) = − I (cid:48) (cid:54) = I (cid:48) =
0. If we consider the case that f = f , f = f = f = f , f = f =
0, the currentsbecome I (cid:48) = − I (cid:48) (cid:54) = I (cid:48) = I (cid:48) = − I (cid:48) (cid:54) = I (cid:48) =
0, respectively. Hence we can selectively determine the direction of currentsflowing through a trijunction by threading a magnetic flux into a specific loop in the design of Fig. 1(b), which can realize thecirculator 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. Asshown in Fig. 4(a) we introduce three pairs of MZMs in the semiconducting nanowire with p-wave-like superconductivityinduced from s-wave superconducting branch via proximity effect. For the quantum computing the scheme for quantum gateoperation should be provided. Hence we consider the system of Fig. 1(b) because for the system of Fig. 1(a) the MZMs areinside of the loop so that the MZMs cannot interact with MZMs outside. (f -f )/2 F | I ' | / p E J I' I' I' F | | / p E M (a)(b) Figure 3. (a) Currents I (cid:48) i in branches as a function of ( f − f ) /
2. When f starts from f = .
42 with f = f =
0, thecurrents | I (cid:48) | = | I (cid:48) | (cid:54) = | I (cid:48) | =0. As f increases while f decreases to zero, the current flow changes so that | I (cid:48) | = | I (cid:48) | (cid:54) = | I (cid:48) | =0. (b) Currents I i carried through MZMs across trijunction. For f = .
42 and f = f = I has largeramplitude than | I | = | I | , but for f = .
42 and f = f = | I | becomes larger, so the asymmetry is changed.In Fig. 3(a) we show the currents I (cid:48) = I − I , I (cid:48) = I − I , and I (cid:48) = I − I of the system in Fig. 1(b) as a function of f − f .If f = f = .
42 with f = f =
0, the current direction is determined such that I (cid:48) =
0, but I (cid:48) = I (cid:48) (cid:54) =
0. In this case the currentflows between the branch with γ and the branch with γ . This is the initial state of the system shown in Fig. 4(b), where thethree MZMs, γ (cid:48) , γ (cid:48) and γ (cid:48) , are tunnel-coupled with each other through the Hamiltonian H T = iE M ( γ (cid:48) γ (cid:48) cos ϕ (cid:48) + γ (cid:48) γ (cid:48) cos ϕ (cid:48) + γ (cid:48) γ (cid:48) cos ϕ (cid:48) ) + i α ∑ i = γ i γ (cid:48) i (21)with Majorana Josephson energy E M and coupling energy α . Then the current carried through MZMs across trijunction isgiven by I i = e (cid:125) ∂∂ ϕ (cid:48) i H T = − π E M Φ i γ (cid:48) i + γ (cid:48) i + sin ϕ (cid:48) i π -periodic behavior. Actually we have ϕ (cid:48) / π ≈ .
246 and ϕ (cid:48) / π = ϕ (cid:48) / π ≈ − .
123 at the minimum of theeffective potential U eff ( ϕ p , ϕ m ) in Fig. 2(a). Then the current I has a larger amplitude than I = I as shown in Fig. 3(b),which is denoted as a solid (dotted) line for I ( I and I ) in the trijunction of Fig. 4(b). As shown in Eq. (22) the currentmediated by MZMs I i ∝ sin ϕ (cid:48) i /
2, while the Cooper pair current ˜ I i ∝ sin ϕ (cid:48) i . If we consider a simplified model such that theJosephson junctions in the three-junction loop in Fig. 1(a) are removed as in the previous study, the boundary conditionbecomes approximately ϕ (cid:48) i − π f i ≈
0. Here, even if we set f i = . ϕ (cid:48) i ≈ π , we cannot switch off the current mediatedby MZMs as I i (cid:54) = I i ≈
0. Hence, instead of switching off I i we change the current direction by using circulatingfunction to perform the braiding operation.In general, for f i = .
42 with f i ± = ϕ (cid:48) i / π ≈ .
