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ARTICLE OPEN Realizing universal quantum gates with topological bases in quantum-simulated superconducting chains

Yong Hu1,2, Y. X. Zhao2, Zheng-Yuan Xue3,2 and Z. D. Wang2

One-dimensional time-reversal invariant topological superconducting wires of the symmetry class DIII exhibit exotic physics which can be exploited to realize the set of universal operations in topological quantum computing. However, the verification of DIII-class physics in conventional condensed matter materials is highly nontrivial due to realistic constraints. Here we propose a symmetry- protected hard-core boson simulator of the one-dimensional DIII topological superconductor. By using the developed dispersive dynamic modulation approach, not only the faithful simulation of this new type of spinful superconducting chains is achieved, but also a set of universal quantum gates can be realized with the computational basis formed by the degenerate ground states that are topologically protected against random local perturbations. Physical implementation of our scheme based on a Josephson quantum circuit is presented, where our detailed analysis pinpoints that this scheme is experimentally feasible with the state-of- the-art technology. npj Quantum Information (2017) 3:8 ; doi:10.1038/s41534-017-0009-3

INTRODUCTION Kitaev model,19–21 the string breaking dynamics of quark pairs,22 23 Topological band theory has become the focus of recent research the electron-electron scattering, and the delocalization of Dirac due to its significant importance in fundamental physics and fermions in disordered 1D wires.24 However, methods in the potential applications in novel devices.1–3 In each spatial existing studies are restricted to spinless fermions, where the dimension, topological insulators and topological superconduc- forms of the concerned Hamiltonians remain unchanged during tors can well be classified in terms of the presence or absence of the bosonization. When bosonizing a spinful 1D fermionic system, time-reversal symmetry (TRS) and particle-hole symmetry.4, 5 an exotic particle-density-dependent U(1) gage phase factor Typical examples include the integer quantum Hall effect6 emerges from fermionic statistics among fermions with opposite belonging to the A class of 2D and the Kitaev chain belonging spin orientations, leading to one of main challenges in the 7 to the D class of 1D. Among the various symmetry-protected simulation of the spinful DIII models. topological phases, increasing research interests have been drawn In this paper, we reveal unambiguously that a 1D HCB lattice to the theoretical and experimental studies of 1D topological – can be exploited to faithfully simulate the proposed DIII model 8 14 12 superconducting wires in the symmetry class DIII. In contrast which realizes the nontrivial Z2 topological phase. Most to the Kitaev chain in the class D with one Majorana zero mode remarkably, with the odd-parity Kramers doublet ground states (MZM) at each end and two-fold degenerate ground states in its being used as the basis of topological qubits, we demonstrate for Z 7 Zð2Þ 2 topological phase, the DIII superconductor wire in the 2 the first time that a set of universal quantum gates can in principle topological phase possesses the TRS, and has a Kramers doublet of 15–17 be achieved using these topological bases with the help of this MZMs at each end and four-fold degenerate ground states. HCB architecture, where we develop a dispersive dynamic Moreover, the DIII superconductor wire exhibits an impactful modulation (DDM) approach to mediate the exotic U(1) gage response to an effective magnetic field and long-range spin- field configuration in the bosonized DIII model. Moreover, we correlation at the two ends with the fixed fermionic parity, which propose a physical implementation scheme based on super- can be utilized to realize universal quantum operations of qubits 25, 26 that are topologically protected against random local perturba- conducting quantum circuit (SQC). Taking the advantages fl 27, 28 tions.9, 12, 13 However, despite the extensive theoretical efforts, still advantage of exibility and scalability of SQC, we encode Zð2Þ HCBs by superconducting transmon qubits and design the the 1D TRS-associated 2 topological phase has not been tested experimentally in electronic systems as it is limited by realistic nontrivial coupling through the inductive connection between 29–32 reasons, e.g. the constraints of materials, the lack of controllability, transmon qubits. Our estimation also implies that the and the co-existing complicated mechanisms. effective coupling strengths can be several orders larger than On the other hand, it has been indicated that a kind of one- that corresponding to the coherence time the decoherence rates dimensional (1D) hard-core boson (HCB) chain may be able to of the transmon qubits,33, 34 and therefore the proposed simulate fermionic physics,18 with interesting phenomena being topological operations may be tested with the current level of investigated including the pursuit of Majorana fermions in the technology.

