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Interface Between Topological and Superconducting Qubits University of Pennsylvania ScholarlyCommons Department of Physics Papers Department of Physics 3-28-2011 Interface Between Topological and Superconducting Qubits Liang Jiang California Institute of Technology Charles L. Kane University of Pennsylvania, [email protected] John Preskill California Institute of Technology Follow this and additional works at: https://repository.upenn.edu/physics_papers Part of the Physics Commons Recommended Citation Jiang, L., Kane, C. L., & Preskill, J. (2011). Interface Between Topological and Superconducting Qubits. Retrieved from https://repository.upenn.edu/physics_papers/129 Suggested Citation: Jiang, L., Kane, C.L. and Preskill, J. (2011). Interface between Topological and Superconducting Qubits. Physical Review Letters. 106, 130504. © 2011 The American Physical Society http://dx.doi.org/10.1103/PhysRevLett.106.130504 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/129 For more information, please contact [email protected]. Interface Between Topological and Superconducting Qubits Abstract We propose and analyze an interface between a topological qubit and a superconducting flux qubit. In our scheme, the interaction between Majorana fermions in a topological insulator is coherently controlled by a superconducting phase that depends on the quantum state of the flux qubit. A controlled-phase gate, achieved by pulsing this interaction on and off, can transfer quantum information between the topological qubit and the superconducting qubit. Disciplines Physical Sciences and Mathematics | Physics Comments Suggested Citation: Jiang, L., Kane, C.L. and Preskill, J. (2011). Interface between Topological and Superconducting Qubits. Physical Review Letters. 106, 130504. © 2011 The American Physical Society http://dx.doi.org/10.1103/PhysRevLett.106.130504 This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/129 week ending PRL 106, 130504 (2011) PHYSICAL REVIEW LETTERS 1 APRIL 2011 Interface between Topological and Superconducting Qubits Liang Jiang,1 Charles L. Kane,2 and John Preskill1 1Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA 2Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA (Received 29 October 2010; published 28 March 2011) We propose and analyze an interface between a topological qubit and a superconducting flux qubit. In our scheme, the interaction between Majorana fermions in a topological insulator is coherently controlled by a superconducting phase that depends on the quantum state of the flux qubit. A controlled-phase gate, achieved by pulsing this interaction on and off, can transfer quantum information between the topological qubit and the superconducting qubit. DOI: 10.1103/PhysRevLett.106.130504 PACS numbers: 03.67.Lx, 03.65.Vf, 74.45.+c, 85.25.Àj pffiffiffi Ày 2 Introduction.—Topologically ordered systems are in- ij ¼ð i À ijÞ= , defining a two dimensional Hilbert trinsically robust against local sources of decoherence, Ày À 0 1 space labeled by nij ¼ ij ij ¼ ; . The two basis and therefore hold promise for quantum information pro- states for the topological qubit, each with an even cessing. There have been many intriguing proposals for number of Dirac fermions, are j0itopo ¼j012034i and topological qubits, using spin lattice systems [1], p þ ip j1itopo ¼j112134i. superconductors [2], and fractional quantum Hall states with filling factor 5=2 [3]. The recently discovered topo- The MFs can be created on the surface of a TI patterned logical insulators [4] can also support topologically pro- with s-wave superconductors [5]. Because of the proximity tected qubits [5]. Meanwhile, conventional systems for effect [16], Cooper pairs can tunnel into the TI; hence the quantum information processing (e.g., ions, spins, photon effective Hamiltonian describing the surface includes a Á i c y c y polarizations, superconducting qubits) are steadily pro- pairing term, which has the form V ¼ 0e " # þ H c c y c y gressing; recent developments include high fidelity : : (where " , # are electron operators), assuming operations using ions [6] and superconducting qubits that the chemical potential is close to the Dirac point [7], long-distance entanglement generation using single [17]. Here is the SC phase of the island. Each MF is photons [8,9], and extremely long coherence times using localized at an SC vortex that is created by an SC trijunc- nuclear spins [10]. tion [i.e., three separated SC islands meeting at a point, Interfaces between topological and conventional quan- see Fig. 1(a)]. The MFs can interact via a superconductor- tum systems have also been considered recently [11,12]. TI-superconductor (STIS) wire [Fig. 1(a)] that separates Hybrid systems [13,14] may allow us to combine the advantages of conventional qubits (high fidelity