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

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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.

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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.

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.j

pffiffiffi y 2 Introduction.—Topologically ordered systems are in- ij ¼ði ijÞ= , defining a two dimensional Hilbert trinsically robust against local sources of decoherence, y 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 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 propose 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 superconducting (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) 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 onic anticommutation relation fi;jg¼ij. A Dirac d ¼ c 4 with the choice of 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=0, the effective Hamiltonian is H12 couples the flux qubit and the topological qubit. 0 1 STIS x z Assuming a small phase separation " " " H ¼ivF @x þ " ; (1) MF 0;1 =2, we can switch off the coupling H12 by tuning " 0;1 2 where vF is the effective Fermi velocity, " ¼ to satisfy vF=L 0 " <= [5], so that the MFs x;z 0 cosðd uÞ=2 ¼0 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=L0 to induce a non-negligible 0 1 separated by distance L; these are two of the four MFs jEð" ÞEð" Þj 0". 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 " =2 ~MF characterized by the Hamiltonian H12 ¼ iEð"Þ12=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 1 2 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 with Josephson potential energy U ¼ i¼1;2;3;4EJ;ið less parameter 0L sin" .For 1 and 0 <"< " vF 2 " cos Þ, and capacitive charging energy T ¼ P i =2 [5], the energy splitting Eð"Þ2j je " 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 x=L wire, as the wave functions are proportional to e " capacitance, and V is the voltage across the junction [15,19]. i "ðLxÞ=L & 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], 0 _ voltage across each junction Vi ¼ð2Þi [16]. The parame- which enables us to switch the coupling on and off. ters are chosen as follows: the first two JJs have equal Flux qubit.—The SC island d can also be part of an SC 0 1 Josephson coupling energy EJ;1 ¼ EJ;2 ¼ EJ, the third JJ flux qubit [Fig. 1(b)], with d ¼ " ¼ " or " depending has EJ;3 ¼ EJ with 0:5 <<1, and the fourth JJ has EJ;4 ¼ EJ with 1. For JJs with the same thickness but different junction area fAig, EJ;i / Ai and Ci / Ai.The charging energies can be defined as EC;1 ¼ EC;2 ¼ EC ¼ e2 1 1 , E 3 ¼ E and E 4 ¼ E . For these parame- 2C1 C; C C; C ters and f 1=2, the system has two stable states with persistent circulating current of opposite sign. We identify the flux qubit basis states with the two potential minima 0 1 2 j iflux ¼jfi gi and j iflux ¼ jfi gi (modulo ), as illus- trated in Figs. 2(b) and 2(c). When !1, we may neglect the fourth junction and this system reduces to the previous flux qubit design with 1 three JJs [15,19]. For , there is a small phasepffiffiffiffiffiffiffiffiffiffiffi differ- 421 1 ence across the fourth JJ [18], 4 ¼4 2 , where the choice of sign depends on the direction of the circulating current. We may write 4 ¼ Zflux , with FIG. 2 (color online). (a) The energy splitting Eð"Þ (in units of 4 Zflux ¼ðj0ih0jj1ih1jÞflux. The fourth JJ connects SC 0L sin 2 E ¼ vF=L) as a function of " ¼ v "= . (b) A contour F islands c and d, and if we fix c relative to u with a plot of potential energy U as a function of f1;2;4g with 0 phase controller [20], then d will be " ¼ c þ 4 or 3 ¼ 1 2 4. There are two potential minima asso- 1 " ¼ depending on the state of the flux qubit. The ciated with flux qubit states j0i and j1i. (c) A contour plot of U as c 4 separation a function of f1;4g with 1 ¼ 2 and 3 ¼ 21 4. