Superconducting Quantum Simulator for Topological Order and the Toric Code

PHYSICAL REVIEW A 95, 042330 (2017)

Superconducting quantum simulator for topological order and the toric code

Mahdi Sameti,1 Anton Potocnik,ˇ 2 Dan E. Browne,3 Andreas Wallraff,2 and Michael J. Hartmann1 1Institute of Photonics and Quantum Sciences, Heriot-Watt University Edinburgh EH14 4AS, United Kingdom 2Department of Physics, ETH Zürich, CH-8093 Zürich, Switzerland 3Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom (Received 19 August 2016; published 21 April 2017)

Topological order is now being established as a central criterion for characterizing and classifying ground states of condensed matter systems and complements categorizations based on symmetries. Fractional quantum Hall systems and quantum spin liquids are receiving substantial interest because of their intriguing quantum correlations, their exotic excitations, and prospects for protecting stored quantum information against errors. Here, we show that the Hamiltonian of the central model of this class of systems, the toric code, can be directly implemented as an analog quantum simulator in lattices of superconducting circuits. The four-body interactions, which lie at its heart, are in our concept realized via superconducting quantum interference devices (SQUIDs) that are driven by a suitably oscillating flux bias. All physical qubits and coupling SQUIDs can be individually controlled with high precision. Topologically ordered states can be prepared via an adiabatic ramp of the stabilizer interactions. Strings of qubit operators, including the stabilizers and correlations along noncontractible loops, can be read out via a capacitive coupling to read-out resonators. Moreover, the available single-qubit operations allow to create and propagate elementary excitations of the toric code and to verify their fractional statistics. The architecture we propose allows to implement a large variety of many-body interactions and thus provides a versatile analog quantum simulator for topological order and lattice gauge theories.

DOI: 10.1103/PhysRevA.95.042330

Topological phases of quantum matter [1,2] are of scientific excitations, called anyons, is of great interest as they neither interest for their intriguing quantum correlation properties, show bosonic nor fermionic but fractional statistics [4,5]. exotic excitations with fractional and non-Abelian quantum In two dimensions, the toric code can be visualized as 1 statistics, and their prospects for physically protecting quan- a square lattice where the spin- 2 degrees of freedom are tum information against errors [3]. placed on the edges [see Fig. 1(a)]. Stabilizer operators can The model which takes the central role in the discussion of be defined for this lattice as products of σ x operators for all topological order is the toric code [4–6], which is an example of four spins around a star [orange diamond in Fig. 1(a)], i.e., = x y a Z2 lattice gauge theory. It consists of a two-dimensional spin As j∈star(s) σj , and products of σ operators for all four lattice with quasilocal four-body interactions between the spins spins around a plaquette [green diamond in Fig. 1(a)], i.e., g = y and features 4 degenerate, topologically ordered ground states Bp j∈plaq(p) σj . The Hamiltonian for this system reads as for a lattice on a surface of genus g. The topological order of   g these 4 ground states shows up in their correlation properties. HTC =−Js As − Jp Bp, (1) The topologically ordered states are locally indistinguishable s p (the reduced density matrices of a single spin are identical for all of them) and only show differences for global properties where Js (Jp) is the strength of the star (plaquette) interactions (correlations along noncontractible loops). Moreover, local and the sums s ( p) run over all stars (plaquettes) in the perturbations cannot transform these states into one another lattice [9]. and can therefore only excite higher-energy states that are Despite the intriguing perspectives for quantum computa- separated by a finite-energy gap. tion and for exploring strongly correlated condensed matter In equilibrium, the toric code ground states are therefore physics that a realization of the toric code Hamiltonian would protected against local perturbations that are small compared offer, progress towards its implementation has to date been to the gap, which renders them ideal candidates for storing inhibited by the difficulties in realizing clean and strong quantum information [4]. Yet, for a self-correcting quantum four-body interactions while at the same time suppressing memory, the mobility of excitations needs to be suppressed as two-body interactions to a sufficient extent. Realizations of well, which so far has only been shown to be achievable in topologically encoded qubits have been reported in supercon- four or more lattice dimensions [7,8]. Excitations above the ducting qubits [10,11], photons [12], and trapped ions [13]. ground states of the toric code are generated by any string The toric code is also related to a model of geometrically of spin rotations with open boundaries. The study of these frustrated spins on a honeycomb lattice [14]. Yet, in contrast to the toric code itself, this honeycomb model only features two-body interactions, which can naturally occur in spin lattices, and evidence for its realization in iridates [15] and Published by the American Physical Society under the terms of the ruthenium-based materials [16] has been found in neutron Creative Commons Attribution 4.0 International license. Further scattering experiments. distribution of this work must maintain attribution to the author(s) There is, moreover, an important difference in the use and the published article’s title, journal citation, and DOI. and purpose of an implementation of the Hamiltonian (1)

2469-9926/2017/95(4)/042330(20) 042330-1 Published by the American Physical Society MAHDI SAMETI et al. PHYSICAL REVIEW A 95, 042330 (2017)

(a) constraints [25–27]. We anticipate that the approach we describe here may also be relevant for quantum annealing, however, our focus in this paper is on the realization of Hamiltonians supporting topological order. Superconducting circuits have recently made tremendous advances in realizing engineered quantum dynamics for quan- tum simulation [28–32] and quantum information processing [17–19,33–37]. Their quantum features originate from inte- grated Josephson junctions that behave like ideal nonlinear inductors. They are typically built as superconducting quantum interference devices (SQUIDs) that can be tuned and driven via magnetic fluxes. Currently, superconducting circuits are as- sembled in ever larger connected networks [17,18,34–36,38]. Here, we make use of these exquisite properties of supercon- ducting circuits for a scheme implementing the Hamiltonian of the two-dimensional toric code in a direct way by creating = x four-body interactions of the form As j∈star(s) σj and (b) ϕn+1,m  = y Bp j∈plaq(p) σj between the respective qubits and fully suppressing all two-body interactions. Superconducting circuits are an ideal platform for exper- ϕ b;n,m L imentally exploring topological order since high-precision control and measurements of both individual qubits as well 2C ϕa;n,m J as multiqubit clusters have been demonstrated [17,30,39–41]. Importantly, this platform is also suitable for realizing periodic ϕn,m L L ϕn+1,m+1 boundary conditions since conductors can cross each other via air bridges [42,43]. This property, which is essential for Φext exploring topological order, is so far not accessible in other EJ ϕn,m+1 platforms unless one resorts to digitized approaches. L Our approach is designed to keep the necessary circuitry as simple as possible and only employs standard superconducting Cq;n,m+1 EJq;n,m+1 qubits that are coupled via dc-SQUIDs [44,45]. To generate the desired interactions, we make use of the nonlinear nature ϕn,m+1 of the coupling SQUIDs and the possibilities to bias them with constant and oscillating magnetic fluxes. In particular, the four- FIG. 1. Toric code lattice and its implementation. (a) Lattice body stabilizer interactions of the toric code are generated by for the toric code. Small rectangles on the lattice edges indicate driving the SQUIDs with suitably oscillating magnetic fluxes. spin- 1 systems (qubits). Star interactions A are marked with 2 s They can therefore be switched on and off at desired times B orange diamonds and plaquette interactions p with green diamonds. or even be smoothly tuned in strength. In this way, the toric The transition frequencies of the qubits form a periodic pattern in both code Hamiltonian is realized in a frame where the qubits rotate lattice directions such that no nearest neighbors and no next-nearest at their transition frequencies. Importantly, the transformation neighbors of spins have the same transition frequency and, therefore, all four spins at the corners of a physical cell have different transition between the rotating and the laboratory frame is composed of frequencies. (b) Circuit of a physical cell. The rectangles at the single-qubit unitaries only, so that the entanglement properties corners represent qubits with different transition frequencies (see of the states are identical in both frames. Yet, for implementing inset for the circuit of each qubit). Qubits (n,m)and(n,m + 1) the toric code Hamiltonian (1), considering a rotating frame has [(n + 1,m)and(n + 1,m + 1)] are connected via the inductances the pivotal advantages that the effective transition frequencies L to the node (a; n,m)[(b; n,m)]. The nodes (a; n,m)and(b; n,m) of the individual spins become arbitrarily small and that the in turn are connected via a dc-SQUID with an external magnetic approach becomes largely insensitive to disorder in the lattice. flux ext threaded through its loop. The Josephson junctions in the In contrast to earlier approaches to multibody interactions dc-SQUID have Josephson energies EJ and capacitances CJ ,which based on gate sequences [46–49], the four-body interactions in include shunt capacitances. our approach are directly implemented as an analog quantum simulator and are therefore time independent, which avoids any errors routed in the Trotter sequencing. Moreover, our compared to error detection via stabilizer measurements as approach suppresses two-body interactions to an extent that pursued in [12,17–20]. In particular, two-body interactions are they are negligible compared to the four-body interactions. all that is needed for arbitrary stabilizer measurements, while A minimal two-dimensional lattice covering the surface of lattice gauge theories [21–24] and Hamiltonians such as (1) a torus requires eight qubits connected via superconducting require k-body (with k>2) effective interactions. We note wires such that the lattice has periodic boundary conditions that the realization of k-body interactions is also important in both directions and is thus realizable with available in quantum annealing architectures, where implementations technology [17,18,34]. We show that the topologically have been proposed utilizing local interactions and many-body ordered states of the toric code can be prepared via an

