Superconducting Quantum Simulator for Topological Order and the Toric Code

Mahdi Sameti,1 Anton Potoˇcnik,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 (Dated: March 20, 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 non-contractible 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.

Topological phases of quantum matter [1, 2] are of sci- In two dimensions, the Toric Code can be visualized as entific interest for their intriguing quantum correlation a square lattice where the spin-1/2 degrees of freedom are properties, exotic excitations with fractional and non- placed on the edges, see Fig. 1a. Stabilizer operators can Abelian quantum statistics, and their prospects for phys- be defined for this lattice as products of σx-operators for ically protecting quantum information against errors [3]. all four spins around a star (orange diamond in Fig. 1a), Q x y i.e. As = j star(s) σj , and products of σ -operators The model which takes the central role in the discus- ∈ sion of topological order is the Toric Code [4–6], which for all four spins around a plaquette (green diamond in Q y is an example of a lattice gauge theory. It consists Fig. 1a), i.e. Bp = j plaq(p) σj . The Hamiltonian for Z2 ∈ of a two-dimensional spin lattice with quasi local four- this system reads, body interactions between the spins and features 4g de- X X generate, topologically ordered ground states for a lattice HTC = −Js As − Jp Bp, (1) on a surface of genus g. The topological order of these s p 4g ground states shows up in their correlation proper- where Js(Jp) is the strength of the star (plaquette) in- ties. The topologically ordered states are locally indis- P P teractions and the sums s ( p) run over all stars (pla- tinguishable (the reduced density matrices of a single spin quettes) in the lattice [9]. are identical for all of them) and only show differences Despite the intriguing perspectives for quantum com- for global properties (correlations along non-contractible putation and for exploring strongly correlated condensed loops). Moreover local perturbations cannot transform matter physics that a realization of the Toric Code these states into one another and can therefore only ex- Hamiltonian would offer, progress towards its implemen- cite higher energy states that are separated by a finite arXiv:1608.04565v2 [quant-ph] 17 Mar 2017 tation has to date been inhibited by the difficulties in energy gap. realising clean and strong four-body interactions while In equilibrium, the Toric Code ground states are there- at the same time suppressing two-body interactions to a fore protected against local perturbations that are small sufficient extent. Realisations of topologically encoded compared to the gap, which renders them ideal candi- qubits have been reported in superconducting qubits dates for storing quantum information [4]. Yet for a self- [10, 11], photons [12] and trapped ions [13]. The Toric correcting quantum memory, the mobility of excitations Code is also related to a model of geometrically frustrated needs to be suppressed as well, which so far has only been spins on a honeycomb lattice [14]. Yet, in contrast to the shown to be achievable in four or more lattice dimensions Toric Code itself, this Honeycomb model only features [7, 8]. Excitations above the ground states of the Toric two-body interactions, which can naturally occur in spin Code are generated by any string of spin rotations with lattices, and evidence for its realization in iridates [15] open boundaries. The study of these excitations, called and ruthenium-based materials [16] has been found in anyons, is of great interest as the neither show bosonic neutron scattering experiments. nor fermionic but fractional statistics [4, 5]. There is moreover an important difference in the use 2

(a) effective interactions. We note that the realisation of k-body interactions is also important in quantum annealing architectures, where implementations have been proposed utilising local interactions and many-body con- straints [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 realisation of Hamiltonians supporting topological order. Superconducting circuits have recently made tremen- dous advances in realizing engineered quantum dynamics for quantum simulation [28–32] and quantum information processing [17–19, 33–37]. Their quantum features originate from integrated Josephson junctions that behave like ideal nonlinear inductors. They are typically built as SQUIDs that can be tuned and driven via magnetic fluxes. Currently superconducting circuits are as- sembled in ever larger