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (d) Marginal probability distributions of 4 associated with states 42 1 1 j0i (blue solid line) and j1i (red dashed line). The parameters are " (4) EJ=EC ¼ 80 and fEJ;i=EJgi¼1;2;3;4 ¼f1; 1;¼ 0:8;¼ 10g. 130504-2 week ending PRL 106, 130504 (2011) PHYSICAL REVIEW LETTERS 1 APRIL 2011 between the two possible values of becomes small, as g d H ¼ ZfluxZtopo (7) we desired, when is large. I 4 Aside from this small phase separation, there are also 1 0 with coupling strength g ¼hE1ihE0i½Eð" ÞEð" Þþ quantum fluctuations in 4 due to the finite capacitance. 1½E00ð"1ÞE00ð"0Þ2 þOð3Þ. The first term arises from Near its minimum at fg, the potential energy is ap- 4 i the phase separation and the second term from the quantum proximately quadratic; therefore, for 1, the dynamics fluctuations; corrections of higher order in 1 are of 4 can be well described by a harmonic oscillator (HO) small. Hamiltonian Because the energy splitting function Eð"Þ is highly 2 nonlinear, we may tune to off such that v =L0 HO p4 EJ;4 2 c F Z 0;1 H ¼ þ ð4 flux4Þ ; (5) " ¼ off "=2 <=2 and switch off the coupling 2M4 2 joff j0L=2v g 0"e F 0. On the other hand, we may 1 where the effective mass is M4 ¼ and the canonical & 8EC;4 adiabatically ramp c to on vF=L 0, which effec- @ 1 g 0" momentum p4 satisfies ½4;p4 ¼i (with ). We tively switches on the coupling . By adiabati- HO y 1 2 Z Rcally changing c from off ! on ! off with may rewritepffiffiffiH ¼ða a þ = Þ! and 4 ¼ flux4 þ y 2 gðtÞdt ¼ , we can implement the controlled-phase pðffiffiffiffiffiffiffiffiffiffiffiffiffiffia þ aÞ= , where the oscillator frequency is ! ¼ (CPHASEt;f) gate between the topological (t) and flux (f) 8EJEC and the magnitude of quantum fluctuations is 8 CNOT ¼ðECÞ1=41=2. Figure 2(d) shows the probability qubits. With controlled-NOT gates t;f and EJ 2 2 Had CNOT Had 1 ð4 Þ = Hadamard gates f, we can achieve t;f ¼ f distribution functions p0 1ð4Þ pffiffiffi e 4 associ- = CPHASE Had , which flips the flux qubit conditioned 0 1 t;f f ated with j if and j if. The magnitude of the quantum on j1i and can be used for quantum nondemolition mea- t fluctuations is comparable to the phase separation "; surement of the topological qubit [11,22]. Furthermore, / 1=2 indeed may even dominate the phase separation with Hadamard gates Hadt (implemented by exchanging 1 " / for large (Fig. 3). [21] Therefore, we should two MFs [3,5]), we can achieve the swap operation 3 consider both the phase separation and the quantum SWAPt;f ¼ðHadt Hadf CPHASEt;fÞ . Finally, with fluctuations. CPHASEt;f, Hadt, and single-qubit rotations Uf, we can Hybrid system.—The Hamiltonian for the hybrid system achieve arbitrary unitary transformations for the two- of topological and flux qubits [Fig. 1(c)] is: qubit hybrid system of flux and topological qubits 1 [13,14]. ¼ HO þ MF ¼ð y þ 1 2Þ ð ÞZ H H H12 a a = ! 2 E " topo (6) Imperfections.—There are four relevant imperfections pffiffiffi for the coupled system of flux and topological qubits Z y 2 where " ¼ c þ 4 ¼ c þ flux4 þ ða þ aÞ= .In [23]. The first imperfection is related to the tunneling both flux qubit basis states, the oscillator is in its ground 0 1 between j iflux andpffiffiffiffiffiffiffiffiffiffiffiffiffiffij iflux of the flux qubit, with tunneling state with hayai¼0. To first order in the small parameter rate t ! expð EJ=ECÞ. The coupling between flux dEðÞ j 1, the Hamiltonian becomes ! d ¼c and topological qubits should be strong enough, 1 g t, to suppress the undesired tunneling probability ¼ HO ðh ij0ih0j 2 H H 2 E0 tunnel ðt=gÞ . 