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

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.

Introduction-In exploration of symmetry-protected topological phases, more and more attention has been paid to theoretical and experimental studies of superconductor wires in the symmetry class DIII [1][2][3][4][5][6][7]. In contrast to the Kitaev model in class D with only one Majorana zero-mode at each end and two-fold degenerate ground states in its Z (1) 2 topological phase [8], these DIII superconductor wires in the Z (2) 2 topological phase protected by the time-reversal symmetry(TRS) have a Kramer doublet of Majorana zero-modes at each end and four-fold degenerate ground states. In addition, these DIII superconductor wires exhibit a novel response to external weak magnetic field and long-range spin-correlation at two ends with fermionic parity fixed, which together with the four-fold ground-state degeneracy may be utilized to realize the universal quantum manipulation of a topological qubit [2,5,6]. Although being well studied theoretically, still the one-dimensional TRS protected Z 2 topological phase has not been confirmed experimentally so far.
Utilizing the superconducting quantum circuit(SQC), in this manuscript we provide a faithful simulation of the DIII model constructed in Ref. [5], which realizes the nontrivial topological phase through a special p-wave spin-triplet pairing. Comparing with the traditional atomic and photonic systems, the SQC has been seen to be a promising candidate to simulate lattice models with the advantage of flexibility and scalability [9][10][11][12][13][14], allowing us to engineer superconducting devices as basic unit cells and wire them up to design nontrivial many-body couplings . Recently SQCs have been poposed to simulate one-dimensional (1D) fermionic systems [15], with interesting phenomena being investigated including the pursuit of Majorana fermions in the Kitaev model [16][17][18], the string breaking dynamics of quark pairs [19], and the electron-electron scattering [20]. However methods in the existing works are restricted to spinless fermions, where the form of a Hamiltonian remains unchanged after bosonization. When bosonizing a spinful 1D fermionic system, an exotic particle-densitydependent U(1) gauge field emerges from fermionic statistics among fermions with opposite spin orientations, leading to one of main challenges in the simulation of the spinful DIII model. Here we develop the dispersive dynamic modulation(DDM) approach to mediate the exotic U(1) gauge field configuration in the bosonized DIII model, and present in detail a physical implementation in a chain of inductively coupled transmon qubits [21][22][23][24], with strong tunability and scalability. Moreover a careful analysis shows that the coupling constants in the proposed SQC system are several orders larger than the lifetime of the qubit excitation, implying the dissipative effect of SQCs is negligible in the detection of topological Majorana zero-modes through the state-of-art cross-correlation measurements [25].

Spin
The hard-core boson realization of DIII model-An intriguing and important model Hamiltonian of fermionic 1D topological superconductor in class DIII reads [5], where a † j = a † j,↑ , a † j,↓ and a j = (a j,↑ , a j,↓ ) and ∆ is real. Under the basis given by the eigenstates of σ 1 , H DIII can be rewritten as with i.e., H DIII is decoupled into two independent Kitaev-like chains with opposite signs of their superconducting order parameters. Although simulating the standard Kitaev model with hard-core bosons (HCBs) was proposed in coupled nonlinear cavity lattices and in superconducting qubit arrays [16][17][18], the simulation of the H DIII is much more challenging than superficially simulating H + and H − by two independent HCB chains. Explicitly, the DIII model (4) can be bosonized as with the bosonization along the zig-zag path in Fig.[1] being given by where b j,± and b † j,± are the ladder operators of the HCBs, P s,± = exp(−iπb † j,± b j,± ) [26]. It is observed from Eq.(5) that the anti-commutation relation between fermions with spin ± is converted to a gauge field after bosonization, which in one chain is given by the particle density of the other chain in a site wise sense.
In the following we implement the bosonization of the DIII model (5) using the DDM approach. It is assumed that the HCB sites on the upper and lower array of Fig.[1] have energy splits Ω + and Ω − with δ = Ω + − Ω − ∼ 10 −1 Ω + . Such HCB sites can be simulated either by nonlinear cavities with strong on-site Hubbard repulsion or by multilevel atoms with large anharmonicity [27][28][29][30] . Moreover, the HCB sites are assumed to be coupled with their nearest-neighbors along the zig-zag path, taking the general form where V jσ,kγ ∈ [10 −2 , 10 −3 ]δ that can be modulated harmonically and in situ, and σ and γ are spin indexes. In addition, for each HCB a harmonic driving is added, leading to being the external harmonic driving imposed on the site (j, σ).