1School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China; 2Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China and 3Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China Correspondence: Yong Hu ([email protected]) Received: 25 February 2016 Revised: 18 January 2017 Accepted: 26 January 2017

Published in partnership with The University of New South Wales Bosonic quantum simulator of DIII topological superconductors YHuet al 2 RESULTS modulated harmonically and in situ. We further introduce a η 1δ ; The HCB realization of DIII model dispersive energy threshold ’ 10 such that jVjðtÞj jVjðtÞj η δ An intriguing model Hamiltonian of fermionic 1D topological and modulate Vj(t) and VjðtÞ as 12 η δ η ε superconductor in the class DIII reads, VjðtÞ¼2sj cosð Þt þ 2qj cosð Þt X ð4Þ y Δ : : μ y η δ η ε HDIII ¼ ðwcj;αcjþ1;α i cjþ1;αcj;α þ h c Þ ðcj;αcj;α 1Þ ð1Þ VjðtÞ¼2sj cosð Þt þ 2qj cosð þ Þt jα with ε ¼ Ω þ Ω. Based on the parameter choice where cy is the creation operator of the spin-α fermion on the j;α j1 ; j1 jth site with α = ↑,↓, w is the real-value tight-binding hopping sj ¼ð1Þ s qj ¼ð1Þ q ð5Þ strength, μ is the chemical potential, Δ is the real-value real j1 j1 sj ¼ð1Þ s; q ¼ð1Þ q; q=s ¼q=s amplitude of p-wave pairing parameter, and α denotes the spin j α component oppositepffiffiffi to . Under the basis transformationpffiffiffi aj ¼ we can directly reproduce the hopping and pairing terms in Eq. (3) iπ=4 = iπ=4 = =η Δ =η e ðcj;" þ cj;#Þ 2 and aj ¼ e ðcj;" cj;#Þ 2, HDIII can be with w ¼P ss andP ¼ sq from the dispersive coupling L L rewritten as between jHj and jHj (For detailed derivation we refer to Δ HDIII ¼ H þ H Methods). The obtained amplitudes of w and can be independently tuned in the range [10−3,10−2]δ by the parameters with ; ; ; X ðs q s qÞ. In addition, the chemical potential term in Eq. (3) y Δ : : μ y = can beP produced by adding an on-site dispersive radiation H ¼ ðwaj ajþ1 þ ajþ1aj þ h c Þ ðaj aj 1 2Þ D y y j H ¼ jDjðtÞðbj þ bjÞþDjðtÞðbj þ bjÞ. A convenient choice of X ð2Þ i(Ω±3η) −i(Ω±3η) y y Dj(t) (and similarly DjðtÞ)isDj(t)=dj[e +e ] which results H ¼ ðwa ajþ1 Δajþ1aj þ h:c:Þμða aj 1=2Þ 2 y = η η j j in the a.c. Stark shift ± dj bj bj 3 . The 3 setting of the radiation j frequency is to avoid the crosstalk between HD and HL, and the η i.e., HDIII is decoupled into two Kitaev chains differed by the signs choice of the 3 sign depends on whether positive or negative of their superconducting order parameters (hereafter we denote correction of the chemical potential μ is needed. the quantities related with the (+)-chain as A and their counter- parts for the (−)-chain as A). Although the topology-preserved MZMs and the universal quantum gates simulation of a standard Kitaev model with HCBs has been μ fi 19–21 The region |w|>| /2| can be identi ed as the topological phase proposed, to simulate HDIII with HCBs is much more 7 + − where the nontrivial MZMs emerge. Especially, we focus on the challenging than the separate simulations of H and H by two ideal case w = Δ and μ = 0 where the DIII model (2) can simply be independent HCB chains, because such straightforward scenario reduced to cannot result in the fermionic anti-commutation relation of inter- X γ γ γ γ ; species. For this, here we exploit a quasi-1D HCB chain shown in HDIII ¼ iwð j;B jþ1;A þ j;B jþ1;AÞ ð6Þ Fig. 1(a), where the upper/lower array plays the role of the +/− j chain, respectively. The DIII model (2) can be bosonized as γ y γ y with Majorana operators j;A ¼ iðaj aj Þ, j;B ¼ aj þ aj , X 1 γ y γ y y Δ : : μ y j;A ¼ aj þ aj , and j;B ¼iðaj aj Þ. The Hamiltonian is fully H ¼ ðwPjbj bjþ1 þ Pjbjþ1bj þ h c Þ ðbj bj Þ 2 diagonalized through the pairing of γj,B and γj+1,A (and simulta- j γ γ γ γ X ð3Þ neously j;B and jþ1;A) in the bulk, leaving four MZMs A = 1,A, y y 1 Δ : : μ γ ¼ γ ; , γB = γN,B, and γ ¼ γ ; unpaired at the two ends (N H ¼ ðwPjþ1bj bjþ1 Pjþ1bjþ1bj þ h c Þ ðbj bj Þ A 1 A B N B 2 being the length of the DIII chain).