246 and ϕ (cid:48) i ± / π ≈ − . I i at the trijunction. In next stage we adiabatically apply theflux f , while decreasing f (See Eq. (S27) of Supplementary Information for general f i ). In Fig. 3(a), then, | I (cid:48) | increaseswhile | I (cid:48) | decreases. In the meanwhile, | I (cid:48) | decreases to zero and then grows up to the maximum value. Finally for f = . f = f =
0, we have I (cid:48) =
0, but I (cid:48) = I (cid:48) (cid:54) =
0. Hence the current direction is changed: the current I i flows between thebranch with γ and the branch with γ but there is no current in the branch with γ as shown in Fig. 4(c), and meanwhile the j j j ' j ' j ' f1 f2 f X X t =0 t = t t =2 t (a) (b) (c)(d) f f ' ' ' f X X ' ' ' I ' I ' I ' I I I f t =3 t (e) Figure 4. (a) Three MZM (red circle) pairs are introduced at the end of branches where three MZMs, γ (cid:48) i , are coupled througha Josephson trijunction. Braiding sequence of system in (a): by applying adiabatically the fluxes (b) f , (c) f , (d) f , and finally(e) f again, the green and yellow MZMs are exchanged with each other to complete a non-Abelian braiding procedure. In thebranches represented as dotted line there is no current flowing. In trijunction thick red line corresponds to a large currentamplitude of I i .green MZM loses its weight in γ and gains weight in γ . Here the current I has a larger amplitude than I = I , and thusthe asymmetry in the amplitude of I i is changed. In this way, between t = τ and t = τ , the yellow MZM loses its weight in γ and gains weight in γ as shown in Fig. 4(d). At the last stage the green MZM loses its weight in γ and gains weight in γ . Asa result, the green and yellow MZMs are exchanged with each other as shown in Fig. 4(e), completing the braiding operation.In Fig. 5 we show an architecture for a scalable design for a superconducting circuit with MZMs. Two MZMs belongto different trijunctions (the green box in Fig. 5) can be coupled or fused to perform quantum gate operations and quantummeasurements. For the green box operation, for example, we can introduce a gate voltage applied to the sector between twoMZMs to control the chemical potential of the nanowire. Though the system in Fig. 5 is one-dimensional, we can extend it totwo-dimensional lattice straightforwardly. x x x f1 f X x x x x x x x x x f X j ' j ' j ' f1f1 f1 f X f X f X f X f X f X x x x x xxx xx x xx Figure 5.
A scalable design for a superconducting circuit with MZMs. Two MZMs in each circuit of Fig. 4(a) can be coupledin the green box to form a one-dimensional lattice structure. iscussion
In conclusion, we proposed a scheme for the circulator function in a superconducting circuit consisting of three small loopsand branches which meet at a trijunction. Usually the effective potential in the Hamiltonian for superconducting circuit isphenomenologically obtained. However in this study we obtained the boundary conditions from the fundamental fluxoidquantization condition for the superconducting loop to derive the effective potential of the system analytically, which is requiredfor accurate and systematic study for the quantum information processing applications. We expect that this kind of study can beapplied to other systems.At the minimum of the effective potential we can see that two branches carry current while the other does not. By applyinga magnetic flux into one of the loops we can determine which branches among three carry the current, achieving the circulatorfunction. For the NISQ computing we need to perform the circulator function in a scalable design. We thus introduced animproved model where the trijunction is extracted out from the outmost loop to interact with other external branches. For theimproved design we obtained the ground state of the system from the effective potential, and showed that it can perform thecirculator 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 currentdirections to give rise to MZM braiding. We thus use the circulator function to achieve a non-Abelian braiding operation byintroducing three pairs of MZMs in the branches that meet at a trijunction in the improved model where MZMs are introducedoutside 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 adiabaticallya flux into another loop while decreasing the previous flux we are able to gain the weight of MZM while losing in the previousbranch. Consecutive executions in this way can perform the braiding operation between two MZMs. This scheme could beextended to a scalable design to implement braiding operations in one- or two-dimensional circuits.
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education(2019R1I1A1A01061274), 2020 Hongik University Research Fund, and Korea Institute forAdvanced Study(KIAS) grant funded by the Korea government.
Author contributions statement
M.D.K. solely developed the ideas, performed calculations, and wrote the manuscript.
Competing interests
The author declares no competing interests.
Additional information
Correspondence and requests for materials should be addressed to M.D.K.