042330-2 SUPERCONDUCTING QUANTUM SIMULATOR FOR . . . PHYSICAL REVIEW A 95, 042330 (2017) adiabatic sweep from a product state for experimentally phase at the auxiliary node (a/b; n,m). The modes ϕ+;j realizable parameters and that measurements of individual behave as harmonic oscillators described by the Hamiltonians = −2 2 + 2 qubits and qubit-qubit correlations can be performed to H+;j 4EC+h¯ π+;j ELϕ+;j , where π±;j are the canoni- 2 demonstrate the topological order of these states. Moreover, cally conjugate momenta to ϕ±;j and EC+ = e /Cg, with Cg the robustness of the topologically ordered states against local the capacitance between each of the nodes (a; j)or(b; j) perturbation that would change the energy of the system by a and ground. The modes ϕ−;j in turn are influenced by the limited amount can be verified experimentally. dc-SQUID. Their Hamiltonians read as   EC− 2 2 ϕext I. PHYSICAL LATTICE H−;j = 4 π− + ELϕ− − 2EJ cos cos(ϕ−;j ), h¯ 2 ;j ;j 2 For realizing the toric code lattice, star and plaquette inter- (3) actions, indicated by orange and green diamonds in Fig. 1(a), = are both realized via the circuit shown in Fig. 1(b) but with where ϕext ext/φ0 is the phase associated to the external different external fluxes ext applied to the coupling SQUID. flux through the SQUID loop, EJ the Josephson energy of 2 The lattice resulting from the orange and green diamonds each junction in the SQUID, and EC− = e /2CJ with CJ the in Fig. 1(a) with spins on the lattice vertices rather than capacitance of one junction in the SQUID, including a shunt edges is a convenient representation introduced in [50] (see capacitance parallel to the junction. also [51,52]). The small black squares in Figs. 1(a) and 1(b) We choose the indices j = (n,m) for the auxiliary nodes indicate spins formed by the two lowest-energy eigenstates of such that node (a; n,m) couples to the qubits at nodes (n,m) and superconducting qubits. For the transition frequencies between (n,m + 1) whereas node (b; n,m) couples to the qubits at nodes these two lowest states, we assume a periodic pattern in (n + 1,m) and (n + 1,m + 1) [see Fig. 1(b)]. The interaction both lattice directions such that no nearest neighbors and terms between the qubit variables ϕj and the SQUID modes no next-nearest neighbors of qubits have the same transition ϕ±;j due to the inductances L thus read as frequency and, hence, all qubits in each four-spin cell have H =−E ϕ + ϕ+ − E ϕ − ϕ− , (4) different transition frequencies (see Sec. III for details). L;j L q ;j ;j L q ;j ;j For each cell, the four qubits on its corners are connected where ϕq±;j = ϕn,m + ϕn,m+1 ± ϕn+1,m ± ϕn+1,m+1. Here, in- by a dc-SQUID formed by two identical Josephson junctions ductive couplings via mutual inductances are an alternative with Josephson energy EJ [see Fig. 1(b) (an asymmetry option [54]. The full Hamiltonian of the lattice is therefore between the junctions causes a negligible perturbation only)]. given by The dc-SQUID is threaded by a time-dependent external flux (N,M) ext and its ports are connected to the four qubits via four H = [Hq j + H+ j + H−,j + HL j ]. (5) inductors, each with inductance L. There are thus auxiliary ; ; ; j=(1,1) nodes at both ports of the SQUID. As we show in Sec. II, these auxiliary nodes (a; n,m) and (b; n,m) can be eliminated from A detailed derivation of the Hamiltonian H, starting from a the description while, together with the flux bias ext on the Lagrangian for the circuit, is provided in Appendix A.We SQUID, they mediate the desired four-body interactions of the now turn to explain how the four-body interactions of the stabilizers. stabilizers emerge in our architecture.

Hamiltonian of the physical lattice II. STABILIZER OPERATORS

We describe the lattice in terms of the phases ϕj and The guiding idea for our approach comes from the following their conjugate momenta πj for each node, which obey the approximate consideration (we present a quantitatively precise commutation relations [ϕj ,πl] = ihδ¯ j,l. The nodes are labeled derivation in Sec. II A). In each single cell, as sketched in by composite indices j = (n,m), where n labels the diagonals Fig. 1(b), the auxiliary nodes (a; n,m) and (b; n,m)have extending from top left to bottom right and m labels the qubits very small capacitances with respect to ground and therefore along each diagonal in Fig. 1(a). cannot accumulate charge. As a consequence, the node phases Each qubit is characterized by a Josephson energy EJq;j ϕa;n,m and ϕb;n,m are not dynamical degrees of freedom but 2 + and a charging energy ECq;j = e /(2Cq;j ), where e is the instantly follow the phases at nodes (n,m) and (n,m 1) or + + + elementary charge and Cq;j the qubit’s capacitance. Including (n 1,m) and (n 1,m 1) as given by Kirchoff’s current 2 a contribution 4ELϕj that arises from the coupling to its law saying that the sum of all currents into and out of four adjacent auxiliary nodes via the inductances L,the such a node must vanish. Therefore, the Josephson energy Hamiltonian of the qubit at node j reads as of the coupling SQUID, 2EJ cos(ϕext/2) cos(ϕa;n,m − ϕb;n,m), becomes a nonlinear function of the node phases ϕn,m, ϕn,m+1, H = 4E h¯ −2π 2 − E cos(ϕ ) + 4E ϕ2, (2) q;j Cq;j j Jq;j j L j ϕn+1,m, and ϕn+1,m+1, which contains four-body terms. The external flux applied through the SQUID is then chosen where E = φ2/(2L) and φ = h/¯ (2e) is the rescaled quantum L 0 0 such that it triggers the desired four-body interactions forming of flux. We consider transmon qubits [17,18,29,30,34,35,53] the stabilizers. To this end, it is composed of a constant part because of their favorable coherence properties and moderate with associated phase ϕ and an oscillating part associated charging energies (cf. Appendix D1). dc with phase amplitude ϕ , For the description of the dc-SQUIDs it is useful to ac consider the modes ϕ±;j = ϕa;j ± ϕb;j , where ϕa/b;j is the ϕext(t) = ϕdc + ϕacF (t), (6)

042330-3 MAHDI SAMETI et al. PHYSICAL REVIEW A 95, 042330 (2017) where the oscillation frequencies of F (t) are linear combina- tions of the qubit transition frequencies such that the desired four-body spin interactions of the Hamiltonian (1) are enabled anharmonic spring and all other interaction terms are suppressed. Moreover, ω we choose ϕ = π to suppress the critical current I = qubit 1 dc c −iω1t −1 q1 e 2EJ φ0 cos(ϕdc/2) of the dc-SQUID [55] and thus unwanted interactions of excitations between qubits at opposite sides of cos(ωdt) it [cf. Fig. 1(b)]. Assuming furthermore that the oscillating SQUID part of the flux bias is small compared to the constant part, we approximate cos(ϕ /2) ≈−ϕ F (t)/2. ext ac ω− The above consideration uses approximations and only ω4 ω2 −iω4t −iω2t serves us as a guide. To obtain a quantitatively precise picture, q4 e q2 e we here take the small but finite capacitances of the Josephson ω junctions in the SQUID into account and derive the stabilizer 3 −iω3t interactions in a fully quantum mechanical approach. Our harmonic spring q3 e derivation makes use of methods for generating effective four-body Hamiltonians in the low-energy sector of a two-body Hamiltonian [14,56–58] by employing a Schrieffer-Wolff transformation [59] and combines these with multitone driving FIG. 2. Macroscopic analogy providing an intuitive explanation in order to single out specific many-body interactions while of our approach to generate the stabilizer interactions. The qubits fully suppressing all two-body interactions. are nonlinear oscillators so that one can think of them as masses Further intuition for our approach can be obtained from (corresponding to the qubits’ capacitances) attached to anharmonic the macroscopic analogy explained in Fig. 2. In short, the springs (corresponding to the qubits Josephson energy). We here plasma frequency of the coupling SQUIDs differs form all sketch four nonlinear oscillators representing four qubits as green transition frequencies of the qubits they couple to as well as spheres connected by black (anharmonic) springs to the support from all frequencies contained in the oscillating flux that is frame. Note that all four qubits have different transition frequencies threaded through their loops. They thus decouple from the so that all four spheres representing them here would have different qubits but mediate the desired four-body interactions via masses and their springs different spring constants. For vanishing the drives applied to them. Further details of this analogy critical current, the relevant SQUID modes ϕ−;j behave as harmonic are explained in the caption of Fig. 2. We now turn to present oscillators and in our macroscopic analogy we can represent them the quantitatively precise derivation. by a mass (here sketched as a red sphere, corresponding to the capacitance 2CJ ) that is attached via four springs (sketched in orange, corresponding to the inductors L) to the four masses representing the A. Elimination of the SQUID degrees of freedom neighboring qubits. For vanishing critical current, the only potential energy of the modes ϕ−;j comes from their inductive coupling to For the parameters of our architecture, the oscillation the neighboring qubits, and hence the mass representing the SQUID frequencies of the SQUID modes ϕ±;j differ from the transition mode (red sphere) has no spring connecting it to a support frame. We frequencies of the qubits and the frequencies of the time- now imagine that we shake the mass representing the SQUID mode dependent fluxes threaded through the SQUID loops by an with a driving term that is proportional to the fourth power of the amount that exceeds their mutual couplings by more than 4 mass’ position variable ϕ−;j . This corresponds to the oscillating flux an order of magnitude. These modes will thus remain in bias ∝ϕacF (t). If we shake this mass at a frequency ωd that is distinct their ground states to very high precision. To eliminate them from its eigenfrequency ω−;j but equals the sum of the oscillation from the description and obtain an effective Hamiltonian for frequencies of all four neighboring masses (neighboring qubits), we † † † † † the qubits only, it is useful to consider the Schrieffer-Wolff drive a term proportional to q1 q2 q3 q4 ,whereqj creates an excitation transformation in qubit j. If, on the other hand, we shake the SQUID mass at a frequency ωd = ω1 + ω2 − ω3 − ω4, we drive a term proportional † † − to q q q q . The selective driving of specific four-body terms, as H → H˜ = eS He S with 1 2 3 4 we consider it here, of course relies on choosing the transition (N,M) i  frequencies of all four qubits to be different. Our approach considers S = (ϕ + π+ + ϕ − π− ), (7) 2¯h q ;j ;j q ;j ;j a superposition of various drive frequencies such that the sum of the j=(1,1) triggered four-body terms adds up to the desired stabilizer interaction (see text). which eliminates the qubit-SQUID interactions described in Eq. (4). The effective Hamiltonian for the qubits is then obtained by projecting the resulting Hamiltonian onto the a truncation of the expansion is here well justified since ground states of the SQUID modes ϕ±;j . The details of this both fields ϕq±;j and π±;j have low enough amplitudes [see derivation are discussed in Appendix B. Eq. (B16)], so that higher-order terms can safely be ignored. For the transformed Hamiltonian H˜ [see Eq. (7)], we The transformed Hamiltonian can be written as consider terms up to fourth order in S to get the leading contributions of the terms which are in resonance with the oscillating part of ϕext(t) [cf. Eq. (6)]. On the other hand, H˜ = H0q + HIq + HS + HqS, (8)