ÈÉ The ground states are thus j four-fold degenerate with a basisji 1 1 ; ji0 1 ; ji1 0 ; ji0 0 with the bosonizationQ alongQ the zig-zag pathQ in Fig.Q 1(a) being − γ γ 〉 〉 γ γ ; ; satisfying i A B|0,1 = ±|0,1 and i A B 0 1 ¼ ± 0 1 . Here we given by bj ¼ aj squbit. Such subspace has the odd parity π y π y ji chain, Ps ¼ expði as asÞ½¼expði bs bsÞ, and distinguishing it from the other ground states. For comparison, we π y Ps ¼ expði as asÞ. Although this is a nonlocal transformation, recall that the ground state of the Kitaev model with only one the locality of operators of physical noises (or random perturba- MZM at each end have the opposite (even) fermionic parity.7 tions) is preserved, and so does the topological stability (This fl It is shown that a single Kitaev chain can be minimally simulated bosonization inherits topological protection against local uctua- by a 3–HCB array.20 Therefore, for an easy experimental setup but tions seen from the original fermionic system of Eq. (2), because without loss of generality, we consider the 12–HCB lattice the locality of the three forms of physical perturbations y ; ; y sketched in Fig. 1(b) to demonstrate the universal quantum ðbj bjþ1 bjþ1bj bj bjÞ in the simulated bosonic system remains operations in the topological bases. We notice that two DIII chains unchanged after fermionization. To be concrete in other words, if can be constructed by “cutting” the proposed HCB array into two the transformed terms, which correspond to possible random pieces (i.e. by tuning s3 ¼ q3¼ 0 and q4 = 0. Note that turning on local (perturbation) terms in the simulated system, are still local or off these coupling parameters does not have any impact on the ones in the original fermonic picture of topological model after total ground state energy as MZMs have no contributions to the application of the non-local transformation, one can deduce that energy, namely, the total energy is conserved under these the simulated system is also topologically protected.). It is noticed operations in the ground state, with the TRS being also preserved.) that the fermionic anti-commutation relation has been converted fi These two subchains, labeled by L and R, have consequently 8 to the gage eld factors after bosonization, which in one chain is emerged MZMs at the ends (4 for each subchain, see Fig. 1(b)). given by the particle density of the other chain in a site-wise For each subchain, the previously established formalism of sense. defining topological qubits can still be exploited. For universal To implement the bosonization version of the DIII model (3), we single qubit operation, we take the left subchain as an example by set that the HCBs on the upper/lower array of Fig. 1(a) have energy B S^L B fi Ω=Ω δ Ω Ω : Ω considering HS ¼ with as an effective magnetic eld, splits with ¼ 0 3 , andP the inter-HCB coupling which is very weak compared with the bulk gap, and L L L L P takes the nearest-neighbor form H ¼ jHj þ Hj with Hj ¼ L 3 T L y y L y y S^ y; y σ^ ; S^ ¼ j 1ðaj aj Þ ðaj ajÞ . It has been shown that can be VjðtÞðbj þ bjÞðbj þ bjÞ and Hj ¼ VjðtÞðbj þ bjÞðbjþ1 þ bjþ1Þ where ¼ 2 jVjðtÞj; jVjðtÞj ’ 10 δ are the coupling strengths which can be regarded as an effective single spin operator acting on the left