connected networks [17, 18, 34– (b) 'n+1,m 36, 38]. Here we make use of these exquisite properties of superconducting circuits for a scheme implementing the Hamiltonian of the two-dimensional Toric Code in 'b;n,m L a direct way by creating four-body interactions of the Q x Q y form As = σ and Bp = σ between 2C j star(s) j j plaq(p) j 'a;n,m J the respective∈ qubits and fully suppressing∈ all two-body interactions. ' ' n,m L L n+1,m+1 Superconducting circuits are an ideal platform for ex- ext perimentally exploring topological order since high preci- E J ' sion control and measurements of both, individual qubits n,m+1 as well as multi-qubit clusters have been demonstrated L [17, 30, 39–41] . Importantly, this platform is also suit- Cq;n,m+1 able for realizing periodic boundary conditions, since con- EJq;n,m+1 ductors can cross each other via air-bridges [42, 43]. This 'n,m+1 property, which is essential for exploring topological order, is so far not accessible in other platforms unless one correlations are built up Tuesday, June 16, 15 FIG. 1. Toric Code lattice and its implementation. (a) Lat- resorts to digitized approaches. tice for the Toric Code. Small rectangles on the lattice edges Our approach is designed to keep the necessary cir- indicate spin-1/2 systems (qubits). Star interactions As are cuitry as simple as possible and only employs standard marked with orange diamonds and plaquette interactions Bp superconducting qubits that are coupled via dc-SQUIDs with green diamonds. The transition frequencies of the qubits [44, 45]. To generate the desired interactions we make use form a periodic pattern in both lattice directions such that no nearest neighbors and no next nearest neighbors of spins have of the nonlinear nature of the coupling SQUIDs and the the same transition frequency and therefore all four spins at possibilities to bias them with constant and oscillating the corners of a physical cell have different transition frequen- magnetic fluxes. In particular the four-body stabilizer in- cies. (b) Circuit of a physical cell. The rectangles at the cor- teractions of the Toric Code are generated by driving the ners represent qubits with different transition frequencies, see SQUIDs with suitably oscillating magnetic fluxes. They inset for the circuit of each qubit. Qubits (n, m) and (n, m+1) can therefore be switched on and off at desired times or [(n + 1, m) and (n + 1, m + 1)] are connected via the induc- even be smoothly tuned in strength. In this way the tances L to the node (a; n, m) [(b; n, m)]. The nodes (a; n, m) Toric Code Hamiltonian is realized in a frame where the and (b; n, m) in turn are connected via a dc-SQUID with an qubits rotate at their transition frequencies. Importantly, external magnetic flux Φext threaded through its loop. The the transformation between the rotating and the labora- Josephson junctions in the dc-SQUID have Josephson energies tory frame is composed of single qubit unitaries only, so E and capacitances C , which include shunt capacitances. J J that the entanglement properties of the states are identical in both frames. Yet for implementing the Toric Code Hamiltonian (1), considering a rotating frame has the and purpose of an implementation of the Hamiltonian pivotal advantages that the effective transition frequen- (1) compared to error detection via stabilizer measure- cies of the individual spins become arbitrarily small and ments as pursued in [12, 17–20]. In particular, two-body that the approach becomes largely insensitive to disorder interactions are all that is needed for arbitrary stabiliser in the lattice. In contrast to earlier approaches to multi- measurements, while lattice gauge theories [21–24] and body interactions based on gate sequences [46–49], the Hamiltonians such as (1) require k-body (with k > 2) four-body interactions in our approach are directly imple- 3 mented as an analog quantum simulator and are therefore the commutation relations [ϕj, πl] = i~δj,l. The nodes time-independent, which avoids any errors routed in the are labelled by composite indices j = (n, m), where n la- Trotter sequencing. Moreover, our approach suppresses bels the diagonals extending from top left to bottom right two-body interactions to an extent that they are negligi- and m labels the qubits along each diagonal in Fig. 1a. ble compared to the four-body interactions. Each qubit is characterized by a Josephson energy 2 A minimal two-dimensional lattice covering the surface EJq;j and a charging energy ECq;j = e /(2Cq;j), where of a torus requires eight qubits connected via supercon- e is the elementary