042330-4 SUPERCONDUCTING QUANTUM SIMULATOR FOR . . . PHYSICAL REVIEW A 95, 042330 (2017)  = (N,M) B. Effective Hamiltonian of one cell where H0q j=(1,1) Hq;j with Hq;j as in Eq. (2) describes the individual qubits and We now focus on the interactions between the four qubits in one cell only [see Fig. 1(b)]. To simplify the notation, we N,M label these four qubits clockwise with indices 1, 2, 3, and =− HIq 2EL ϕn,mϕn,m+1 (9) 4, i.e., (n,m) → 1, (n + 1,m) → 2, (n + 1,m + 1) → 3, and n,m=1 (n,m + 1) → 4. We thus divide one term, say for (n,m) = (n0,m0), in the sum of Eq. (11) into the following parts: interactions between neighboring qubits that are ineffective  since their transition frequencies differ by an amount that (0) Hcell = H = H1 + H2 + H3, (12) exceeds the relevant coupling strength by more than an order qS cell of magnitude (see Appendix D1). where The remaining two terms H and H in Eq. (8) describe S qS χϕacEJ the SQUIDs and the residual couplings between the qubits and H1 = F (t)ϕ1ϕ2ϕ3ϕ4 16 theSQUIDs[seeEqs.(B7) and (B8) in Appendix B]. Since the oscillation frequencies of the SQUID modes are chosen contains the four-body interactions of the stabilizers, such that they cannot be excited by the applied driving fields χϕ E or via their coupling to the qubits, an effective Hamiltonian, H =− ac J F (t)(ϕ + ϕ − ϕ − ϕ )2 2 8 1 4 2 3 describing only the qubits, can be obtained by projecting HqS onto the ground states of the SQUID modes contains all terms that are quadratic in the node phases ϕj , and  → (0) = | |  H =−χϕ E F t C ϕ ϕ ϕ ϕ HqS HqS 0SQUIDs HqS 0SQUIDs . (10) 3 ac J ( ) ij kl i j k l (i,j,k,l) As the SQUID modes remain in their ground states, their contains all terms that are fourth order in the node phases ϕ Hamiltonian H may be discarded. The projected Hamiltonian j S except for the four-body interactions. These are terms where H (0) contains two-body interactions and four-body interactions qS at least two of the indices i, j, k, and l are the same as between the qubits,  indicated by the notation (i,j,k,l). Details of the derivation of the Hamiltonian (12) are given in Appendix C, where we χϕ E N,M (0) ac J 2 also state the exact form of the coefficients Cij kl . H =− F (t)ϕ − qS 8 q ;n,m The Hamiltonian (12) can be written in second quan- n,m=1 = + † † tized form using ϕj ϕj (qj qj ), where qj (qj ) creates 1 χϕ E N,M + ac J 4 (annihilates) a bosonic excitation at node j with energy F (t)ϕq−;n,m (11) hω¯ j = 8EC;j (EJq;j + 4EL) and zero-point fluctuation am- 4! 16 = n,m 1 = + 1/4 plitude ϕj [2EC;j /(EJq;j 4EL)] . Note that the effec- tive Josephson energy EJq j + 4EL includes here contribu- with χ =0−;n,m| cos(ϕ−;n,m)|0−;n,m, where |0−;n,m is the ; tions of the attached inductances (see Appendix B1for ground state of the mode ϕ−;n,m. We emphasize here that our approach requires nonlinear details). Due to the qubit nonlinearities we can restrict the coupling circuits, such as the considered SQUIDs. The terms description to the two lowest-energy levels of each node by → − † → + in the second line of Eq. (11), which give rise to the desired replacing qj σj (qj σj ) and get for the four-body four-body interactions, only emerge because the Hamiltonian interactions of the coupling SQUIDs contains terms that are at least fourth 4  χϕacEJ power in ϕ−;n,m (see Appendix B). H = F (t) ϕ σ aσ bσ cσ d . (13) 1 16 i 1 2 3 4 The oscillation frequencies contained in the external flux, i=1 a,b,c,d∈{+,−} i.e., in ϕext, can be chosen such that selected terms in the second  line of Eq. (11) are enabled as they become resonant. For Here, the sum a,b,c,d∈{+,−} runs over all 16 possible strings + − suitable amplitudes of the frequency components of ϕext, these of σ and σ operators, which are listed in Table II in terms add up to the desired four-body stabilizer interactions Appendix C. whereas all other interactions contained in Eq. (11) are for our The explicit form of the Hamiltonians H2 in second choice of parameters strongly suppressed in a rotating wave quantized form is given in Eq. (D15). The terms contained approximation. in H2 only lead to frequency shifts for the qubits and Before discussing this point in more detail in the next negligible two-body interactions. The frequency shifts can section, let us here emphasize the versatility of our approach: be compensated for by modified frequencies for the driving With suitably chosen Fourier components of ϕext,almostany fields and the residual two-body interactions can be made two-, three-, or four-body interaction can be generated. The two orders of magnitude smaller than the targeted four-body only limitation here is that terms with the same rotation interactions (see Sec. II C and Appendix D). Moreover, the frequency can not be distinguished, e.g., one can not choose terms contained in H3 only lead to corrections that are at least z z z z = = between generating σj σk or σj σl for j k l which are both an order of magnitude smaller than those of H2 and can safely nonrotating. On the other hand, straightforward modifications be discarded (see Appendix D3). Here, we therefore focus on of the circuit allow to implement higher than four-body (e.g., the part H1 which gives rise to the dominant terms in the form five- and six-body) interactions. of four-body stabilizer interactions.

042330-5 MAHDI SAMETI et al. PHYSICAL REVIEW A 95, 042330 (2017)

C. Triggering four-body interactions In a rotating frame where each qubit rotates at its tran- + + − − − = iωj t = iωj t sition frequency σj (t) e σj [σj (t) e σj ], the a b c d strings σ1 σ2 σ3 σ4 that appear in Eq. (13) rotate at the frequencies ωa,b,c,d = aω1 + bω2 + cω3 + dω4. Hence, if we thread a magnetic flux, which oscillates at the frequency ωa,b,c,d, through the SQUID loop, the corresponding operator a b c d σ1 σ2 σ3 σ4 acquires a prefactor with oscillation frequency ωa,b,c,d, which renders the term nonoscillating and hence enables this particular four-body interaction. Importantly, all qubits of a cell have different transition frequencies such that a drive with frequency ωa,b,c,d only enables the particular a b c d interaction σ1 σ2 σ3 σ4 , whereas all other interactions are suppressed by their fast oscillating prefactors. x x x x As the star and plaquette interactions Js σ1 σ2 σ3 σ4 and y y y y FIG. 3. Lattice for minimal realization on a torus with eight Jpσ1 σ2 σ3 σ4 can be written as linear combinations of operator a b c d physical qubits including four star interactions (orange diamonds) products σ1 σ2 σ3 σ4 , their generation requires magnetic fluxes that are a superposition of oscillations at the respective and four plaquette interactions (green diamonds). The qubits are frequencies. To enable the specific four-body interactions of indicated by small colored squares, where the coloring encodes the toric code we thus consider their transition frequencies and the numbering indicates the periodic boundary conditions. Here, eight different transition frequencies are Fs = FX + FY , needed. F (t) = (14) Fp = FX − FY , our choice of qubit transition frequencies, where no nearest where FX = f1,1,1,1 + f1,1,−1,−1 + f1,−1,1,−1 + f1,−1,−1,1 neighbors or next-nearest neighbors of qubits have the same and FY = f1,1,1,−1 + f1,1,−1,1 + f1,−1,1,1 + f−1,1,1,1 with transition frequencies, the leading corrections to the toric fa,b,c,d = cos(ωa,b,c,dt). The driving field Fs generates a star code Hamiltonian of Eq. (1), aside from local frequency interaction As and Fp a plaquette interaction Bp. Applying Fs shifts, are two-body interactions between qubits that are or Fp to a cell, its Hamiltonian (13) thus reads as further apart (i.e., neither nearest neighbors nor next-neighrest  (1)  =− x x x x + neighbors). For suitable parameters as discussed in Sec. III, Hcell(t) F Js σ1 σ2 σ3 σ4 r.t. for Fs , (15a)  s these interactions are about two orders of magnitude weaker (1)  =− y y y y + than the stabilizers and thus even weaker than dissipation Hcell(t) Jpσ1 σ2 σ3 σ4 r.t. for Fp, (15b) Fp processes in the device (see Appendix D). where r.t. refers to the remaining fast rotating terms, and In this context, we also note that the leading contribution to the perturbations of the Hamiltonian (1) is proportional 4 χE ϕ to the charging energy ECq;j of the employed qubits (cf. J = J = J ac ϕ . (16) s p 16 i Appendix D). Hence, the considered transmon qubits are a i=1 well-suited choice for our aims. Hence, star and plaquette interactions are realized with the same coupling circuit [shown in Fig. 1(b)] and only differ D. Including dissipative processes by the relative sign of the FX and FY contributions, i.e., a relative phase of π in the respective Fourier components. As we have shown, the dynamics of the architecture we Consequently, a driving field with Fs (Fp) is applied to the envision can be described by the Hamiltonian (1) in a rotating = + − coupling circuit of every orange (green) diamond in Fig. 1(a) frame with respect to H0 j ωj σj σj , where we assume (see also Fig. 3) and the resulting checkerboard pattern of that all frequency shifts discussed in the previous section (see driving fields generates the toric code Hamiltonian of Eq. (1). also Appendix D) have been absorbed into a redefinition of Note that the signs and magnitudes of the interactions Js and the ωj . In an experimentally realistic scenario, the qubits will, Jp can for each cell be tuned independently via the respective however, be affected by dissipation. The full dynamics of the choices for the drive amplitudes ϕac. circuit can therefore be described by the Markovian master Aside from the stabilizer interactions, there are further equation terms in the Hamiltonian (8). As we show in Appendix D, =− −1 + D + D all these other interactions are strongly suppressed by their ρ˙ ih¯ [H,ρ] r [ρ] d [ρ], (17) fast oscillating prefactors (the oscillation frequencies of the prefactors exceed the interaction strengths by more than where ρ is the state of the qubit excitations, H is as in Eq. (1), D = κ − + − + − − + − an order of magnitude in all cases). Their effect can be r [ρ] 2 j (2σj ρσj σj σj ρ ρσj σj ) describes re- D = z z − estimated using time-dependent perturbation theory. We find laxation at a rate κ, and d [ρ] κ j (σj ρσj ρ) pure for our parameters that they lead to frequency shifts for dephasing at a rate κ . Note that both the relaxation and the individual qubits, which can be compensated for by a dephasing terms Dr [ρ] and Dd [ρ] are invariant under the modification of the frequencies of the driving field ϕext.For transformation into the rotating frame.