2 The next imperfection comes from undesired excitations þhE1ij1ih1jÞflux Ztopo þ Oð Þ of the oscillators. According to the Hamiltonian H for the R hybrid system, the oscillators may be excited via interac- where hE0=1i d4Eðc þ 4Þp0=1ð4Þ. y a^ þa^1 Z dE 1pffiffi Up to a single-qubit rotation, the effective Hamiltonian tion Eðc þ 4Þ¼Eðc þ flux4Þþd" 2 þ. coupling the flux and topological qubits is The excitation probability can be estimated as excite dE 2 dE 8EC 1=4 1=2 ð2 Þ . Since j j & 0, ð Þ , and ! d" ffiffiffiffiffiffiffiffiffiffiffiffiffiffi d" EJ qffiffiffiffiffi p 1 0.30 8 & 0 2 EJ ! ¼ EJEC, we estimate excite 20 ðE Þ E . 0.25 J C 0.20 The third imperfection is due to the finite length of the 0.15 STIS wire, which limits the fidelity for the topological 0.10 qubit itself. When we switch off the coupling between 0.05 the flux and topological qubits by having c ¼ off and 1 0.00 off for the STIS wire, there is an exponentially 5 10 15 20 25 30 35 40 small energy splitting E 0e off . The last relevant imperfection is associated with the FIG. 3 (color online). Comparison between the phase separa- excitation modes of the quantum wire, with excitation 1 0 tion " / (black solid line) and the magnitude of energy E vF=L [5]. Occupation of these modes can quantum fluctuations / 1=2 (purple dashed line), assuming potentially modify the phase separation of the flux qubit. EJ=EC ¼ 80. Therefore, we need sufficiently low temperature to 130504-3 week ending PRL 106, 130504 (2011) PHYSICAL REVIEW LETTERS 1 APRIL 2011 exponentially suppress the occupation of these modes by flux qubit has two possible tunneling pathways, exploits 0 the factor eE =kBT. the observation that whether two tunneling paths interfere Physical parameters.—We may choose the following destructively or constructively can be controlled by the design parameters for the flux qubit: ¼ 0:8, ¼ 10, state of the topological qubit. In contrast, our proposal, 1 EJ=EC ¼ 80, and EJ ¼ 200ð2Þ GHz. Both phase separa- which applies in the parameter regime < where the tion and quantum fluctuations depend sensitively on (see flux qubit has only one tunneling pathway, exploits the Fig. 3), with " 0:16 and 0:18. Meanwhile, the flux nonlinearity of the energy splitting Eð"Þ to achieve a qubit has plasma oscillation frequency ! 60ð2Þ GHz, controlled-phase coupling between the topological and 0 26 flux qubits. Recently, the related work [28] appeared energyffiffiffiffiffiffiffiffiffiffiffiffi barrier U : EJ, tunneling rate t p 1=2 1:8 EJEC exp½0:7ðEJ=ECÞ 70ð2Þ MHz; these parameters only marginally depend on [18]. For mesoscopic aluminum junctions with critical current 500 A cm2 density = , the largest junction (EJ;4 ¼ EJ) [1] A. Y. Kitaev, Ann. Phys. (N.Y.) 303, 2 (2003). 2 has an area of about 1 m [15]. For the topological qubit, [2] N. Read and D. Green, Phys. Rev. B 61, 10 267 (2000). it is feasible to achieve the parameters 0 0:1 meV [3] C. Nayak et al., Rev. Mod. Phys. 80, 1083 (2008). 5 25ð2Þ GHz, vF 10 m=s, L 5 m, and T ¼ 20 mK. [4] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 For the interface, the effective coupling is g 0" (2010). 2ð2Þ GHz. Therefore, we have imperfections tunnel [5] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 3 3 20j sinoff =2j 3 (2008). 10 , excite & 10 , e off e < 10 0 3 [6] R. Blatt and D. Wineland, Nature (London) 453, 1008 4 E =kBT 10 (assuming off = ), and e < [24]. (2008). Phase qubit.