Let us take the building of the (j, +) ←→ (j + 1, +) coupling through the modulation of V j+,j− and V j−,(j+1)+ as an example. The main obstacle towards the realization of Eq.(5) is to synthesize the exotic population-dependent phase factors. Here we exploit the dispersive coupling mechanism which states that, for a system governed by a fast oscillating Hamiltonian Ae iωt + Be −iωt with A , B ≪ ω, its evolution can be described by the effective For this purpose we adopt the rotating frame for which H L j+,j− and H L j−,(j+1)+ take the forms with ǫ = Ω + + Ω − . As being noticed already, the four terms in H L j+,j− and H L j−,(j+1)+ oscillate with frequencies ǫ, δ, −δ and − ǫ, respectively. Meanwhile, the dynamic modulation of V j+,j− and V j−,(j+1)+ can synthesize the dispersive coupling by shifting the frequencies of the terms in Eq.(8) upward and downward. We further introduce the dispersive energy scale If the terms in H L j+,j− and H L j−,(j+1)+ oscillate with frequency η, they can still induce dispersive couplings with strength on the level of such that H P it oscillates with frequency η) and its Hermitian conjugate All the other terms in H L j+,j− are de-activated because their frequencies are far away from the dispersive active region [−η, η] at least by the order of δ. Also, we modulate V j−,(j+1)+ as V j−,(j+1)+ = T j e i(η−δ)t + Q j e i(η+ǫ)t + h.c.
to positively activate H P j−,(j+1) Similarly, the dispersive coupling between the (j, −) ←→ (j + 1, +) and (j + 1, +) ←→ (j + 1, −) links provides the effective (j, −) ←→ (j + 1, −) coupling Notice that all the effective hopping and pairing constants in Eqs. (10) and (11) can be independently controlled by the pumping parameters (T j , Q j ) and (t j , q j ). By the suitable choice t j = (−1) n−1 t 1 , q j = (−1) n−1 ∆t 1 /w, we can directly reproduce from Eqs. (10) and (11)  Superconducting circuit implementation-We now describe in detail how to realize the theoretical model described by SQCs. For a particular site (j, σ), let us consider a superconducting transmon qubit consisting of a symmetric superconducting quantum interference device (SQUID) with the maximal Josephson energy E jσ being penetrated by an external flux bias Φ S jσ , and shunted by a large capacitance C jσ [21,22], as shown in Fig.[2]. The two characteristic energy scales of the transmon qubit are given by its level splitting Ω jσ ≃ 8E C jσ E J jσ and its anharmonicity E C jσ , where E C jσ = e 2 /2C jσ is its charging energy and E J jσ = E jσ cos(πΦ S jσ /Φ 0 ) is its effective Josephson-tunnel energy with Φ 0 = h/2e being the flux quantum. As shown in recent experiments, the typical range of Ω jσ /2π is 5 − 20 GHz [32][33][34]. Without lost of generality here we choose j-independent Ω j+ /2π = Ω + /2π = 20 GHz and Ω j− /2π = Ω − /2π = 15 GHz. In addition, the anharmonicity can be set at the level of E C jσ /2π ≃ 500 MHz. Therefore, the presence of the higher levels can only slightly modify the effective parameters derived below and the transmon qubit can be consequently modeled by a two-level HCB Hamiltonian For the link between neighboring qubits, we exploit the current divider technique recently studied in experiments [23,24]. As shown in Fig.[2], the transmon qubit design is slightly Since the critial current of the grounding SQUID is much larger than the current flowing through the transmon qubit, we can regard the grounding SQUID as a small linear inductance L D jσ = Φ 2 0 /4π 2 E JD jσ which creates a low voltage node on the qubit (Fig.[2]). We further connect the neighboring jσ, kγ nodes by a coupling SQUID with the maximal Josephson coupling energy E I jσ,kγ and the external penetrating flux Φ I jσ,kγ . The effective Josephson coupling energy of the coupling SQUID is set as E JI jσ,kγ = E I jσ,kγ cos(πΦ I jσ,kγ /Φ 0 ) ≃ E J jσ but its capacitance C I jσ,kγ is chosen to be much smaller than C jσ . In addition, a compensation flux bias Φ C jσ,kγ is added to the (jσ, kγ) coupling loop. While the Hamiltonian of this lattice may be derived rigorously by the method of circuit quantization [35], we can explain the inter-qubit coupling mechanism in an intuitive way. For each qubit, the presence of the grounding SQUID slightly renormalizes the qubit's eigenenergy and leaves the qubit's anharmonicity largely unaffected. Meanwhile, this grounding SQUID can divide the current from one qubit to flow through its neighboring qubits due to its finite inductance. An excitation current from the (j, +) qubit, which may be written as I j+ ≃ Ω + /2L J j+ (b † j,+ + b j+ ) with L J j+ = Φ 2 0 /4π 2 E J jσ , will mostly flows through L D jσ to the ground, with a small fractions I j+,j− and I j+,(j−1)− flowing to the neighboring qubits (j − 1, −) and (j, −) through the two coupler SQUIDs. The dividing currents which depend on the inductances connecting to the (j, +) node can be written as −1 being the total grounding inductance through the (j, +) node and L I jσ,kγ = Φ 2 0 /4π 2 E JI