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TABLE I. Possible transition frequencies for an eight-qubit toric Furthermore, the inductances coupling the qubits to the code in natural frequency units. The qubit numbers correspond to SQUIDs need to be chosen large enough so that the resulting the numbering chosen in Fig. 3. These frequencies are assumed to coupling term remains weak compared to the qubit nonlinear- already include the frequency shifts calculated in Appendix D. ities. This requires inductances of about 100 nH which can be built as superinductors in the form of Josephson junction Trans. frequency Trans. frequency arrays [61,62] or with high kinetic inductance superconducting Qubit ωj /2π Qubit ωj /2π nanowires [63]. We note that our concept is compatible 1 13.8 GHz 5 13.1 GHz with nonlinearities Josephson junction arrays may have. = 2 13.0 GHz 6 13.7 GHz The amplitude of the drive is ϕac 0.1 rad corresponding × 3 3.5 GHz 7 4.2 GHz to an oscillating flux bias of amplitude 0.1 0 at the 4 4.1 GHz 8 3.4 GHz SQUID. These parameters give rise to a strength for star and plaquette interactions of Js = Jp = h×1 MHz, which exceeds currently achievable dissipation rates of transmon In the following, we turn to discuss realistic experimental qubits significantly [32,64–67]. The strength of the stabilizer parameters that lead to an effective toric code Hamiltonian, interactions can be enhanced further by increasing the Joseph- son energies of the SQUIDs E and/or the amplitudes of the the preparation and verification of topological order in ex- J oscillating drives ϕ . periments, and protocols for probing the anyonic statistics of ac For the above parameters, the total oscillation frequency, excitations in a realization of the toric code with our approach. i.e., the sum of the frequencies of the qubit operators and In all these discussions, we fully take the relaxation and the frequency of the external flux, is for any remaining dephasing processes described by Eq. (17) into account. two-body interaction term at least 10–30 times larger than the corresponding coupling strength. To leading order, these III. TOWARDS AN EXPERIMENTAL REALIZATION terms give rise to frequency shifts of the qubits which can be compensated for by a modification of the drive frequencies. An experimental implementation of our approach is feasible All interaction terms contained in higher-order correction are with existing technology as it is based on transmon qubits negligible compared to the four-body interactions forming the which are the current state of the art in many laboratories. stabilizers. We note that the experimental verification of the Moreover, periodic driving of a coupling element at fre- topological phase is further facilitated by its robustness against quencies comparable to the qubit transition frequencies is a local perturbations, such as residual two-body interactions or well-established technique [41,60]. The minimal lattice of a shifts of the qubit transition frequencies (cf. [68,69]). toric code with periodic boundary conditions in both lattice As an alternative to our approach, one could aim at directions is the eight-qubit lattice sketched in Fig. 3.Anim- implementing the Honeycomb model [14,70], which allows to portant benefit of our rotating frame approach is its inherent in- obtain the toric code via a fourth-order perturbative expansion sensitivity to disorder due to largely spaced qubit transition fre- in the limit where one of the spin-spin couplings strongly quencies. The parameters we discuss in the sequel do thus not exceeds the two others [5]. This strategy would, however, need to be realized exactly but only with moderate precision. require twice the number of qubits as our approach. For typical qubit-qubit couplings of ∼100 MHz, it would moreover only A. Parameters lead to stabilizer interactions that are two orders of magnitude weaker than in our approach. A working example for this eight-qubit realization with experimentally realistic parameters is provided by the qubit transition frequencies given in Table I, which are realized B. Control and read-out in qubits with Josephson energies in the range 4.9 GHz  In common with all two-dimensional lattices, the architec- EJq/h  94 GHz (h is Planck’s constant) and capacitances ture we envision requires an increasing number of control lines in the range 52 fF  Cq;j  77 fF. Note that the frequencies for the qubits, SQUIDs, and the read-out as one enhances the are chosen such that neighboring qubits are detuned from each lattice size. Whereas solutions to this complexity challenge other by several GHz, whereas the transition frequencies of are already being developed, e.g., by placing the control lines next-nearest-neighbor qubits differ by ∼100 MHz. A practical on a second layer [35,71,72], a minimal realization of the advantage of our approach with respect to frequency crowding Hamiltonian (1) requires only eight qubits that form four stars is that it only requires a small set of continuous single- and four plaquettes (see Fig. 3). Moreover, these eight physical frequency driving fields, whereas the frequency width of the qubits are arranged on a 4×2 lattice so that every qubit is pulses in digitized schemes grows inversely proportional to located on its boundary, facilitating the control access. their durations. Controlling single-qubit states and two-qubit correlation For the coupling SQUIDs, the Josephson energy of each measurements would require a charge line and a read-out junction should be EJ /h ≈ 10 GHz and the associated capac- resonator for each qubit. Our scheme, where a single cell itance 3.47 fF. Since high-transparency Josephson junctions features four qubits with different transition frequencies, has typically have capacitances of 1 fF per 50 GHz of EJ , each the beneficial property that four read-out resonators can be SQUID is shunted by a capacitance of a few fF. For these connected to a single transmission line and the qubit states parameters, the modes ϕ−;j oscillate at ω−/2π = 8.45 GHz, read-out using frequency division multiplexing [73–75]. Such whereas the oscillation frequency of the modes ϕ+;j is much a joint read-out is capable of measuring the state of all four higher since Cg CJ . qubits and qubit pairs even when their transition frequencies

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IV. ADIABATIC PREPARATION OF TOPOLOGICAL ORDER On the surface of a torus, i.e., for a two-dimensional lattice with periodic boundary conditions in both lattice directions, the Hamiltonian (1) has four topologically ordered, degenerate eigenstates |ψa (a = 1,2,3,4), which are joint eigenstates of the stabilizers As and Bp with eigenvalue +1 (see Appendix E for a possible choice of the |ψa for an eight-qubit lattice). A state in this manifold of topologically ordered states can either be prepared via measurements of the stabilizer operators or via an adiabatic sweep that starts from a product state [80]. Measurements of stabilizers can in our architecture be performed via a joint dispersive read-out of all qubits in a star or plaquette (see Sec. V). For the adiabatic sweep, one starts in our realization with the oscillating driving fields turned off, ϕac = 0 and the qubit FIG. 4. Pattern of transition frequencies for an implementation transition frequencies blue-detuned by a detuning with of a large lattice. The qubits are indicated by small colored squares, respect to their final values ωj . In the cryogenic environment where the coloring encodes their transition frequencies. Here, four of an experiment, this initializes the system in a product state, principal transition frequencies are needed. These differ by a few GHz where all qubits are in their ground state, |ψ0= |0j . and are indicated by the filling colors brown, blue, red, and purple. j Then, the amplitudes of the external oscillating fluxes ϕac are To suppress two-body interactions between next-nearest-neighbor gradually increased from zero to their final values ϕ , i.e., qubits, each pair of these needs to be detuned from each other by ac,f ϕ (t) = λ(t) ϕ , where λ(t = 0) = 0, λ(T ) = 1, and T is ∼100 MHz. The resulting frequency differences are indicated by the ac ac,f S S the duration of the sweep. The qubit transition frequencies colored frames around the respective squares. The gray diamond − = − highlights the frequency set which is periodically repeated in both are decreased to ωj , i.e., ωj (t) ωj [1 λ(t)] > 0, with − = − = lattice directions. ωj (0) ωj and ωj (TS) ωj 0. For this sweep, the Hamiltonian of the circuit can be written in the rotating frame as are separated by several GHz. An improvement over the    read-out through a common transmission line would be to H (λ) = (1 − λ) σ z − λJ A − λJ B , (18) 2 j s s p p employ a common Purcell filter [76,77] with two modes, one j s p mode placed at 3.5 GHz and the second at 12.5 GHz. Such where A and B are as in Eq. (1). Thus, the initial Hamiltonian scheme would not only allow a joint read-out of four qubits, s p is H (λ = 0) = σ z/2 and the unique initial ground state but also protect them against the Purcell decay. Frequency j j is the vacuum with no qubit excitations (all spins down), division multiplexing can also be used for qubit excitation | = |  where a single charge line can be split or routed to four qubits i.e., ψ0 j 0j . The final Hamiltonian is the targeted toric code Hamiltonian given in Eq. (1). Whereas the initial in the cell. If a common transmission line is used for the |  read-out, the same line can also be used as a feedline for qubit detuning ensures that the state ψ0 is also an eigenstate in the = = state preparation [18]. rotating frame for Js Jp 0, the sweep will drive the system adiabatically into the manifold of protected states, provided the parameters are tuned sufficiently slow. C. Scalability The spectrum of the minimal eight-qubit lattice (cf. Fig. 3) Importantly, our scheme is scalable as it also allows to isshowninFig.5 as a function of the tuning parameter λ. There realize the toric code on larger lattices. As already mentioned is one unique ground state at λ = 0 which eventually becomes in Secs. I and II C, the requirement for the choice of the qubit degenerate with three other states to form the degenerate four- transition frequencies is that no nearest-neighbor qubits and no dimensional manifold of topologically ordered ground states at next-nearest-neighbor qubits have the same transition frequen- λ = 1. In finite-size lattices, there is, however, for any value of cies. The resulting pattern of transition frequencies for a large λ, a finite-energy separation of the order of Js or Jp between lattice is sketched in Fig. 4. It involves 16 qubit transition fre- the ground state and all states which do not end up in the quencies and can be realized with circuit parameters similar to topologically ordered manifold at λ = 1. We have numerically the ones given above. These 16 qubit transition frequencies are confirmed this for 8, 12, and 16 qubits. This spectral gap periodically repeated in both lattice directions and, therefore, makes the sweep robust because it is not important which in principle, sufficient for building any lattice size. state in this manifold is prepared. All states in the manifold Larger and more realistic toric code realizations will and any linear combination of them are equivalent. Therefore, obviously suffer from increased complexity. This could be we use the probability of ending up in the topologically = 4  | |  drastically simplified when adding on-chip RF switches [78] ordered manifold, F a=1 ψa ρ(TS) ψa , as a criterion for routing microwave signals to different cells and using for a successful preparation. Moreover, in the absence of selective broadcasting technique which would also reduce the dissipation, one will end up in a pure state since a unitary number of microwave devices such as signal generators and evolution cannot change the purity of a state. To quantify how arbitrary waveform generators [79]. unavoidable dissipative processes in a real experiment would

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|+  x |+ =+ |+  into states j with σj j 1 j via a π/2 pulse. |+  Since the resulting state j j is an eigenstate of all star stabilizers As , these may then be switched on abruptly together − x |+  withaterm Bx j σj which stabilizes the state j j . Subsequently, the plaquette interactions are turned on and the − x term Bx j σj is turned off in a sweep that is adiabatic since

MHz the relevant transition matrix elements of the instantaneous Hamiltonian always vanish [80]. λ Once they are prepared, it is of course desirable to stabilize the topologically ordered states as long as possible. Several approaches to this task have been put forward (see [81,82] and references therein), which are compatible with the rotating frame implementation we consider. Most approaches consider FIG. 5. The energy eigenvalues of H (λ) as a function of λ a shadow lattice of auxiliary qubits that are coupled to the = × = = for the eight-qubit lattice with 2π 5 MHz and Js Jp primary lattice. In our architecture, such auxiliary systems may × | = 2π 2 MHz. The initial ground state is the unique vacuum ψ0 not be required as the coupling SQUIDs could be employed 8 |  ∼ j=1 0j .Atλ 0.6 the ground state starts to become degenerate for this task. but the degenerate ground-state manifold is always separated by a gap ∼Js ∼ Jp from higher excited states. V. MEASURING TOPOLOGICAL CORRELATIONS affect the preparation, we also compute the purity of the final Our scheme allows to experimentally explore and verify state P = Tr(ρ2). topological order. To this end, one prepares the lattice in one Figure 6(a) shows the fidelity F for being in the topologi- of the topologically ordered states |ψa and measures the state cally ordered qubit subspace as a function of the sweep time TS of individual physical qubits as well as a correlation such as  x  and the initial detuning . Here, ρ(t) is the actual state of the l∈v σl for a noncontractible path v along one of the lattice system in the rotating frame and the |ψa (a = 1,...,4) form directions, which is closed in a loop to implement the periodic the subspace of degenerate protected ground states. To test that boundary conditions. For the minimal eight-qubit realization, the prepared state is pure, we also plot the purity P as a function this is merely a measurement of a two-spin correlation, i.e., = = ×  x x  y of TS and in Fig. 6(b). We choose Js Jp h 2MHz(see σmσn . Noncontractible loops of σ operations, such as = = y |  Sec. II D), relaxation and dephasing times T1 T2 50 μs, l∈v σl , transform the topologically ordered states ψa into 6 6 and a ramp profile λ(t) = (1 + ζ )(t/TS ) /[1 + ζ (t/TS) ] with one another [5]. Here, these can be realized via strings of π ζ = 100. For these values, a protected state can successfully pulses applied to the qubits along the loop. After applying a be prepared. string of π pulses along one lattice direction, which transforms For larger lattices, one may also follow the scheme proposed |ψa into a different topologically correlated state |ψb, one in [80] which does not rely on a finite-energy separation again measures individual spins as well as the correlation  x  between the ground states and the higher excited states that l∈v σl . Whereas the states of the individual spins should do not become topologically ordered in the sweep. In our be the same for |ψa and |ψb because the topologically case, the scheme of [80] proceeds as follows. All qubits, correlated states are locally indistinguishable, the results for which are initially in their states |0j , are first transformed the correlations should be distinct.