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The design of Fig. 1 in the main manuscript can be simplified as Fig. 6. In the figures we denote the currents I i and I (cid:48) i in theloop whose direction is opposite to the Cooper pair wave vector k i and k (cid:48) i , respectively. In this Supplementary Information weconsider the more general case of Fig. 6(b). Here, we consider that f = f and f = f =
0. The boundary conditions for thescheme in Fig. 6(b) including the phase differences across the Josephson junctions are represented as, k (cid:48) l (cid:48) − k (cid:48) l (cid:48) + k l + ϕ + ϕ (cid:48) = π ( m + f + f ind , ) , (23) − k (cid:48) l (cid:48) + k (cid:48) ˜ l − k l − ϕ − ϕ (cid:48) = π ( − m + f ind , ) , (24) k (cid:48) l (cid:48) − k (cid:48) ˜ l − k l − ϕ − ϕ (cid:48) = π ( − m + f ind , ) , (25)with integers m i .Equation (23) describes the boundary condition for the outmost loop containing the Josephson junctions with phasedifferences ϕ and ϕ (cid:48) , and Eqs. (24) and (25) the left and right loop in Fig. 6(b). The induced flux, f ind , i = Φ ind , i / Φ , can bewritten as f ind , = ( / Φ )( I + I + I ) L s / , (26) f ind , = ( / Φ )( − L (cid:48) s I (cid:48) + ˜ L s I (cid:48) − L s I / ) , (27) f ind , = ( / Φ )( L (cid:48) s I (cid:48) − ˜ L s I (cid:48) − L s I / ) , (28)where the Cooper pair current I is given by I i = − ( n c Aq c / m c ) (cid:125) k i . (29)With the kinetic inductances of side branches, central branch, and the three-Josephson junction loop being L (cid:48) K = m c l (cid:48) / An c q c , ˜ L K = m c ˜ l / An c q c and L K = m c l / An c q c , respectively, the induced fluxes become f ind , = − π [( L (cid:48) s / L (cid:48) K )( k (cid:48) − k (cid:48) ) l (cid:48) + ( L s / L K ) k l / ] , (30) f ind , = − π [ − ( L (cid:48) s / L (cid:48) K ) k (cid:48) l (cid:48) + ( ˜ L s / ˜ L K ) k (cid:48) ˜ l − ( L s / L K ) k l / ] , (31) f ind , = − π [( L (cid:48) s / L (cid:48) K ) k (cid:48) l (cid:48) − ( ˜ L s / ˜ L K ) k (cid:48) ˜ l − ( L s / L K ) k l / ] . (32)Then the boundary conditions are represented as (cid:18) + L s L K (cid:19) k l + (cid:18) + L (cid:48) s L (cid:48) K (cid:19) k (cid:48) l (cid:48) − (cid:18) + L (cid:48) s L (cid:48) K (cid:19) k (cid:48) l (cid:48) = π (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) (33) (cid:18) + L s L K (cid:19) k l + (cid:18) + L (cid:48) s L (cid:48) K (cid:19) k (cid:48) l (cid:48) − (cid:18) + ˜ L s ˜ L K (cid:19) k (cid:48) ˜ l = π (cid:18) m − ϕ + ϕ (cid:48) π (cid:19) (34) (cid:18) + L s L K (cid:19) k l + (cid:18) + ˜ L s ˜ L K (cid:19) k (cid:48) ˜ l − (cid:18) + L (cid:48) s L (cid:48) K (cid:19) k (cid:48) l (cid:48) = π (cid:18) m − ϕ + ϕ (cid:48) π (cid:19) (35)with ˜ L s and L (cid:48) s being the self inductance of the central and the side branch, respectively.The current conservation conditions, I = I + I (cid:48) , I = I + I (cid:48) , and I = I + I (cid:48) , at the nodes of three-Josephson junction loopgive rise to the relations, k = k + k (cid:48) , k = k + k (cid:48) , k = k + k (cid:48) . (36) k ' k ' k ' l ' I ~ I ~ I I ' ' I ' I I x xx x j ' j ' j ' x k j j k x k j I ~ f2 f j j j k k k k ' k ' k ' l ' l ' l ~ x x x x xx f1 f2 f I I I j ' j ' j ' I ~ I ' I ' I ' I ~ I ~ (a) (b) I l ' l ' XX Figure 6.