npj Quantum Information (2017) 8 Published in partnership with The University of New South Wales Bosonic quantum simulator of DIII topological superconductors YHuet al 3

Fig. 1 a Schematic view of the 1-D HCB simulator of HDIII. The solid red line describes the path along which the HCB sites are physically coupled and the bosonization is performed. Through the DDM of the nearest neighbor coupling (the green wavelet boxes) the effective intra- species hopping and pairing (the black dashed lines) can be induced. The 12-HCB lattice shown is minimal for the demonstration of universal quantum operations. b Realization of nontrivial two-qubit gate in the 12-HCB lattice. In the situation w = Δ and μ = 0, each HCB site (the ellipse) can be decomposed into two Majorana modes (the filled and the hollow dots) pairing in the bulk. Two topological qubits can be “ ” − ↔ γL γR γL γR prepared by cutting the whole DIII chain into two pieces. Through the modulation of (3, ) (4, + ) coupling, the effective B A þ B A coupling can be induced. c Schematic plot of the array of coupled transmon qubits topological qubit, with |•〉L and |×〉L being the two eigenstates To implement the nontrivial inter-qubit quantum gate, we SL 12 B η δ of z. Thus through the control of the universal single-qubit modulate V3ðtÞ as V3ðtÞ¼2hðtÞ cosð Þt to induce the operation can be achieved. Notably, since the qubits are effective (3, +)↔(4, +) and (3, −) ↔ (4, −) hopping which can be constructed by the ground states of the whole system, only the written as global change of the direction of B may manipulate them, while random local variations are actually irrelevant (Note that the TRS is s HT ¼i M ðγL γR þ γL γRÞ; ð7Þ weakly broken during a generic adiabatic operation of the single M 2 B A B A qubit gate, but the operation, similar to the case of the Kitaev chain, is still topologically protected by the particle-hole symmetry with sM ¼ shðtÞ=η. The coupling strength sM can be controlled by of the superconductor, such that this type of gate is no doubt a ≪ Δ the slow-varyingR envelope hðtÞ such that |sM| , sM(0) = sM(tf)= symmetry-protected topological one unless the superconducting tf π SL 0, and t sMdt ¼ . Consequently, we get an unitary transforma- gap vanishes). The x rotation can be implemented by the i γL γRγL γR resonant dynamic modulation approach: for each (j,+)↔(j, −) link, tion U ¼ B A B A transforming the four basis states as RX δ we add a resonant tone Vj ðtÞ¼2BxðtÞ cos t to Vj(t) with Bx(t) L R L R L R L R the slow-varying (adiabatic) pulse amplitude satisfying Bx(ti)= ji ji ! ji´ ji´ ; ji´ ji !jiji´ ; BPx(tf) = 0 to induce the resonant inter-species hopping L ´ R ´ L R; ´ L ´ R L R; 3 y y SL ji ji!jiji jiji!jiji j¼1BxðtÞðbj bj þ bjbj Þ which is equivalent to HSX ¼ BxðtRÞ x in θSL θ tf the fermionic picture. The rotation expði xÞ with ¼ t Bxdt SL i SL SR can be realized after tf. The y rotation can be similarly realized by i.e., U acts as a nontrivial two-qubit gate y y which can be changing the initial phase of the resonant dynamic modulation as exploited to achieve a set of universal quantum gates in RY δ π= Vj ðtÞ¼2ByðtÞ cosð t 2Þ. combination with the single-qubit gates mentioned previously.