0.95 0.95

0.9 0.9 0.85 FP 0.8 0.85 0.75

0.7 0.8 1.5 1.5 1000 1000 1 800 1 800 TS [μs] 600 TS [μs] 600 400 Δ [MHz] 400 Δ [MHz] 0.5 200 0.5 200

FIG. 6. Adiabatic preparation of an eight-qubit lattice in the protected subspace. (a) Fidelity F for being in the protected subspace and (b) purity P of the state. Js = Jp = h×2 MHz and coherence times T1 = T2 = 50 μs (see Sec. II D).

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The property of only being distinguishable via spin-spin For the prepared topologically ordered state |ψ,the correlations, but not by single-spin quantities, is not unique protocol considers applying a single-qubit rotation σ y to to topologically ordered states but also holds for Bell states. one of the qubits. This operation, which in our approach In contrast, a property that is unique to topologically or- can be implemented via a microwave pulse in a charge line dered states is that local perturbations cannot transform one to the qubit, creates a pair of so-called e particles [5]on topologically ordered state |ψa into another one |ψb.This the neighboring vertices and puts the system into a state crucial property can be experimentally tested with our ap- |ψ,e. Alternatively,√ half a σ y rotation creates the state proach. (|ψ+|ψ,e)/ 2. Similarly, with a σ x rotation on another In the rotating frame we consider, single-spin perturbations qubit, one creates a pair of m particles. Then, one can move on lattice site j can be taken to read as one of these m particles around one of the e particles via x α successive σ rotations on the qubits along the chosen path Hpert,j (t) = g(t)σ for α =+, − or z, (19) j and fuse both m particles again. After√ this sequence, the α | −|  where oscillating prefactors of σj have been included into the system is in the state ( ψ ψ,e )/ 2 due to the fractional time dependence of g(t). In this rotating frame, the excited phase π acquired from braiding the e and m particles.√ This states of the toric code Hamiltonian (1) are separated from phase flip can for instance be detected√ with another σ y the degenerate topologically ordered states by an energy operation√ that maps (|ψ−|ψ,e)/ 2 →|ψ and (|ψ+ = = 4J (assuming Js Jp J ), irrespective of the lattice size. |ψ,e)/ 2 →|ψ,e and a subsequent measurement of the Therefore, the perturbations Hpert,j (t) cannot transform a corresponding stabilizer As . We note that in our approach all topologically ordered state into a state outside the topologically single-qubit operations can be implemented via pulses applied ordered manifold provided |g(t)| 4J . Yet, because of through charge lines and the stabilizer measurement can be the topological order, no local perturbation is capable of realized via the technique described in Sec. V. transforming a topologically ordered state |ψa into another one |ψb. As a consequence, provided their strength fulfills the | | condition g(t) 4J , no local perturbation of the form (19) VII. CONCLUSION AND OUTLOOK can affect a topologically ordered state |ψa. This robustness can be tested experimentally in our approach by verifying that We have introduced a scheme and architecture for a the prepared topologically ordered states are insensitive to any quantum simulator of topological order that implements the local operation, provided it is weak enough. central model of this class of quantum many-body systems, However, as indicated above, the topologically ordered the toric code, on a two-dimensional superconducting circuit states are in our rotating frame implementation not robust lattice. Our approach is based on a scheme that allows to realize against local dissipation because the coupling to an environ- a large variety of many-body interactions by coupling multiple ment can change the energy in the toric code lattice by an qubits via dc-SQUIDs that are driven by a suitably oscillating amount that exceeds 4J . flux bias. The advanced status of experimental technology for This experimental verification of topological correlations superconducting circuits makes available an ample toolbox for can even be performed for the minimal lattice consisting of exploring the physics of topological order with our scheme. eight qubits, where it only requires measuring individual qubits All physical qubits and coupling SQUIDs can be individ- and two-qubit correlations. An example of possible outcomes ually controlled with high precision, topologically ordered for these measurements is given in Table III and discussed in states can be prepared via an adiabatic ramp of the stabilizer Appendix E. interactions, multiqubit correlations including stabilizers and In recent years, several strategies for state tomography in correlations along noncontractible loops can be measured, and superconducting circuits have been demonstrated experimen- anyons can be braided and their fractional statistics can be tally [18,39,40,83]. These either proceed by reading-out the verified. Moreover, superconducting circuit technology allows qubit excitations via an auxiliary phase qubit or resonator. Both for realizing periodic boundary conditions, a feature that is not concepts are compatible with our circuit design. In particular, easily available in other platforms but crucial for exploring correlations between multiple qubits, including the stabilizers topological order. We thus expect our work to open up a As and Bp, can be measured in a joint dispersive read-out realistic path towards an experimental investigation of this via a common coplanar waveguide resonator [18,30,39,40]or intriguing area of quantum many-body physics that at the same several read-out resonators connected to a single transmission time will enable access for detailed measurements. line using frequency multiplexing [73–75]. Our scheme is highly versatile and not restricted to implementations of the toric code only. Several directions for generalizations appear very intriguing at this stage. VI. BRAIDING AND DETECTION OF ANYONS (i) By tuning the time-independent flux bias of the coupling Our approach is also ideally suited for probing the fractional SQUIDs away from the operating point for the toric code, statistics of excitations in the toric code. Once the lattice has one can generate additional two-body interactions of the z z been prepared in a topologically ordered state, for example via form σj σl between neighboring qubits. These can become the sweep described in Sec. IV, the anyonic character of the stronger than the stabilizer interactions. The ability to tune excitations can be verified in an experiment using single-qubit the transition frequencies of the qubits makes our approach a rotations and the measurement of one stabilizer only. We here very versatile quantum simulator for exploring quantum phase discuss a protocol that has originally been proposed for cold transitions between topologically ordered and other phases of atoms [84] but is very well suited for superconducting circuits. two-dimensional spin lattices.

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(ii) In our concept, the many-body interactions mediated (iv) Considering regimes where the degrees of freedom of by SQUIDs are not limited to four-body interactions. By the coupling SQUID can no longer be adiabatically eliminated, connecting one port of a coupling SQUID to kl qubits and the one can modify our approach towards quantum simulations of other to kr qubits, k-body interactions for k = kl + kr can be dynamical gauge fields theories, where the coupling SQUID implemented in kth order of the Schrieffer-Wolff expansion (7) would represent the dynamical degrees of freedom of the gauge together with a suitable choice of frequencies for the flux bias field. ϕext. Thus, three-body interactions, for example, that are an order of magnitude larger than the stabilizer interactions of Eq. (16) can be implemented. For the parameters considered ACKNOWLEDGMENTS in Sec. III, also six-body interactions that are stronger than M.S. and M.J.H. are grateful for discussions with M. Leib dissipation would be feasible and even interactions among at an early stage of this project. A.P. and A.W. acknowledge a larger number of qubits are within reach with some support by ETH Zürich. M.J.H. was supported by the EPSRC improvement of circuit parameters. These capabilities of our under Grant No. EP/N009428/1. M.S. and M.J.H. were scheme open the door to realizations of higher-dimensional supported in part by the National Science Foundation under toric codes [7,8] and other topological codes such as color Grant No. NSF PHY11-25915 for a visit to the program codes [85]. A useful property here is that the implementation “Many-Body Physics with Light” at KITP Santa Barbara of the sum of a star and plaquette interaction in one cell would where a part of this work has been done. only require half the frequency components for the driving fields. (iii) If two qubits that are connected to the same port APPENDIX A: LAGRANGIAN AND HAMILTONIAN of a coupling SQUID are identified with one lattice site OF ENTIRE CIRCUIT of the simulated model, our approach can straightforwardly be generalized to the analog quantum simulation of non- In this Appendix, we present a detailed derivation of the Abelian lattice gauge theories, where the engineered four-body Hamiltonian in Eq. (5) that describes the considered circuit interactions play the role of the coupling matrix between the lattice. We start with the full Lagrangian describing the lattice color degrees of freedom of the adjacent lattice sites. depicted in Fig. 1, which reads as

N,M M,N 2 2 Cq;n,m 2 φ0 2 2 2 L = φ ϕ˙ + E cos(ϕ ) − [(ϕ − ϕ ) + (ϕ − ϕ + ) + (ϕ − ϕ + ) 0 2 n,m Jq;n,m n,m 2L a;n,m n,m a;n,m n,m 1 b;n,m n 1,m n,m=1 n,m=1

M,N 2 2 Cg 2 Cg 2 2 + (ϕ − ϕ + + ) ] + φ ϕ˙ + ϕ˙ + C (˙ϕ − ϕ˙ ) b;n,m n 1,m 1 0 2 a;n,m 2 b;n,m J a;n,m b;n,m n,m=1   M,N ϕ + 2E cos ext cos(ϕ − ϕ ). (A1) J 2 a;n,m b;n,m n,m=1

As explained in the main text, ϕn,m is the phase at node (n,m), As there is neither capacitive coupling between the qubits Cq;n,m is the capacitance, and EJq;n,m the Josephson energy in the lattice nor between the qubits and the coupling SQUIDs, of the qubit at that node. ϕa;n,m and ϕb;n,m are the phases the kinetic energy can be decomposed into the contribution of at the auxiliary nodes next to the SQUID located between individual qubits and that of the individual coupling SQUIDs, nodes (n,m), (n,m + 1), (n + 1,m), and (n + 1,m + 1). The N,M φ2  Cg are the capacitances of these auxiliary nodes with respect K = 0 2 + ˙ T ˙ Cq;n,mϕ˙n,m s;n,mCs s;n,m , (A3) to ground and CJ is the capacitance of each junction in 2 n,m=1 the SQUID including a shunt capacitance parallel to the T junctions. EJ is the Josephson energy of each junction in the where s;n,m = (ϕ+;n,m,ϕ−;n,m) and SQUID.   + Cg As the Josephson junctions of the dc-SQUIDs only couple 2CJ 2 0 Cs = (A4) to the difference between the phases of the two adjacent 0 Cg auxiliary nodes, we define the modes 2 is the capacitive matrix associated with one SQUID. The Hamiltonian of the lattice is obtained in the standard ϕ±;n,m = ϕa;n,m ± ϕb;n,m, (A2) way via a Legendre transform of the Lagrangian L in Eq. (A1) by defining the qubit and SQUID momenta and describe the dc-SQUIDs in terms of ϕ+;n,m and ϕ−;n,m from now on. We now turn to derive the Hamiltonian associated to ∂L ∂L πn,m = and π±;n,m = . (A5) the Lagrangian in Eq. (A1). ∂ϕ˙n,m ∂ϕ˙±;n,m

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The kinetic energy thus reads as in Eq. (7) with the generator   N,M 2 2 2 N,M πn,m π−;n,m π+;n,m i K = + + , (A6) S = (ϕq+;n,mπ+;n,m + ϕq−;n,mπ−;n,m), (B1) 2¯hφ2C 4¯hφ2C hφ¯ 2C 2¯h n,m=1 0 q;n,m 0 J 0 g n,m=1 and expand the transformed Hamiltonian up to fourth order wherewehaveusedthatCg CJ in inverting Cs . We thus arrive at the Hamiltonian in S, S −S N,M H˜ = e He ECq;n,m 2 2 H = 4 π + 4ELϕ − EJq;n,m cos(ϕn,m) 1 1 h¯ 2 n,m n,m = H + [S,H] + [S,[S,H]] + [S,[S,[S,H]]] n,m=1 2! 3!