charge and Cq;j the qubit’s capaci- 2 ducting wires such that the lattice has periodic boundary tance. Including a contribution 4ELϕj that arises from conditions in both directions and is thus realizable with the coupling to its four adjacent auxiliary nodes via the available technology [17, 18, 34]. We show that the topo- inductances L, the Hamiltonian of the qubit at node j logically ordered states of the Toric Code can be prepared reads, via an adiabatic sweep from a product state for exper- 2 2 2 imentally realizable parameters and that measurements Hq;j = 4ECq;j~− πj − EJq;j cos(ϕj) + 4ELϕj , (2) of individual qubits and qubit-qubit correlations can be where E = φ2/(2L) and φ = /(2e) is the rescaled performed to demonstrate the topological order of these L 0 0 ~ quantum of flux. We consider transmon qubits [17, 18, states. Moreover the robustness of the topologically or- 29, 30, 34, 35, 53] because of their favorable coherence dered states against local perturbation that would change properties and moderate charging energies, c.f. appendix the energy of the system by a limited amount can be ver- D 1. ified experimentally. For the description of the dc-SQUIDs it is useful to consider the modes ϕ ;j = ϕa;j ± ϕb;j, where ϕa/b;j is the phase at the auxiliary± node (a/b; n, m). The modes I. THE PHYSICAL LATTICE ϕ+;j behave as harmonic oscillators described by the 2 2 2 Hamiltonians H+;j = 4EC+~− π+;j + ELϕ+;j, where For realizing the Toric Code lattice, star and plaquette π ;j are the canonically conjugate momenta to ϕ ;j and ± 2 ± interactions, indicated by orange and green diamonds in EC+ = e /Cg, with Cg the capacitance between each of Fig. 1a, are both realized via the circuit shown in Fig. 1b the nodes (a; j) or (b; j) and ground. The modes ϕ ;j in − but with different external fluxes Φext applied to the cou- turn are influenced by the dc-SQUID. Their Hamiltoni- pling SQUID. The lattice resulting from the orange and ans read green diamonds in Fig. 1a with spins on the lattice vertices rather than edges is a convenient representation in- EC 2 2 ϕext H ;j = 4 − π ;j +ELϕ ;j −2EJ cos cos(ϕ ;j), troduced in [50], see also [51, 52]. The small black squares − ~2 − − 2 − (3) in Figs. 1a and 1b indicate spins formed by the two low- where ϕ = Φ /φ is the phase associated to the ex- est energy eigenstates of superconducting qubits. For the ext ext 0 ternal flux through the SQUID loop, E the Josephson transition frequencies between these two lowest states, we J energy of each junction in the SQUID and E = e2/2C assume a periodic pattern in both lattice directions such C J with C the capacitance of one junction in the− SQUID, that no nearest neighbors and no next nearest neighbors J including a shunt capacitance parallel to the junction. of qubits have the same transition frequency and hence We choose the indices j = (n, m) for the auxiliary all qubits in each four-spin cell have different transition nodes such that node (a; n, m) couples to the qubits at frequencies, see Sec. III for details. nodes (n, m) and (n, m + 1) whereas node (b; n, m) cou- For each cell the four qubits on its corners are con- ples to the qubits at nodes (n + 1, m) and (n + 1, m + 1), nected by a dc-SQUID formed by two identical Joseph- see Fig. 1b. The interaction terms between the qubit son junctions with Josephson energy E , see Fig 1b (an J variables ϕ and the SQUID modes ϕ due to the in- asymmetry between the junctions causes a negligible per- j ;j ductances L thus read, ± turbation only). The dc-SQUID is threaded by a time dependent external flux, Φext and its ports are connected HL;j = −ELϕq+;jϕ+;j − ELϕq ;jϕ ;j (4) to the four qubits via four inductors, each with induc- − − tance L. There are thus auxiliary nodes at both ports where ϕq ;j = ϕn,m+ϕn,m+1±ϕn+1,m±ϕn+1,m+1. Here, ± of the SQUID. As we show in Sec. II, these auxiliary inductive couplings via mutual inductances are an alter- nodes (a; n, m) and (b; n, m) can be eliminated from the native option [54]. The full Hamiltonian of the lattice is description while, together with the flux bias Φext on the therefore given by SQUID, they mediate the desired four-body interactions (N,M) of the stabilizers. X H = [Hq;j + H+;j + H ,j + HL;j] (5) − j=(1,1) A. Hamiltonian of the physical lattice A detailed derivation of the Hamiltonian H, starting from a Lagrangian for the circuit, is provided in appendix A. We describe the lattice in terms of the phases ϕj and We now turn to explain how the four-body interactions their conjugate momenta πj for each node, which obey of the stabilizers emerge in our architecture. 4