jσ,kγ the effective inductance of the jσ, kγ coupling SQUID. The current I j+,j− generates a flux Φ j+,j− = L D j− I j+,j− in the (j, −) qubit. Therefore, the interaction between the two qubits can be written as with the coupling constant The first factor comes from the r.m.s current flowing through the two qubits, and the latter one consisting of two terms is the mutual inductance between the two qubits, with the second term accounting for the contribution of I j−,j+ in the L D j+ grounding SQUID. As being noticed already, the coupling constant g j+,j− depends on the inductances of the SQUIDs connecting to the (j, +) and (j, −) nodes. Modulating the penetrating flux biases harmonically results in the oscillation of the ratios of current division in the nodes and in turn the oscillation of the inter-qubit coupling constants. However, it is noted that the modulation frequencies of the SQUIDs should not be higher than their plasma frequencies, otherwise complicated quasiparticle excitations would happen. To fulfill this requirement, we may tune the effective Josephson coupling energies of the coupling and grounding SQUIDs to non-zero DC values and thereby establish the nonzero static inter-qubit coupling. As being mentioned previously, the maximal modulating frequency of the DDM approach is of the order Ω + + Ω − . Meanwhile, with the parameter ranges being chosen before, the grounding SQUIDs have effective Josephson energies much larger than that of the transmon SQUIDs, and the coupling SQUIDs have capacitive energies larger than that of the the transmon capacitances. Thus their plasma frequencies are all much higher than Ω + + Ω − . Based on Eq. (16), it is estimated that the DC bias of the SQUIDs leads to the static qubit-qubit couplings on the level of 150 MHz, much smaller than the energy difference between the neighboring qubits on the level of 5 GHz. Consequently its influence is corrections of the hopping and pairing parameters derived from the DDM method. The couplings beyond the nearest neighbors can also be estimated. Since the divided current I j+,j− from (j, +) is further divided in the (j, −) node before flowing to (j + 1, +) inductively, it is found that couplings decay exponentially with respect to the site distance with the decay speed determined by the ratio between the grounding and coupling SQUIDs. The next nearest neighbor coupling is estimated to be of the order 10 MHz. To suppress its effect, we may use two sub-lattices for each of the two legs. A simple strategy is that the eigenfrequencies of the four qubits are modified to be (15,20,14,19)2πGHz, respectively, as shown in Fig.[2]. To let the performance of the DDM be unaffected by such modification, we merely need to adjust the modulating frequencies accordingly. However, the next nearest neighbor coupling is significantly suppressed by the 1 GHz energy difference between the next nearest neighboring qubits. The higher order couplings are even smaller, and therefore can be safely neglected.
Based on the static bias, we can add the AC biases T j , Q j , t j , q j at the order of 10 ∼ 20 2πMHz. The strength of the AC bias is very small compared with the anharmonicity of the qubits such that the requirement of two-level HCBs is satisfied. Choosing the dispersive active region as η/2π = 100 MHz, the resulting dispersive tunneling w/2π and the pairing constant ∆/2π are estimated to be on the level of 2 ∼ 3 MHz, which is two orders larger than the decoherence rates of the qubit 20 kHz ≤ κ jσ /2π ≤ 30 kHz [34]. Such large coupling strength allows us to detect the properties of the system through spectral method developed in recent experiments [25]. It is also noticed that the tuning of coupling inductances lead to nonlocal effects. For instance, the tuning of the (j+, j−) coupling SQUID changes the current dividing ratios of the (j, +) and (j, −) nodes. In turn, it will influence not only the (j+, j−) coupling but also the ((j − 1)−, j+) and (j−, (j + 1)+) couplings. However, the 2N − 1 coupling SQUIDs for a chain of N spinful fermionic sites can be synchronically controlled such that the 2N − 1 coupling constants can be modulated independently. The synchronized tuning of the grounding SQUIDs can also be utilized to keep the currentdividing ratio through each node invariant. In this way there are 4N − 1 controlling flux biases (2N ground SQUIDs and 2N − 1 coupling SQUIDs) and 2N constraints (the current dividing ratios through the 2N nodes should be kept unchanged), thus we still have 2N − 1 degrees of freedom to control each of the inter-qubit link independently. The modulation of the coupling and grounding SQUIDs may also change the parameters of qubits individually and add on the qubit AC drivings. However, the DDM frequency of the SQUIDs is in the neighborhood of either Ω + + Ω − or Ω + − Ω − . Neither of the two frequencies is in resonance with any of the qubits. Therefore the qubits are largely unaffected by the DDM pulses.