N,M 1 E + E − + C 2 + C 2 + 2 + [S,[S,[S,[S,H]]]] +··· . (B2) 4 π+;n,m 4 π−;n,m ELϕ+;n,m 4! h¯ 2 h¯ 2 n,m=1   Using the canonical commutation relations (A9), the commu- 2 ϕext tators in the expansion (B2) read as + E ϕ− − 2E cos cos(ϕ− ) L ;n,m J 2 ;n,m [S,H] M,N  N,M − EL[ϕq+;n,mϕ+;n,m + ϕq−;n,mϕ−;n,m], (A7) = ELϕq+;n,mϕ+;n,m + ELϕq−;n,mϕ−;n,m n,m=1 n,m=1 where we have introduced the notation N,M − 4 ECq;n,mπn,m(πS−;n,m + πS+;n,m) ϕ ± = ϕ + ϕ + ± ϕ + ± ϕ + + . (A8) q ;n,m n,m n,m 1 n 1,m n 1,m 1 n,m=1 = 2 N,M EL φ0 /(2L) is the inductive energy associated with an   EL 2 2 inductor L and the charging energies of the qubits and SQUID − ϕ + + ϕ − 2 q ;n,m q ;n,m modes are n,m=1 2 2 2   e e e N,M ϕ = + = − = ext ECq;n,m ,EC , and EC , + EJ cos sin(ϕ−;n,m)ϕq−;n,m, 2Cq;n,m Cg 4CJ 2 n,m=1 where e is the elementary charge. The first line in Eq. (A7) [S,[S,H]] describes the qubits, the second line the SQUIDs, and the N,M third line the interactions between qubits and SQUIDs via the E  = L ϕ2 + ϕ2 inductances L. q+;n,m q−;n,m = 2 The Hamiltonian (A7) is quantized in the standard way by n,m 1 imposing the canonical commutation relations N,M 2 + 2 E (π − + π + ) = = Cq;n,m S ;n,m S ;n,m [ϕj ,πl] ihδ¯ j,l and [ϕ±;j ,π±;l] ihδ¯ j,l. (A9) n,m=1   The external flux applied through the SQUID loops and hence N,M E ϕ + J ext 2 the associated phase is composed of a constant dc part and a cos cos(ϕ−;n,m)ϕq−;n,m, = 2 2 time-dependent ac part as given in Eq. (6). We choose ϕdc = π n,m 1 and ϕac ϕdc = π, and obtain to leading order in ϕac, [S,[S,[S,H]]]     ϕext ϕac N,M cos ≈− F (t). (A10) EJ ϕext 3 =− cos sin(ϕ− )ϕ − , 2 2 4 2 ;n,m q ;n,m n,m=1 [S,[S,[S,[S,H]]]] APPENDIX B: EFFECTIVE TIME-DEPENDENT   N,M HAMILTONIAN EJ ϕext 4 =− cos cos(ϕ− )ϕ − , 8 2 ;n,m q ;n,m In this Appendix, we present details of the derivation of n,m=1 the effective Hamiltonian for the qubits only, which will be where we have here introduced the abbreviations dominated by the four-body interactions corresponding to the stabilizer operators in the toric code Hamiltonian [see Eq. (1)]. πS±;n,m = π±;n,m + π±;n,m−1 ± π±;n−1,m ± π±;n−1,m−1. As the modes ϕ±;n,m of the SQUIDs oscillate at frequencies (B3) that strongly differ from the transition frequencies of the qubits and the oscillation frequencies of the applied flux biases, they We note that the terms [S,[S,[S,H]]] and [S,[S,[S,[S,H]]]] remain in their ground states to a high degree of precision and do not contain any contributions from the modes ϕ+;n,m can be adiabatically eliminated from the description. To this which are just harmonic oscillators. They would vanish for end, we apply the Schrieffer-Wolf transformation introduced an entirely harmonic coupling circuit and only the Josephson

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  terms of the coupling SQUIDs that are nonlinear in ϕ−;n,m N,M 1 EJ ϕext 3 lead to contributions. − cos sin(ϕ− )ϕ − 3! 4 2 ;n,m q ;n,m Truncating the expansion (B2) at fourth order is a good n,m=1   approximation since the amplitudes of the terms ϕn,mπ±;n,m/h¯ N,M are much smaller than unity [see Eq. (B16) for details]. 1 EJ ϕext 4 − cos cos(ϕ−;n,m)ϕ − , (B8) Moreover, as we explain in the sequel, the frequencies 4! 8 2 q ;n,m n,m=1 contained in the flux bias φext that is applied to the SQUIDs can be chosen such that interactions contained in the fourth-order where again we have neglected terms ∝πn,mπS+;n,m because terms [S,[S,[S,[S,H]]]] become the dominant terms of the |ω+ − ωn,m||ω− − ωn,m| [see Eq. (B15)], which makes expansion (B2). these terms highly off resonant so that we only keep the As stated in Eq. (8) of Sec. II A, the transformed Hamil- dominant contributions ∝πn,mπS−;n,m. tonian H˜ can be written as a sum of a part H0q describing the individual qubits, a part HIq describing pure qubit-qubit 1. Second-quantized form interactions, a part H describing the coupling SQUIDs, and S The free Hamiltonian of the qubits can be written in second- a part HqS describing the residual interactions between qubits † and SQUIDs quantized form by defining the ladder operators qi and qi via ˜ = + + + = + † H H0q HIq HS HqS. (B4) ϕj ϕj (qj qj ),

ih¯ † The Hamiltonian of the individual qubits, H0q , and the qubit- =− − πj (qj qj ), with qubit interactions, HIq, are given in Eqs. (8) and (9). We here 2ϕj   repeat them for completeness: 1/4 = 2ECq;j N,M ϕj . (B9) EJq;j + 4EL H0q = Hq;n,m with † n,m=1 Here, qj (qj ) is the annihilation (creation) operator for qubit = E j [j (1,1),...,(N,M)] and the effective Josephson energy = Cq;n,m 2 + 2 − + Hq;n,m 4 2 πn,m 2ELϕn,m EJq;n,m cos(ϕn,m) of the qubits is EJq;j 4EL due to the contributions of the four h¯ adjacent inductors. In terms of these creation and annihilation (B5) operators, H0q reads as and (N,M) = † − † † N,M H0q [¯hωj qj qj hU¯ j qj qj qj qj ], (B10) = HIq =−2EL ϕn,mϕn,m+1. (B6) j (1,1) n,m=1 where we have approximated cos(ϕj ) by its expansion up The term describing the SQUIDs reads as to fourth order in ϕj and the transition frequencies ωj and nonlinearities Uj are given by M,N E˜ + = C ;n,m 2 + 2  HS 4 2 π+;n,m ELϕ+;n,m −1 h¯ ωj = h¯ 8ECq;j (EJq;j + 4EL) and (B11) n,m=1

M,N ˜ ECq;j EJq;j EC−;n,m 2 2 Uj = . (B12) + 4 π− + ELϕ− 2¯h E + 4E h¯ 2 ;n,m ;n,m Jq;j L n,m=1   For the SQUID modes we introduce the annihilation (creation) M,N † ϕext operators s± (s±) according to − 2EJ cos cos(ϕ−;n,m), (B7) 2 † n,m=1 = + ϕ±;j ϕ±(s±;j s±;j ), where E˜ + and E˜ − are the renormalized charging en- C ;n,m C ;n,m =− ih¯ − † ergies of the + and − modes, E˜ ± = E ± + (E + π±;j (s±;j s±;j ), with C ;n,m C ;n,m q;n,m 2ϕ± Eq;n,m+1 + Eq;n+1,m + Eq;n+1,m+1)/4. The qubit-SQUID in-  1/4 teractions read as E˜C± ϕ± = (B13) N,M EL ECq;n,m HqS =−4 πn,mπS−;n,m h¯ 2 and get n,m=1   (N,M) N,M ϕ † † ext HS = [¯hω+s+ s+;j +hω¯ −s− s−;j ], (B14) + EJ cos sin(ϕ−;n,m)ϕq−;n,m ;j ;j 2 j=(1,1) n,m=1   N,M where 1 EJ ϕext 2  + cos cos(ϕ−;n,m)ϕ − 2! 2 2 q ;n,m = −1 ˜ n,m=1 ω± 4¯h EC±EL (B15)

042330-13 MAHDI SAMETI et al. PHYSICAL REVIEW A 95, 042330 (2017) and we have neglected the last term of Eq. (B7) since where cos (ϕext/2) ≈−(ϕac/2)F (t) and the frequencies contained in  F (t) strongly differ from ω+ and ω−. Note that CJ Cg and E˜J = χϕac EJ with (C3) thus E˜C+  E˜C−, so that ω+  ω−.   2 ϕ− χ = exp − ;n,m . (C4) 2. Accuracy of truncating the transformation 2