II. STABILIZER OPERATORS precise derivation.

The guiding idea for our approach comes from the following approximate consideration (We present a quantitatively precise derivation in Sec. II A). In each single cell, as sketched in Fig. 1b, the auxiliary nodes (a; n, m) anharmonic spring and (b; n, m) have very small capacitances with respect to ground and therefore cannot accumulate charge. As a qubit !1 i!1t consequence the node phases ϕa;n,m and ϕb;n,m are not q1 e dynamical degrees of freedom but instantly follow the cos(!dt) phases at nodes (n, m) and (n, m + 1) or (n + 1, m) and SQUID (n + 1, m + 1) as given by Kirchoff’s current law say- ing that the sum of all currents into and out of such a node must vanish. Therefore the Josephson energy of the ! !4 !2 coupling SQUID, 2EJ cos(ϕext/2) cos(ϕa;n,m − ϕb;n,m), i!4t i!2t q4 e q2 e becomes a nonlinear function of the node phases ϕn,m, ϕn,m+1, ϕn+1,m and ϕn+1,m+1, which contains four-body !3 i!3t terms. harmonic spring q3 e The external flux applied through the SQUID is then chosen such that it triggers the desired four-body interactions forming the stabilizers. To this end it is composed of a constant part with associated phase ϕdc and an oscillating part associated with phase amplitude ϕac, FIG. 2. Macroscopic analogy providing an intuitive expla- nation of our approach to generate the stabilizer interactions. ϕext(t) = ϕdc + ϕacF (t), (6) The qubits are nonlinear oscillators so that one can think of them as masses (corresponding to the qubits’ capacitances) where the oscillation frequencies of F (t) are linear com- attached to anharmonic springs (corresponding to the qubits binations of the qubit transition frequencies such that Josephson energy). We here sketch four nonlinear oscillators the desired four-body spin interactions of the Hamilto- representing four qubits as green spheres connected by black nian (1) are enabled and all other interaction terms are (anharmonic) springs to the support frame. Note that all four suppressed. Moreover we choose ϕdc = π to suppress the qubits have different transition frequencies so that all four 1 spheres representing them here would have different masses critical current Ic = 2EJ φ0− cos(ϕdc/2) of the dc-SQUID [55] and thus unwanted interactions of excitations be- and their springs different spring constants. For vanishing tween qubits at opposite sides of it, c.f. Fig. 1b. Assum- critical current the relevant SQUID modes ϕ−;j , behave as ing furthermore that the oscillating part of the flux bias harmonic oscillators and in our macroscopic analogy we can represent them by a mass (here sketched as a red sphere, cor- is small compared to the constant part, we approximate responding to the capacitance 2CJ ) that is attached via four cos(ϕext/2) ≈ −ϕacF (t)/2. springs (sketched in orange, corresponding to the inductors L) The above consideration uses approximations and only to the four masses representing the neighboring qubits. For serves us as a guide. To obtain a quantitatively precise vanishing critical current, the only potential energy of the picture, we here take the small but finite capacitances of modes ϕ−;j comes from their inductive coupling to the neigh- the Josephson junctions in the SQUID into account and boring qubits, and hence the mass representing the SQUID derive the stabilizer interactions in a fully quantum me- mode (red sphere) has no spring connecting it to a support chanical approach. Our derivation makes use of methods frame. We now imagine that we shake the mass representing the SQUID mode with a driving term that is proportional to for generating effective four-body Hamiltonians in the 4 low energy sector of a two-body Hamiltonian [14, 56–58] the fourth power of the mass’ position variable, ϕ−;j . This corresponds to the oscillating flux bias ∝ ϕ F (t). If we shake by employing a Schrieffer-Wolff transformation [59] and ac this mass at a frequency ω that is distinct from its eigenfre- combines these with multi-tone driving in order to single d quency ω−;j but equals the sum of the oscillation frequencies out specific many-body interactions while fully suppress- of all four neighboring masses (neighboring qubits), we drive a ing all two-body interactions. † † † † † term proportional to q1q2q3q4, where