—A similar interface can be constructed to [7] J. Clarke and F. K. Wilhelm, Nature (London) 453, 1031 couple the SC phase qubit [7,27] and the topological qubit. (2008). A phase qubit is just a JJ with a fixed dc-current source I. [8] D. L. Moehring et al., Nature (London) 449, 68 (2007). phase phase The phase qubit Hamiltonian is H ¼ T þ U , [9] E. Togan et al., Nature (London) 466, 730 (2010). 1 2 phase where T ¼ 2 ECV and U ¼I0 I00 cos. [10] M. V.G. Dutt et al., Science 316, 1312 (2007). The qubit can be encoded in the two lowest energy states, [11] F. Hassler et al., New J. Phys. 12, 125002 (2010). [12] J. D. Sau, S. Tewari, and S. Das Sarma, Phys. Rev. A 82, j0iphase and j1iphase, with magnitude of quantum fluctua- 052322 (2010). tions 0 and 1, respectively. The coupling strength [13] L. Jiang et al., Nature Phys. 4, 482 (2008). between phase and topological qubits can be estimated as phase 00 2 2 [14] M. Aguado et al., Phys. Rev. Lett. 101, 260501 (2008). g E ð"Þð1 0 Þ. [15] J. E. Mooij et al., Science 285, 1036 (1999). Conclusion.—We have proposed and analyzed a feasible [16] M. Tinkham, Introduction to Superconductivity (McGraw interface between flux and topological qubits. Our proposal Hill, New York, 1996), 2nd ed.. uses a flux qubit design with four JJs, such that the two [17] T. D. Stanescu et al., Phys. Rev. B 81, 241310 (2010). basis states of the qubit have a small phase separation " [18] L. Jiang, C. L. Kane, and J. Preskill, arXiv:1010.5862. on a particular superconducting island, enabling us to [19] T. P. Orlando et al., Phys. Rev. B 60, 15398 (1999). adiabatically switch on and off the coupling between [20] An SC phase controller can fix the phase difference the flux and topological qubits. Such interfaces may enable between two SC islands. It can be implemented by con- us to store and retrieve quantum information using the trolling either the external flux or the current [18]. [21] Despite the large quantum fluctuations in 4 (>"), the topological qubit, to repetitively read out the topological two quantum states of the flux qubit are well localized qubit with a conventional qubit, or to switch between because the two potential minima are widely separated in conventional and topological systems for various quantum the 1 direction [see Fig. 2(c)]. information processing tasks. [22] J. D. Sau et al., Phys. Rev. Lett. 104, 040502 (2010). We are especially indebted to Mikhail Lukin for inspir- [23] We assume that flux qubit parameters can be accurately ing discussions. We also thank Anton Akhmerov, Jason measured, and hence ignore fabrication uncertainties. Alicea, Erez Berg, David DiVincenzo, Garry Goldstein, [24] The SQUID circuit that measures the flux qubit is another Netanel Lindner, and Gil Refael for helpful comments. potential source of error, as it may introduce an additional 0 0 This work was supported by the Sherman Fairchild plasma mode with low frequency ! . Assuming ! 2 2 GHz 200 2 MHz Foundation, by NSF Grants No. DMR-0906175 and ð Þ [25], we may choose g ¼ ð Þ and 10 2 MHz 0 1 No. PHY-0803371, by DOE Grant No. DE-FG03-92- t ¼ ð Þ , so that ! g t =T2. If the flux qubit’s spin-echo coherence time T2 is long ( 4 s ER40701, and by NSA/ARO Grant No. W911NF-09- [26]), a 1% error rate can be achieved. 1-0442. [25] I. Chiorescu et al., Science 299, 1869 (2003). Note added.—It was recently proposed to use the [26] P. Bertet et al., Phys. Rev. Lett. 95, 257002 (2005). Aharonov-Casher effect for quantum nondemolition mea- [27] J. M. Martinis et al., Phys. Rev. Lett. 89, 117901 (2002). surement of a topological qubit [11,12]. This proposal, [28] P. Bonderson and R. M. Lutchyn, following Letter, Phys. which applies in the parameter regime >1 where the Rev. Lett. 106, 130505 (2011).

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