After having introduced the amplitudes ϕj and ϕ± in In deriving Eq. (C2) we have used the ground-state expectation Eqs. (B9) and (B13), we can now estimate the amplitude of the values Schrieffer-Wolff generator S [see Eqs. (7)or(B1)] to check whether a truncation of the expansion (B2) at fourth order 0−;n,m| π−;n,m |0−;n,m=0, provides a good approximation. For the scenario with at most a 0−;n,m| cos(ϕ−;n,m) |0−;n,m=χ, single excitation in each qubit and vanishingly small excitation  | | = numbers in the SQUID modes, which is of interest here, the 0−;n,m sin(ϕ−;n,m) 0−;n,m 0. (C5) generator S has a magnitude given by (0) The Hamiltonian HqS is a sum of contributions from each cell 1 ϕj formed by four qubits at its corners which interact via one (B16) coupling SQUID. 4 ϕ± When considering the terms of one cell only, as described  for each site. Typical parameters of our scheme lead to ϕj in Sec. II A, and labeling its qubits by indices 1, 2, 3,  ≈ → + → + + → 0.7 and ϕ± 1.65 so that ϕj /4ϕ± 0.1 and the truncation and 4, i.e., (n,m) 1, (n 1,m) 2, (n 1,m 1) + → (0) of the expansion in Eq. (B2) is well justified. 3, (n,m 1) 4, one can divide the Hamiltonian HqS for one cell into three parts as described in Eq. (12):  APPENDIX C: EFFECTIVE HAMILTONIAN (0) = = + + HqS cell Hcell H1 H2 H3, (C6) IN THE ROTATING FRAME where To analyze which terms of the transformed Hamiltonian (8), 1 respectively (B4), provide the dominant contributions, we H1 = E˜J F (t)ϕ1ϕ2ϕ3ϕ4, now move to a rotating frame, where all qubits rotate at 16 their transition frequencies ωn,m and the SQUID modes rotate 1 2 H2 =− E˜J F (t)(ϕ1 + ϕ4 − ϕ2 − ϕ3) , at their oscillation frequencies ω±. In transforming to this 8  rotating frame, the raising and lowering operators transform =−˜ as H3 EJ F (t) Cij kl ϕi ϕj ϕkϕl. (ij kl) −iωj t iω±t qj → e qj and s±;j → e s±;j (C1) Here, H1 contains the four-body interactions which will for j = (1,1),...,(N,M). give rise to stabilizer interactions, H2 contains all two-body interactions, and H3 all fourth-order interactions beyond the four-body interaction, i.e., terms where at least two of 1. Adiabatic elimination of the SQUID modes the indices i, j, k, and l are the same as we indicate by For the parameters of our architecture, the SQUID frequen- the notation (ij kl). The coefficients Cij kl have the values =± 2 = = cies ω± differ strongly from the qubit transition frequencies Cij kl 1/32 for terms of the form ϕj ϕkϕl with j k, j l, ωj and the frequencies contained in the driving fields. The = =± 2 2 and k l, Cij kl 1/64 for terms of the form ϕj ϕk with SQUID modes ϕ+;n,m and ϕ−;n,m therefore remain to a very = =± 3 = j k, Cij kl 1/96 for terms of the form ϕj ϕk with j k, good approximation in their ground states |0+  and |0− . ;n,m ;n,m =± 4 + They can thus be adiabatically eliminated from the description, and Cij kl 1/384 for terms of the form ϕj . Here, “ ” signs apply whenever an even number of the indices i,j,k,l refer to where the dominant contributions come from the ϕ− modes ;n,m the same side of the SQUID (i.e., the same diagonal) and “−” because ω+  ω−. The leading order of the adiabatic elimination is obtained signs otherwise. Whereas we discuss perturbations originating from H2 and H3 in Appendix D, we now turn to discuss H1 in by projecting HS andHqS onto the ground states of the | = N,M |  more detail. SQUIDs, 0SQUIDs n,m=1 0−;n,m,0+;n,m . Whereas HS only gives rise to an irrelevant constant, we get for H (0) = qS 2. Four-body interactions 0SQUIDs|HqS|0SQUIDs the expression given in Eq. (11), which we here repeat for completeness Due to the nonlinearities of the qubits, we can here restrict the description to the two lowest-energy levels of N,M → − † → + (0) 1 2 each node by replacing qj σj (qj σj ). The Hamil- H =− E˜J F (t) ϕ − qS q ;n,m tonian H1 contains 16 terms of the form S1S2S3S4, where 8 = − − + n,m 1 ∈{ iωj t iωj t } Sj e σj ,e σj [see (C6)]. Ignoring the oscillating N,M prefactor F (t), these terms rotate at the angular frequencies as 1 1 4 + E˜ F (t) ϕ − , (C2) listed in Table II. Denoting the sum of the terms in the first 16 4! J q ;n,m n,m=1 column of Table II by V1 and the sum of the terms in the second

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TABLE II. The four-body terms contained in H1 of Eq. (C6)and APPENDIX D: PERTURBATIONS their rotation frequencies. In this Appendix, we estimate the effects of the remaining Term Frequency Term Frequency terms of the transformed Hamiltonian H˜ [see Eq. (8)or(B4)]. These include contributions contained in the Hamiltonians HIq σ +σ +σ +σ + ν σ +σ +σ +σ − ν 1 2 3 4 1 1 2 3 4 5 [see Eq. (B6)] and HqS [see Eq. (B8)]. In our approach, the − − − − − + + − + σ1 σ2 σ3 σ4 ν1 σ1 σ2 σ3 σ4 ν6 transition frequencies of the qubits ωn,m and the frequency + + − − + − + + spectrum of the oscillating fluxes νk are chosen such that all σ1 σ2 σ3 σ4 ν2 σ1 σ2 σ3 σ4 ν7 + − + − − + + + undesired terms are highly off resonant and therefore strongly σ σ σ σ ν σ σ σ σ ν 1 2 3 4 3 1 2 3 4 8 suppressed. They, however, lead to a frequency shift + − − + − − − + − σ1 σ2 σ3 σ4 ν4 σ1 σ2 σ3 σ4 ν5 − + + − − − − + − − = (1) + (2) + (3) + (4) σ1 σ2 σ3 σ4 ν4 σ1 σ2 σ3 σ4 ν6 j j j j j (D1) − + − + − − + − − − σ1 σ2 σ3 σ4 ν3 σ1 σ2 σ3 σ4 ν7 (1) (2) (3) (4) − − + + + − − − for each qubit j, where , , , and are given in σ σ σ σ −ν σ σ σ σ −ν j j j j 1 2 3 4 2 1 2 3 4 8 Eqs. (D8), (D11), (D13), and (D18). These frequency shifts can easily be compensated for by modifying the frequency Frequencies ν = ω + ω + ω + ω spectrum of the applied driving fields accordingly, so that 1 1 2 3 4 we did not include them in our preceding analysis. They can ν2 = ω1 + ω2 − ω3 − ω4 be calculated via the following time-dependent perturbation ν3 = ω1 − ω2 + ω3 − ω4 theory. ν4 = ω1 − ω2 − ω3 + ω4 ν = ω + ω + ω − ω In the frame where all qubits rotate at their transition 5 1 2 3 4 ˜ ν6 = ω1 + ω2 − ω3 + ω4 frequencies, the transformed Hamiltonian H can be written ν7 = ω1 − ω2 + ω3 + ω4 as ν =−ω + ω + ω + ω  8 1 2 3 4 iωα t H˜ = HTC + hˆαe , (D2) α column by V2, it can readily be verified that V1 + V2 = AS = where HTC is the toric code Hamiltonian as introduced in x x x x y y y y σ σ σ σ and V1 − V2 = BP = σ σ σ σ . Motivated by Eq. (1) and the remaining terms have been written as a 1 2 3 4 1 2 3 4 ˆ the specific form of the Hamiltonian H1 [see Eq. (C6)], we sum of all their frequency components. Hence, hα is the iωα t thus define two oscillating fields FX and FY : sum of all terms that rotate as e , where ωα contains all = + + + contributing frequencies, i.e., the oscillation frequencies of the FX(t) f1,1,1,1 f1,1,−1,−1 f1,−1,1,−1 f1,−1,−1,1, operators and, for terms with a prefactor F (t), the frequency FY (t) = f1,1,1,−1 + f1,1,−1,1 + f1,−1,1,1 + f−1,1,1,1 (C7) of the oscillating flux bias. Note that both ωα > 0 and ωα < 0 ˜ = + + + contribute to the sum in Eq. (D2) since H is Hermitian. with fa,b,c,d cos[(aω1 bω2 cω3 dω4)t]. Each of the Following approaches outlined in [86–88], the dynamics terms in FX (FY ) brings two of the four-body processes in generated by a Hamiltonian of the form as in Eq. (D2) with the first (second) column of Table II into resonance (makes ||hˆα|| hω¯ α can be approximated by a time-independent them nonrotating). Next to resonant terms, each field also effective Hamiltonian that is an expansion in powers of the generates rotating terms. The rotation frequency of these terms interaction strength over oscillation frequency [87] is the detuning between different components of FX and FY . ∼ As the least detuning is in the GHz range ( 0.4 GHz) and Heff = Heff,0 + Heff,1 + Heff,2 + Heff,3 +··· , (D3) the strength of four-body interaction is in the MHz range (∼1 MHz), the rotating terms can be neglected in a rotating where the individual contributions read as wave approximation (RWA). Hence, the external oscillating H = H , field F = F + F gives rise to the star interaction A and eff,0 TC s X Y s  + the external field Fp = FX − FY gives rise to the plaquette δ(ωα ωβ ) Heff,1 = [hˆα,hˆβ ], interaction Bp, 2¯hωβ α,β x x x x  Fs = FX + FY ⇒ H1 ∝ As = σ σ σ σ , δ(ω + ω ) F (t) = 1 2 3 4 H = α β [[H ,hˆ ],hˆ ] = − ⇒ ∝ = y y y y eff,2 2 TC α β Fp FX FY H1 Bp σ1 σ2 σ3 σ4 α,β 2¯h ωαωβ (C8)  + + + δ(ωα ωβ ωγ ) ˆ ˆ ˆ 2 [[hα,hβ ],hγ ], and the strength of the stabilizer interaction is α,β,γ 3¯h ωβ ωγ 4  δ(ωα + ωβ + ωγ ) = = 1 ˜ H = [[[H ,hˆ ],hˆ ],hˆ ] Js Jp EJ ϕi . (C9) eff,3 3 TC α β γ 16 3¯h ωαωβ ωγ i=1 α,β,γ  δ(ω + ω + ω + ω ) Aside from these stabilizer interactions, there are also two- + α β γ δ [[[hˆ ,hˆ ],hˆ ],hˆ ] body interactions between the qubits which may lead to 3 α β γ δ α,β,γ,δ 4¯h ωβ ωγ ωδ corrections in the effective Hamiltonian (1). We now turn to estimate their effect. and δ(ω)istheDiracδ function.