qj creates an excitation Further intuition for our approach can be obtained in qubit j. If, on the other hand we shake the SQUID-mass from the macroscopic analogy explained in Fig. 2. In at a frequency ωd = ω1 +ω2 −ω3 −ω4 we drive a term propor- † † short, the plasma frequency of the coupling SQUIDs dif- tional to q1q2q3q4. The selective driving of specific four-body fers form all transition frequencies of the qubits they cou- terms, as we consider it here, of course relies on choosing the ple to as well as from all frequencies contained in the os- transition frequencies of all four qubits to be different. Our cillating flux that is threaded through their loops. They approach considers a superposition of various drive frequen- thus decouple from the qubits but mediate the desired cies such that the sum of the triggered four-body terms adds four-body interactions via the drives applied to them. up to the desired stabilizer interaction, see text. Further details of this analogy are explained in the cap- tion of Fig. 2. We now turn to present the quantitatively 5

(0) A. Elimination of the SQUID degrees of freedom Hamiltonian HqS contains two-body interactions and four-body interactions between the qubits,

For the parameters of our architecture, the oscillation N,M frequencies of the SQUID modes ϕ ;j differ from the (0) χϕacEJ X 2 ± HqS = − F (t)ϕq ;n,m transition frequencies of the qubits and the frequencies of 8 − n,m=1 the time dependent fluxes threaded through the SQUID (11) N,M loops by an amount that exceeds their mutual couplings 1 χϕacEJ X 4 + F (t)ϕq ;n,m by more than an order of magnitude. These modes will 4! 16 − thus remain in their ground states to very high precision. n,m=1 To eliminate them from the description and obtain an with χ = h0 ;n,m| cos(ϕ ;n,m)|0 ;n,mi, where |0 ;n,mi is effective Hamiltonian for the qubits only, it is useful to − − − − the ground state of the mode ϕ ;n,m. consider the Schrieffer-Wolff transformation We emphasize here that our− approach requires non- S S H → H˜ = e He− with (7) linear coupling circuits, such as the considered SQUIDs. The terms in the second line of Eq. (11), which give rise to (N,M) i X the desired four-body interactions, only emerge because S = (ϕq+;jπ+;j + ϕq ;jπ ;j), 2 − − the Hamiltonian of the coupling SQUIDs contains terms ~ j=(1,1) that are at least fourth power in ϕ ;n,m, see appendix B. − which eliminates the qubit-SQUID interactions described The oscillation frequencies contained in the external in Eq. (4). The effective Hamiltonian for the qubits is flux, i.e. in ϕext, can be chosen such that selected terms then obtained by projecting the resulting Hamiltonian in the second line of Eq. (11) are enabled as they become resonant. For suitable amplitudes of the frequency com- onto the ground states of the SQUID modes ϕ ;j. The details of this derivation are discussed in appendix± B. ponents of ϕext, these terms add up to the desired four- For the transformed Hamiltonian H˜ , see Eq. (7), we body stabilizer interactions whereas all other interactions consider terms up to fourth order in S to get the leading contained in Eq. (11) are for our choice of parameters contributions of the terms which are in resonance with strongly suppressed in a rotating wave approximation. Before discussing this point in more detail in the next the oscillating part of ϕext(t), c.f. Eq. (6) . On the other hand, a truncation of the expansion is here well section, let us here emphasize the versatility of our approach: With suitably chosen Fourier components of ϕext justified since both fields ϕq ;j and π ;j have low enough amplitudes, see Eq. (B16), so± that higher± order terms can almost any two-, three- or four-body interaction can be safely be ignored. The transformed Hamiltonian can be generated. The only limitation here is that terms with the same rotation frequency can not be distinguished, e.g. written as z z z z one can not choose between generating σj σk or σj σl for H˜ = H0q + HIq + HS + HqS, (8) j =6 k =6 l which are both non-rotating. On the other hand, straightforward modifications of the circuit allow P(N,M) where H0q = j=(1,1) Hq;j with Hq;j as in Eq. (2) de- to implement higher than four-body (e.g. five- and six- scribes the individual qubits and body) interactions.