Published in partnership with The University of New South Wales npj Quantum Information (2017) 8 Bosonic quantum simulator of DIII topological superconductors YHuet al 4 Physical implementation with superconducting circuits modification dose not influence the performance of the DDM We now elaborate in detail how to implement the present method as we merely need to adjust the modulating frequencies theoretical scheme by SQCs. For the site (j, +) we consider a of Vj(t) and VjðtÞ accordingly. However, the effect of the d.c. NNN superconducting transmon qubit consisting of a superconducting coupling is significantly suppressed by the 0.5 GHz energy quantum interference device (SQUID) with effective Josephson difference between the NNN transmon qubits. J 29, 30 energy Ej shunted by a large capacitance Cj, as shown in Based on the static bias, we can add the a.c. modulation pluses Fig. 1(c). The two characteristic energy scales of the transmon on the coupling SQUIDs with s; q; s; q at the order of 2π[10,15] MHz C 2= η π qubit are itsq anharmonicityffiffiffiffiffiffiffiffiffiffiffiffi Ej ¼ e 2Cj and its lowest level and choose the dispersive active region as /2 = 75 MHz. The C J resulting dispersive tunneling is estimated to be on the level w/2π, splitting Ωj ’ 8E E . Here we choose j-independent Ωj/2π = Ω/ j j Δ/2π ∈ [1,5] MHz which is three orders larger than the reported π C= π C= π : 2 = 10GHz and Ej 2 ¼ E 2 ’ 0 75 GHz. The HCB sites on the decoherence rates of the transmon qubits (in the range 2π[1,10] ‘−’ chain can be similarly constructed with Ωj=2π ¼ Ω=2π ¼ 33, 34 : C C kHz). Such strong tunnel/pairing allow us to set the slow 7 5 GHz and Ej ’ Ej . Due to the large anharmonicity, the varying envelopes on the level B (t)/2π, B (t)/2π, s /2π ∈ [0.05,0.5] presence of the higher levels of the transmons can only slightly x y M MHz. Following this setting, the envelopes are all much smaller modify the effective parameters derived below and the transmons than the realized bulk gap such that the high energy excitations can be consequentlyP modeled by the two-level HCB Hamiltonian HCB Ω y Ω y can be omitted, but stronger enough than the decoherence rates H ¼ j bj bj þ bj bj. For the links between neighboring qubits, we exploit the current dividing mechanism recently such that the proposed topological quantum operations can be studied in experiments.31, 32 The transmon qubit design is slightly simulated. fi D J= modi ed by introducing a small grounding inductance Lj ’ Lj 2 J Φ2= π2 J with Lj ¼ 0 4 Ej the effective inductance of the transmon DISCUSSION SQUID at (j, + ). A low voltage node is thus created for each fi transmon qubit (Fig. 1(c)), and the neighboring nodes are The nite anharmonicity of the transmon qubits connected by coupling SQUIDs with tunable Josephson induc- Here we should notice that the proposed superconducting I tances LI; L ’ LJ=4. Moreover, the capacitances of the coupling transmon qubits are not ideal HCBs but anharmonic oscillators. j j j fi SQUIDs is chosen to be much smaller than Cj. The assumption of regarding them as HCBs with in nite The inter-qubit coupling can be established by the current anharmonicity thus needs further investigation. We recall that 29 dividing mechanism. An excitationqffiffiffiffiffiffiffiffiffiffiffiffi current from the (j, + ) qubit the anharmonicity of a transmon qubit is defined by its ω21−ω10 J y C which can be written as Ij ’ Ω=2L ðb þ bjÞ will mostly flow and equals its charging energy E , which is at the level of j j J C D 2π[0.1, 0.8] GHz. The requirement E /E ≫ 1 should be fulfilled through L to the ground, with a small fractions Ij+,j− and Ij+,(j−1)− j because the qubit becomes more and more sensitive to the flowing to the neighboring qubits ( j−1, −) and ( j, −) through the background 1/f charge noise with decreasing EJ/EC, leading to two coupling SQUIDs. The