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Since ||hˆα|| hω¯ α, the first term of Heff,2 and the first the first term of HqS can also lead to interactions between term of Heff,3 are at least two orders of magnitude smaller than two qubits of one cell. These are ineffective provided any pair H and we neglect them. We now turn to discuss the other of qubits in a cell is sufficiently detuned from each other, TC   perturbations to H one by one.   TC  E E   Cq;j Cq;l  1, (D9)  2 − − 2  1. Corrections to the adiabatic elimination of the SQUIDs h¯ (ωl ωj )(ω− ωj )ϕj ϕlϕ− (0) Aside from the terms contained in Hqs that are found by which holds since any pair of qubits in one cell is at least projecting the Hamiltonian H˜ onto the ground states of the detuned by 700 MHz from each other (see Table I). Note that SQUID modes, there are first-order corrections to the adiabatic the detuning between two neighboring qubits on a diagonal elimination. These are processes mediated by the creation n that would couple via two SQUIDs is significantly higher and annihilation of a virtual excitation in a SQUID mode. (∼10 GHz). The next-higher-order nonvanishing term of the To estimate their effect, we first note that terms proportional expansion (D3) is here the second term of Heff,3 which is more (1) to cos(ϕ−;n,m) do not create single excitations in the SQUID than two orders of magnitude smaller than j . modes since 0−;n,m| cos(ϕ−;n,m) |1−;n,m=0 and calculate the In a similar way as second-order terms can lead to qubit- matrix elements qubit interactions in one cell, fourth-order terms could mediate interactions between two qubits in neighboring cells. For  | | = 0−;n,m sin(ϕ−;n,m) 0−;n,m 0, our parameters, these interactions would have a strength of ∼ 0−;n,m| sin(ϕ−;n,m) |1−;n,m=χ, 0.5 MHz but are ineffective as qubits in neighboring cells are always detuned by at least 100 MHz. 1− | sin(ϕ− ) |1− =0. (D4) ;n,m ;n,m ;n,m Via an analogous calculation, one finds for the second term In the subspace of at most one excitation, we may thus replace of HqS that the perturbation theory of Eq. (D3) is applicable if † χ   sin(ϕ−;n,m) → χ(s−;n,m + s− ) = ϕ−;n,m (D5)   ;n,m ϕ  E˜ ϕ  −  J j   ± ±  1 (D10) = − (0) 2¯h[(ω− ωj ) νk] to approximate HqS HqS HqS [cf. Eq. (C6)] as N,M and leads to the frequency shifts (taking into account that each ECq;n,m qubit couples to four SQUIDs) HqS ≈−4 πn,mπS−;n,m h¯ 2 n,m=1 2 2 8  x E˜ ϕ ω− + (−1) ω (2) =−2 J j j (D11) j 2 x 2 2 N,M h¯ [ω− + (−1) ωj ] − ν E˜J k=1 x=0,1 k − F (t) ϕ−;n,mϕq−;n,m, (D6) 2ϕ− n,m=1 for each qubit j, where the frequencies νk that are contained in the driving field F (t) are defined in Table II. For the set of where we have neglected the terms in Eq. (B8) that are pro- parameters as given in Sec. III, we find | (2)|  20 MHz. The portional to ϕ3 as these are a factor ϕ2 /24 1 smaller j q−;n,m n,m next-higher-order nonvanishing term of the expansion (D3)is than the terms that are linear in ϕq−;n,m. We now consider here the second term of Heff,3 which is more than two orders corrections to HTC coming from second-order processes of of magnitude smaller than (2). HqS. j Using the formulation in second quantization [see Eqs. (B9) and (B13)], one finds for the first term of HqS that the 2. Perturbations contained in HIq perturbation theory of Eq. (D3) is applicable if ˜   The transformed Hamiltonian H [see Eq. (8)or(B4)]  E  contains residual interactions between neighboring qubits on  Cq;j  .  −  1 (D7) each diagonal n as described in Eq. (B6). Their strength is h¯(ω− ωj )ϕj ϕ− 2ELϕn,mϕn,m+1 for the interaction between qubits (n,m) and For this regime, the effective Hamiltonian Heff,1 according to (n,m + 1) on diagonal n. Hence, provided that   Eq. (D3) for the first term of HqS just contains a frequency    2ELϕn,mϕn,m+1  shift   1, (D12) h¯(ω − ω + ) 4E2 n,m n,m 1 (1) =− Cq;j j 2 2 2 (D8) the perturbation theory of Eq. (D3) is applicable and leads to h¯ (ω− − ω )ϕ ϕ j j − the frequency shift for each qubit j = (n,m). Here, the prefactor 4 accounts for 4E2 ϕ2 ϕ2 4E2 ϕ2 ϕ2 the fact that each qubit has four neighboring SQUIDs. For the (3) = L n,m n,m−1 + L n,m n,m+1 n,m 2 2 (D13) set of parameters discussed in Sec. III, we find for the qubits h¯ (ωn,m − ωn,m−1) h¯ (ωn,m − ωn,m+1) ∼ with high transition frequencies ECq;j /(hϕj ϕ−) 500 MHz for qubit (n,m). For the parameters discussed in Sec. III, (1) ∼ ≈ leading to j 200 MHz, i.e., a shift of 50 MHz from each we find 2ELϕn,mϕn,m+1/h 300 MHz whereas neighboring coupling to a SQUID. For qubits with low transition frequen- qubits on a diagonal are detuned by almost 10 GHz so that (1) ∼ | (3) |∼ cies, the shifts are j 100 MHz, i.e., a shift of 25 MHz n,m 18 MHz. In a higher order of the perturbation theory from each coupling to a SQUID. Aside from frequency shifts, of Eq. (D3), each qubit (n,m) can also mediate an interaction

042330-16 SUPERCONDUCTING QUANTUM SIMULATOR FOR . . . PHYSICAL REVIEW A 95, 042330 (2017) between its two neighboring qubits on the diagonal n due to Hence, to leading order the Hamiltonian H2 will lead to the couplings 2ELϕn,m−1ϕn,m and 2ELϕn,mϕn,m+1. To suppress shifts of the transition frequencies of the qubits. Taking into these interactions of strength 10 MHz, next-nearest-neighbor account that each cell is embedded into the total lattice and qubits need to be sufficiently detuned from one another, hence each qubit experiences frequency shifts from the two-   body interactions in all four cells it is part of, we find that the  2 2   4E ϕ − ϕ ϕ +   L n,m 1 n,m n,m 1  1, (D14) qubit transition frequencies are shifted by  2  h¯ (ωn,m − ωn,m+1)(ωn,m−1 − ωn,m+1) ˜ 2  8 (4) EJ 4 2ωj which requires |ω − − ω + |  2π×100 MHz as ful- =− ϕ n,m 1 n,m 1 j 8 j 4ω2 − ν2 filled by the frequencies in Table I. {l:|j−l|=1} k=1 j k;j,l Furthermore, there could be interactions between qubits in 2  8 2 2 + E˜ ϕ ϕ [ωj + ωl] neighboring cells, say qubit (n,m) and qubit (n,m 2), since − J j l qubits (n,m) and qubit (n,m + 1) couple via 2E ϕ ϕ + 2 2 L n,m n,m 1 32 [ωj + ωl] − ν and qubits (n,m + 1) and qubit (n,m + 2) couple in a second- {l:|j−l|=1} k=1 k;j,l order process via their interaction with a common SQUID as 2  8 2 2 E˜ ϕ ϕ [ωj − ωl] discussed in Appendix D1. This third-order interactions are + J j l , (D18) strongly suppressed since their strength is ∼3 MHz and thus 30 32 [ω − ω ]2 − ν2 {l:|j−l|=1} k=1 j l k;j,l times smaller than the detuning between the respective qubits.  where the sum {l:|j−l|=1} runs over all qubits l that are nearest (0) neighbors to qubit j and νk;j,l denotes the drive frequencies 3. Residual two-body interactions in HqS at the SQUID that connects qubits j and l. For the set of When projected onto the ground states of the SQUID parameters as given in Sec. III, we find | (4)|  30 MHz. modes in the zeroth order of adiabatic elimination, the Hamil- j (0) = | |  tonian HqS 0SQUIDs HqS 0SQUIDs contains terms that are quadratic in ϕq−;n,m and thus contain two-body interactions APPENDIX E: EXPECTED MEASUREMENT OUTCOMES [see Eq. (C2)]. In the rotating frame, the two-body interaction In this Appendix, we discuss a set of possible measurement Hamiltonian of one cell as given by H in Eq. (C6) can be 2 outcomes for the topologically ordered states on an eight- written as qubit lattice (cf. Fig. 3). One ground state of the toric code 4 8 † Hamiltonian on this lattice can be written as i(ωj −ωl +νk )t i(ωj −ωl −νk )t H2 = Bj,l{[e + e ]q ql j 1   j,l=1 k=1 | = √ + y y y y + y y y y ψ1 1 σ1 σ4 σ5 σ3 1 σ2 σ3 σ6 σ4 + + + − † † 4 2 + [ei(ωj ωl νk )t + ei(ωj ωl νk )t ]q q + H.c.}, (D15) j l   8 =∓˜ × 1 + σ y σ y σ y σ y 1 + σ y σ y σ y σ y |+ , (E1) where the coupling coefficients read as Bj,l EJ ϕj ϕl/8, 5 8 1 7 6 7 2 8 j where the “−” sign applies for j = l and (j,l) = (1,4) or j=1 + (2,3), and the “ ” sign applies otherwise. The perturbation |+  x x |+ = where j is the eigenstate of σj with eigenvalue 1, σj j theory of Eq. (D3) applies to H2 provided that   |+j . The other three ground states can be found by applying   y y y y y y y y  E˜J ϕ ϕ  the loop operators σ5 σ6 , σ4 σ8 , and σ4 σ8 σ5 σ6 :  j l  1 (D16)   y y y y 8¯h(ωj ± ωl ± νk) | = |  | = |  ψ2 σ5 σ6 ψ1 , ψ3 σ4 σ8 ψ1 , for all j, l, and k [cf. Eq. (D15)]. Hence, the spectrum of y y y y |ψ4=σ σ σ σ |ψ1. (E2) the driving fields must be chosen such that conditions (D10) 4 8 5 6 and (D16)aremet. Whereas the reduced density matrices of individual spins are To estimate the corrections due to H2 as written in all completely mixed (i.e., proportional to an identity matrix) Eq. (D15), one thus needs to consider the commutation for all these four states, they can be distinguished via their two relations spin correlations (see Table III). † † = − † − † [qj ql ,qj ql] (1 δj,l)(ql ql qj qj ), † † = + † + † + [qj ql,qj ql ] (1 δj,l)(qj qj ql ql 1), TABLE III. Two-qubit correlations along noncontractible loops † = + [qj ql,qj ql ] (1 δj,l) qj qj , for the four ground states of the toric code on an eight-qubit lattice. † † † = + † † [qj ql ,qj ql ] (1 δj,l) ql ql , (D17)  x x  x x  x x  x x  x x  x x  x x  x x  σ1 σ2 σ3 σ4 σ5 σ6 σ7 σ8 σ3 σ7 σ1 σ5 σ4 σ8 σ2 σ6 as these are the only ones that, together with their oscillating |ψ1 01010101 prefactors Fs or Fp, lead to nonrotating terms. The two |ψ2 01010−10−1 commutators in the last two lines of Eq. (D17) would lead to |ψ3 0 −10−10 10 1 transitions out of the qubit subspace and can thus be ignored |ψ4 0 −10−10−10−1 because of the nonlinearity of the qubits.

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