N,M X HIq = −2EL ϕn,mϕn,m+1 (9) B. Effective Hamiltonian of one cell n,m=1 interactions between neighboring qubits that are ineffec- We now focus on the interactions between the four tive since their transition frequencies differ by an amount qubits in one cell only, see Fig. 1b. To simplify the that exceeds the relevant coupling strength by more than notation we label these four qubits clockwise with in- an order of magnitude, see appendix D 1. dices 1,2,3 and 4, i.e. (n, m) → 1, (n + 1, m) → 2, → → The remaining two terms HS and HqS in Eq. (8) de- (n + 1, m + 1) 3 and (n, m + 1) 4. We thus di- scribe the SQUIDs and the residual couplings between vide one term, say for (n, m) = (n0, m0), in the sum of the qubits and the SQUIDs, see Eqs. (B7) and (B8) in ap- Eq. (11) into the following parts, pendix B. Since the oscillation frequencies of the SQUID (0) modes are chosen such that they cannot be excited by the Hcell = HqS = H1 + H2 + H3, (12) cell applied driving fields or via their coupling to the qubits, an effective Hamiltonian, describing only the qubits, can where be obtained by projecting HqS onto the ground states of χϕ E the SQUID modes, H = ac J F (t)ϕ ϕ ϕ ϕ , 1 16 1 2 3 4 (0) HqS → HqS = h0SQUIDs|HqS|0SQUIDsi. (10) contains the four-body interactions of the stabilizers, As the SQUID modes remain in their ground states, χϕ E H = − ac J F (t)(ϕ + ϕ − ϕ − ϕ )2, their Hamiltonian HS may be discarded. The projected 2 8 1 4 2 3 6 contains all terms that are quadratic in the node phases and hence enables this particular four-body interaction. ϕj and Importantly, all qubits of a cell have different transition frequencies such that a drive with frequency ωa,b,c,d only X a b c d H3 = −χϕacEJ F (t) Cijklϕiϕjϕkϕl enables the particular interaction σ1 σ2σ3σ4 whereas all (i,j,k,l) other interactions are suppressed by their fast oscillating prefactors. x x x x contains all terms that are fourth-order in the node As the star and plaquette interactions Jsσ1 σ2 σ3 σ4 and phases ϕ except for the four-body interactions. These y y y y j Jpσ1 σ2 σ3 σ4 can be written as linear combinations of op- are terms where at least two of the indices i, j, k and l erator products σaσbσcσd, their generation requires mag- P 1 2 3 4 are the same as indicated by the notation (i,j,k,l). De- netic fluxes that are a superposition of oscillations at the tails of the derivation of the Hamiltonian (12) are given respective frequencies. To enable the specific four-body in appendix C, where we also state the exact form of the interactions of the Toric Code we thus consider, coefficients Cijkl. F = F + F The Hamiltonian (12) can be written in second quan- F (t) = s X Y (14) Fp = FX − FY tized form using ϕj = ϕj(qj + qj†), where qj† (qj) creates (annihiliates) a bosonic excitation at node j with energy p where FX = f1,1,1,1 + f1,1, 1, 1 + f1, 1,1, 1 + f1, 1, 1,1 ~ωj = 8EC;j(EJq;j + 4EL) and zero point fluctuation − − − − − − and FY = f1,1,1, 1 + f1,1, 1,1 + f1, 1,1,1 + f 1,1,1,1 with amplitude ϕ = [2E /(E + 4E )]1/4. Note that the − − − − j C;j Jq;j L fa,b,c,d = cos(ωa,b,c,dt). The driving field Fs generates a effective Josephson energy E +4E includes here con- Jq;j L star interaction As and Fp a plaquette interaction Bp. tributions of the attached inductances, see appendix B 1 Applying Fs or Fp to a cell, its Hamiltonian (13) thus for details. Due to the qubit nonlinearities we can re- reads, strict the description to the two lowest energy levels of + (1) x x x x each node by replacing qj → σj− (qj† → σj ) and get for − Hcell(t) = Jsσ1 σ2 σ3 σ4 + r.t. for Fs (15a) the four-body interactions, Fs