Simplified picture of Fig. 1 in the main manuscript.From Eqs. (33), (34), (35), and (36) we can obtain k = π l L K L (cid:48) eff (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) + π l (cid:18) L K L eff − L K L (cid:48) eff (cid:19) (cid:18) n + f − ϕ + ϕ + ϕ π (cid:19) , (37) k = − π l L K L (cid:48) eff (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) − π l L K ˜ L eff (cid:18) m + ϕ + ϕ (cid:48) π − ϕ + ϕ (cid:48) π (cid:19) + π l (cid:18) L K L eff + L K L (cid:48) eff (cid:19) (cid:18) n + f − ϕ + ϕ + ϕ π (cid:19) , (38) k = − π l L K L (cid:48) eff (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) + π l L K ˜ L eff (cid:18) m + ϕ + ϕ (cid:48) π − ϕ + ϕ (cid:48) π (cid:19) + π l (cid:18) L K L eff + L K L (cid:48) eff (cid:19) (cid:18) n + f − ϕ + ϕ + ϕ π (cid:19) , (39) k (cid:48) = π l L K ˜ L eff (cid:18) m + ϕ + ϕ (cid:48) π − ϕ + ϕ (cid:48) π (cid:19) , (40) k (cid:48) = − π l L K ˜ L eff (cid:18) m + ϕ + ϕ (cid:48) π − ϕ + ϕ (cid:48) π (cid:19) − π l L K L (cid:48) eff (cid:20) m − f + ϕ + ϕ (cid:48) π + (cid:18) n + f − ϕ + ϕ + ϕ π (cid:19)(cid:21) , (41) k (cid:48) = − π l L K ˜ L eff (cid:18) m + ϕ + ϕ (cid:48) π − ϕ + ϕ (cid:48) π (cid:19) + π l L K L (cid:48) eff (cid:20) m − f + ϕ + ϕ (cid:48) π + (cid:18) n + f − ϕ + ϕ + ϕ π (cid:19)(cid:21) , (42)where L eff = L K + L s , L (cid:48) eff = L K + L s + ( L (cid:48) K + L (cid:48) s ) , and ˜ L eff ≡ L K + L s + ( L (cid:48) K + L (cid:48) s ) + ( ˜ L K + ˜ L s ) are the effective inductancesof three-Josephson junction loop, side branches, and central branch, respectively. y using the quantum Kirchhoff relation the equation of motion can be represented as Φ π L K l π k i − E J sin φ i = − ∂ U eff ∂ φ i , (43) − Φ π L K l π k (cid:48) i = ∂ U eff ∂ ϕ (cid:48) i + − ∂ U eff ∂ ϕ (cid:48) i + − E (cid:48) J sin ϕ (cid:48) i + + E (cid:48) J sin ϕ (cid:48) i + . (44)We then can obtain the effective potential satisfying Eqs. (43) and (44) as follows, U eff ( { ϕ i , ϕ (cid:48) i } ) = Φ L eff (cid:18) − m + m + ϕ + ϕ (cid:48) π − ϕ + ϕ (cid:48) π (cid:19) + (cid:18) Φ L (cid:48) eff + Φ L eff (cid:19) (cid:18) n + f − ϕ + ϕ + ϕ π (cid:19) − Φ L (cid:48) eff (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19)(cid:18) n + f − ϕ + ϕ + ϕ π (cid:19) + Φ L (cid:48) eff (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) − ∑ i ( E Ji cos ϕ i + E (cid:48) Ji cos ϕ (cid:48) i ) . (45)The third term of Eq. (45) can be rewritten as3 Φ L (cid:48) eff (cid:20)(cid:18) n + f − ϕ + ϕ + ϕ π (cid:19) − (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19)(cid:21) − Φ L (cid:48) eff (cid:18) n + f − ϕ + ϕ + ϕ π (cid:19) − Φ L (cid:48) eff (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) , (46)where by using ϕ (cid:48) = π m (cid:48) − ϕ (cid:48) − ϕ (cid:48) and choosing appropriate m (cid:48) the first term of Eq. (46) can be represented as3 Φ L (cid:48) eff (cid:18) m − ϕ + ϕ (cid:48) π + m − ϕ + ϕ (cid:48) π (cid:19) . (47)As a result, the effective potential U eff ( { ϕ i , ϕ (cid:48) i } ) in Eq. (45) is reexpressed as follows, U eff ( { ϕ i , ϕ (cid:48) i } ) = Φ L (cid:48) eff (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) + (cid:18) Φ L (cid:48) eff + Φ L eff (cid:19)(cid:34)(cid:18) m − ϕ + ϕ (cid:48) π (cid:19) + (cid:18) m − ϕ + ϕ (cid:48) π (cid:19) (cid:35) + (cid:18) Φ L eff − Φ L (cid:48) eff (cid:19) (cid:18) n + f − ϕ + ϕ + ϕ π (cid:19) − ∑ i ( E Ji cos ϕ i + E (cid:48) Ji cos ϕ (cid:48) i ) . + (cid:18) Φ L (cid:48) eff − Φ L eff (cid:19)(cid:18) m − ϕ + ϕ (cid:48) π (cid:19)(cid:18) m − ϕ + ϕ (cid:48) π (cid:19) . (48)If we consider the general case that f (cid:54) = , f (cid:54) = f (cid:54) = f x = f + f + f , the effective potential can be obtainedstraightforwardly as U eff ( { ϕ i , ϕ (cid:48) i } ) = Φ L (cid:48) eff (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) + (cid:18) Φ L (cid:48) eff + Φ L eff (cid:19)(cid:34)(cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) + (cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) (cid:35) + (cid:18) Φ L eff − Φ L (cid:48) eff (cid:19) (cid:18) n + f x − ϕ + ϕ + ϕ π (cid:19) − ∑ i ( E Ji cos ϕ i + E (cid:48) Ji cos ϕ (cid:48) i ) . + (cid:18) Φ L (cid:48) eff − Φ L eff (cid:19)(cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19)(cid:18) m + f − ϕ + ϕ (cid:48) π (cid:19) . (49)(49)