current Ij+,j− in turn generates a flux 29 C Φ D − severe dephasing of the physical qubit. Here we set E /2π = jþ;j ¼ Lj Ijþ;j in the (j, ) qubit. The interaction between the J C two qubits can thus be written as 0.75GHz such that E /E > 10 (previously reported high coherence transmon qubits often work in the region EJ/EC ≈ 20).31, 33 Being L y y ; η π Hj ¼ Vjðbj þ bjÞðbj þ bjÞ ð8Þ larger than the dispersive threshold /2 = 75 MHz by one order of magnitude, such anharmonicity choice prevents the effective where the coupling constant Vj can be estimated as 1=2 excitation of the higher levels of the transmons. ΩΩ= J J D D= I fi Vj ’ð Lj Lj Þ Lj Lj Lj. Therefore, the a.c. modulation of the To evaluate the consequence of the nite anharmonictiy, we penetrating flux bias in the coupling SQUID loop results in the incorporate the higher levels of the transmon qubits into the ideal oscillation of LI and in turn the oscillation of the inter-transmon HCB model by enlarging the Hilbert space of each HCB to three j span ; span ; ; coupling. Here we should notice that the modulation frequencies dimension, i. e. from fji0 j;j ji1 j;jg to fji0 j;j ji1 j;j ji2 j;jg. of the coupling SQUIDs should not be higher than their plasma Following the same essentials of the DDM method, we come to frequencies, otherwise complicated quasiparticle excitations the result that the inclusion of the anharmonicity leads to parasitic would occur. As being mentioned previously, the maximal terms of the proposed ideal Hamiltonian, taking the form modulating frequency of the DDM approach is of the order X y Δ : : ; ε ¼ Ω þ Ω. Therefore, with parameters being chosen before, this Hpara ¼ ðwparaQjbj bjþ1 þ paraQjbjþ1bj þ h c Þ plasma frequency requirement is guaranteed by the small j X ð9Þ capacitance of the coupling SQUIDs. y H ¼ ðw Q b b Δ Q b b þ h:c:Þ; The d.c. bias of the coupling SQUIDs leads to a static nearest para para jþ1 j jþ1 para jþ1 jþ1 j j neighbor qubit-qubit couplings which is estimated to be on the level of 2π[75,100] MHz. Since such nearest-neighbor static coupling strength is much smaller than the energy difference where the strengths of these parasitic terms are of the order between neighboring qubits (on the level of 2π × 2.5 GHz), its ; = C =η; influence is the corrections of the hopping and pairing parameters jwparaj jwparaj’ss E w ¼ ss ð10Þ derived from the DDM method. The couplings beyond the nearest neighbors should also be estimated: we notice that the divided − C current Ij+,j− from (j, +) qubit can be further divided in the (j, ) jΔparaj; jΔparaj’sq=E Δ ¼ sq=η; ð11Þ node to flow through the (j+1, +) node, thus the next-nearest- ↔ neighbor (NNN) (j,+) (j+1, +) static coupling is induced. and Meanwhile, due to the current dividing mechanism at the low- voltage nodes, the non-nearest-neighbor couplings decay expo- π y ; π y ; nentially with respect to the site distance. The NNN coupling is Qj ¼ expði cj cjÞ Qj ¼ expði cj cjÞ ð12Þ estimated to be of the order 2 π[15, 20] MHz. To suppress its effect y we can use the two-sublattice stragegy for each of the two HCB with cj ¼ ji1 jhj2 and cj ¼ ji2 jhj1 . Moreover, as the excitation to legs the eigenfrequencies of the four qubits shown in Fig. 1(c) can the statesji 2 j;j is effectively not excited, we can reduce the be modified to be 2 π(7.5,10,7,10.5) GHz, respectively. Such obtained parasitic terms back to the original 2D HCB Hilbert space