readout, universal gates, distributed quantum communication and computation) with those of topological qubits (robust quantum storage, protected gates). In this Letter, we pro- pose and analyze an interface between a topological qubit based on Majorana fermions (MFs) at the surface of a topological insulator (TI) [5] and a conventional super- conducting (SC) flux qubit based on a Josephson junction device [15]. The flux qubit has two basis states, for which FIG. 1 (color online). On the surface of the TI, patterned SC the SC phase of a particular SC island has two possible islands can form (a) STIS quantum wire, (b) flux qubit, and (c) a values. In our scheme, this SC phase coherently controls hybrid system of topological and flux qubits. (a) Two MFs (red the interaction between two MFs on the surface of the TI. dots) are localized at two SC trijunctions, connected by an STIS This coupling between the MFs and the flux qubit provides quantum wire (dashed blue line). The coupling between the MFs a coherent interface between a topological and conven- is controlled by the SC phases d ¼ " and u ¼. (b) A flux tional quantum system, enabling exchange of quantum qubit consists of four JJs connecting four SC islands (a, b, c, d) È information between the two systems. in series, enclosing an external magnetic flux f 0. (c) The Topological qubit.—The topological qubit can be en- hybrid system consists of an STIS wire and a flux qubit. The STIS wire (between islands d and u) couples the MFs, coded with four Majorana fermion operators f g 1 2 3 4, i i¼ ; ; ; with coupling strength controlled by the flux qubit. The SC which satisfy the Majorana property y ¼ and fermi- i i phase c can be tuned by a phase controller (not shown), and à onic anticommutation relation f i;jg¼ij. A Dirac d ¼ c Æ 4 with the choice of Æ sign depending on the state fermion operator can be constructed from a pair of MFs of the flux qubit. 0031-9007=11=106(13)=130504(4) 130504-1 Ó 2011 American Physical Society week ending PRL 106, 130504 (2011) PHYSICAL REVIEW LETTERS 1 APRIL 2011 the SC islands d and u with d ¼ " and u ¼, on whether the state of the flux qubit is j0iflux or j1iflux as respectively. For a narrow STIS wire with width W shown in Figs. 2(b) and 2(c). Therefore, the Hamiltonian MF vF=Á0, the effective Hamiltonian is H12 couples the flux qubit and the topological qubit. Á 0 1 STIS x z Assuming a small phase separation " " À " H ¼ivF @x þ " ; (1) MF 0;1 =2, we can switch off the coupling H12 by tuning " Á 0;1 2 where vF is the effective Fermi velocity, " ¼ to satisfy vF=L 0 " <= [5], so that the MFs x;z Á0 cosðd À uÞ=2 ¼Á0 sin"=2, and are Pauli are localized and uncoupled with negligible energy matrices acting on the wire’s two zero-energy modes [5]. splitting Eð"0ÞEð"1Þ0. We can also switch on the MF As shown in Fig. 1(a), the STIS wire connects two local- coupling H12 by adiabatically ramping to the parameter 0;1 ized MFs (indicated by two red dots at the trijunctions) regime " & vF=LÁ0 to induce a non-negligible 0 1 separated by distance L; these are two of the four MFs jEð" ÞEð" Þj Á0Á". Because flux qubit designs comprising the topological qubit. The coupling between with three Josephson junctions (JJs) [15,19] are not ame- the MFs (denoted as 1 and 2) via the STIS wire can be nable to achieving a small phase separation Á" =2 ~MF characterized by the Hamiltonian H12 ¼ iEð"Þ 1 2=2, [18], we are motivated to modify the design of the flux with an induced energy splitting Eð"Þ depending on the qubit by adding more JJs. SC phase ". The effective Hamiltonian for the topological AsshowninFig.1(b), our proposed flux qubit consists of a qubit is loop of four Josephson junctions in series that encloses an È 1 2 È 2 Eð"Þ applied magnetic flux f 0 (f = and 0 ¼ h= e MF ¼ Z H12 2 topo; (2) is the SC flux quantum). The Hamiltonian for the flux qubit is flux where Ztopo ¼ðj0ih0jj1ih1jÞtopo. H ¼ T þ U; (3) P In Fig. 2(a), we plot Eð"Þ as a function of a dimension- 1 Á with Josephson potential energy U ¼ i¼1;2;3;4EJ;ið À less parameter à 0L sin" .Forà 1 and 0 <"< " vF 2 " cos Þ, and capacitive charging energy T ¼ à P i =2 [5], the energy splitting Eð"Þ2j jeÀ " 0 is 1 2 " 2 i¼1;2;3;4CiVi .Fortheith JJ, EJ;i is the Josephson coupling negligibly small for localized MFs at the end of the energy, i is the gauge-invariant phase difference, Ci is the Àà x=L wire, as the wave functions are proportional to e " capacitance, and V is the voltage across the junction [15,19]. à i À "ðLÀxÞ=L à & 1 and e . On the other hand, for " , the two In addition, there are relationsP satisfied by the phase accumu- 2 0 mod2 MFs are delocalized and Eð"Þ becomes sensitive to ".We lation around the loop ii þ f ð Þ and the emphasize that Eð"Þ is a nonlinear function of " [18], È0 _ voltage across each junction Vi ¼ð2Þi [16].
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