npj Quantum Information (2017) 8 Published in partnership with The University of New South Wales Bosonic quantum simulator of DIII topological superconductors YHuet al 5 and get noise in a large Josephson junction is even smaller than those of fl 42 X 1 1 1 1 the ux 1/f noise and can then be safely neglected. Summarizing H ¼ ½w ð þ P Þbyb Δ ð þ P Þb b þ h:c:; the analysis above, we then come to the conclusion that our para para 2 2 j j jþ1 para 2 2 j jþ1 j j scheme can survive in the presence of the 1/f noises in SQC ð13Þ devices. X 1 1 y 1 1 H ¼ ½w ð þ P Þb b þ Δ ð þ P Þb b þ h:c:: CONCLUSION para para 2 2 jþ1 j jþ1 para 2 2 jþ1 jþ1 j j In conclusion, we have designed a topolgy-preseved HCB ð14Þ simulator for the TRS-invariant DIII topological superconducting We then observe that the parasitic terms contain two types. The chains and proposed a set of universal quantum gates with first type containing the U(1) gage factors slightly shift the topological bases through the developed DDM technique. proposed ideal hopping and pairing constants. This term can be Physical implementation of our proposal with SQCs has also been compensated by the refined choice of the pumping parameters. explored. The present results may pave the way for realizing The second type that not containing the U(1) gage factors can be universal quantum computation with topological stability. regarded as local perturbation terms. As estimated before, both the two terms have strength smaller than the proposed band gap METHODS by one order of magnitude. Therefore, these parasitic terms results in only the slight correction of the ideal scheme. They cannot The derivation of the DDM pulses destroy the performance of the proposed scheme because they Here we detail the derivation of DDM pulses and the consequent effective Hamiltonian. We exploit the dispersive coupling mechanism which states are too small to change the band topology of the system by ω − ω that, for a system governed by a fast oscillating Hamiltonian Aei t + Be i t closing and reopening the gap. with kAk; kBkω, its evolution can be described by the effective Hamiltonian HEff =[A,B]/ω44, 45 (here the norm kOk of an operator O is Robustness against imperfection factors defined as the square root of the largest eigenvalue of O†O). Our basic idea y L y y is that, if we are able to use the A ¼ b bj term in H as A and the b b In realistic experiments, the imperfection accompanies inescap- y L j j j jþ1 and bj bjþ1 terms in Hj as B, the resulting commutators [A, B] provide the ably with the proposed ideal scheme. The deviation of the circuit y y y required b b Pj and b bj 1Pj terms. For this purpose we adopt the j jþ1 j þ L parameters from their ideal values can be attributed to the errors rotating frame for which HL and H take the forms happened in the fabrication process and the low-frequency noises j j L y y iεt y iδt y iδt iεt ; in the proposed circuit. For the fabrication errors,q theyffiffiffiffiffiffiffiffiffiffiffiffi cause the Hj ¼ Vjðbj bj e þ bj bje þ bjbj e þ bjbje Þ ð16Þ C J deviation of the lowest level splitting Ωj ¼ 8E E of the j j L y ε y δ δ ε H ¼ V ðb by ei t þ b b ei t þ b by ei t þ b b ei tÞ: ð17Þ transmons and the current dividing ratios between neighboring j j j jþ1 j jþ1 j jþ1 j jþ1 L L transmons from their proposed values. In recent experiments the As being noticed already, the four terms in Hj and Hj oscillate with fabrication-induced disorder in SQC lattice systems has been frequencies ε, δ, −δ and −ε, respectively. Meanwhile, the dynamic 35 significantly suppressed by the developing technology. If the modulation of Vj(t) and VjðtÞ can synthesize the dispersive coupling by fabrication errors are not too large such that the resulted level shifting the frequencies of the terms in Eq. (16) upward and downward. For η δ η ε fi the modulation VjðtÞ¼2sj cosð Þt þ 2qj cosð Þt,we nd that splits and current dividing ratios are still around their ideal values P y y y fi Hj ¼ sjbj bj þ qjbj bj is positively activated (i.e. it oscillates with frequency with small disorder, they can be corrected by simple re nement of η N y ) and its Hermitian conjugate Hj ¼ sjbjbj þ qjbjbj is negatively activated. the proposed DDM scheme: The deviation of the level splitting L fi The other terms in Hj are de-activated because their frequencies are far can be compensated by the modi cation of the frequency of the away from the dispersive active region [−η, η] at least by the order of δ. DDM pulses, and the errors of the current dividing ratio can be η δ η ε Also, for the modulation VjðtÞ¼2sj cosð Þt þ 2qj cosð þ Þt, we can compensated by the renormalization of the amplitude of the DDM P y positively activate Hj ¼ sjbjbjþ1 þ qjbjbjþ1 and negatively activate 36–38 N y y y L L pulses. Hj ¼ sjbj bjþ1 þ qjbj bjþ1. The dispersive coupling between Hj and Hj The low-frequency 1/f noises can also induce fluctuations of the results in two effects. The first is the self part circuit parameters.39 Due to their low frequency property, we can P; N P; N =η treat the 1/f noises as quasi-static, i. e. the noises do not vary ð½Hj Hj þ½Hj Hj Þ during a experimental run, but vary between different runs. As the which contains the self-energy corrections of the sites (j, +), (j, −), and (j+1, proposed HCB chain considered consists of only transmon qubits +). The more nontrivial terms emerge from the mutual dispersive coupling and coupling SQUIDs with very small charging energies, it is P; N P; N =η insensitive to the charge type 1/f noise. Such insensitivity has ð½Hj Hj þ½Hj Hj Þ fl already been extensively investigated in Ref. 29. For the ux 1/f which can be simplified as noise penetrated in the loops of the SQUIDs, we notice that s q previous experiments have shown that the flux noise δΦ in a Eff sjsj y j j y y H ¼ b bjþ1Pj þ b b Pj þ h:c: ð18Þ SQUID loop does not vary greatly with the loop size, inductor j η j η j jþ1 value, or temperature, and its strength falls in the range Similarly, the dispersive coupling between the (j, −)↔(j+1, +) and (j+1, −6 −5 40–43 δΦ/Φ0 ∈ [10 ,10 ]. The consequent fluctuation effects can +)↔(j+1, −) links provides the effective (j, −)↔(j+1, −) coupling then be estimated as s s s q Eff j jþ1 y j jþ1 y y : :: 19 δΩ = π 1 ; δ ; δ 3 ; Hj ¼ bj bjþ1Pjþ1 bj bjþ1Pjþ1 þ h c ð Þ j 2 <10 MHz sj qj<10 MHz ð15Þ η η where both of which induce local perturbation to the ideal Notice that all the effective hopping and pairing constants in Eqs. (18), Hamiltonian with amplitudes much smaller than the proposed (19) can be independently controlled by the pumping parameters (sj, qj) ; band gap of the DIII chain. Such small fluctuations cannot destroy and ðsj qjÞ. Through the setting the MZMs as they are too small to change the band topology by j1 ; j1 ; sj ¼ð1Þ s qj ¼ð1Þ q closing and reopening the gap (notice that this argument applies ð20Þ j1 ; j1 ; = = ; to the other parameter fluctuations, and such robustness roots sj ¼ð1Þ s qj ¼ð1Þ q q s ¼q s from the topological nature of our scheme). In addition, the population-dependent phase terms in the bosonic version of HDIII can experiments have shown that the influence of the critical current be directly reproduced from Eqs. (18), (19).

Published in partnership with The University of New South Wales npj Quantum Information (2017) 8 Bosonic quantum simulator of DIII topological superconductors YHuet al 6 ACKNOWLEDGEMENTS 21. Mao, T. & Wang, Z. D. Quantum simulation of topological Majorana bound states We thank T. Mao and D. Zhang for helpful discussions. This work was supported in and their universal quantum operations using charge-qubit arrays. Phys. Rev. A part by the National Science Foundation of China (Grant No. 11374117), the NKRDP of 91, 012336 (2015). China (Grant No. 2016YFA0301802), the GRF (HKU173051/14P & HKU173055/15P), 22. Marcos, D., Rabl, P., Rico, E. & Zoller, P. Superconducting circuits for quantum the CRF (HKU8/11G) of Hong Kong, the fellowship of HongKong Scholars Program simulation of dynamical gauge fields. Phys. Rev. Lett. 111, 110504 (2013). (Grant No. 2012-80), and the National Fundamental Research Program of China 23. Garca-Álvarez, L. et al. 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