(1) − y y y y 4 Hcell(t) = Jpσ1 σ2 σ3 σ4 + r.t. for Fp (15b) χϕacEJ Y X Fp H = F (t) ϕ σaσbσcσd, (13) 1 16 i 1 2 3 4 i=1 a,b,c,d +, where r.t. refers to the remaining fast rotating terms, and ∈{ −} Here, the sum P runs over all 16 possible 4 a,b,c,d +, χEJ ϕac Y + ∈{ −} Js = Jp = ϕ . (16) strings of σ and σ− operators, which are listed in table 16 i II in appendix C. i=1 The explicit form of the Hamiltonians H2 in second Hence star and plaquette interactions are realized with quantized form is given in Eq. (D15). The terms con- the same coupling circuit (shown in figure 1b) and only tained in H only lead to frequency shifts for the qubits 2 differ by the relative sign of the FX and FY contribu- and negligible two-body interactions. The frequency tions, i.e. a relative phase of π in the respective Fourier shifts can be compensated for by modified frequencies components. Consequently, a driving field with Fs (Fp) for the driving fields and the residual two-body interac- is applied to the coupling circuit of every orange (green) tions can be made two orders of magnitude smaller than diamond in figure 1a (see also figure 3) and the resulting the targeted four-body interactions, see Sec. II C and ap- checkerboard pattern of driving fields generates the Toric pendix D. Moreover the terms contained in H3 only lead Code Hamiltonian of equation (1). to corrections that are at least an order of magnitude Note that the signs and magnitudes of the interactions smaller than those of H and can safely be discarded, see 2 Js and Jp can for each cell be tuned independently via appendix D 3. Here, we therefore focus on the part H 1 the respective choices for the drive amplitudes ϕac. which gives rise to the dominant terms in the form of Besides the stabilizer interactions, there are further four-body stabilizer interactions. terms in the Hamiltonian (8). As we show in appendix D, all these other interactions are strongly suppressed by their fast oscillating prefactors (the oscillation frequen- C. Triggering four-body interactions cies of the prefactors exceed the interaction strengths by more than an order of magnitude in all cases). Their In a rotating frame where each qubit rotates at its tran- effect can be estimated using time dependent perturba- + iωj t + iωj t sition frequency, σj (t) = e σj [σj−(t) = e− σj−], the tion theory. We find for our parameters that they lead to a b c d strings σ1 σ2σ3σ4 that appear in Eq. (13) rotate at the frequency shifts for the individual qubits, which can be frequencies ωa,b,c,d = aω1 + bω2 + cω3 + dω4. Hence, if we compensated for by a modification of the frequencies of thread a magnetic flux, which oscillates at the frequency the driving field ϕext. For our choice of qubit transition ωa,b,c,d, through the SQUID loop, the corresponding op- frequencies, where no nearest neighbors or next nearest a b c d erator σ1 σ2σ3σ4 acquires a pre-factor with oscillation fre- neighbors of qubits have the same transition frequencies, quency ωa,b,c,d, which renders the term non-oscillating the leading corrections to the Toric Code Hamiltonian 7 of Eq. (1), besides local frequency shifts, are two-body interactions between qubits that are further apart (i.e. neither nearest neighbors nor next neighrest neighbors). For suitable parameters as discussed in Sec. III, these interactions are about two orders of magnitude weaker than the stabilizers and thus even weaker than dissipation processes in the device, see appendix D. In this context, we also note that the leading contribution to the perturbations of the Hamiltonian (1) is proportional to the charging energy ECq;j of the employed qubits, c.f. appendix D. Hence the considered transmon qubits are a well suited choice for our aims.

D. Including dissipative processes FIG. 3. Lattice for minimal realization on a torus with 8 physical qubits including 4 star interactions (orange diamonds) and 4 plaquette interactions (green diamonds). The As we have shown, the dynamics of the architecture qubits are indicated by small colored squares, where the col- we envision can be described by the Hamiltonian (1) in a oring encodes their transition frequencies and the numbering P + rotating frame with respect to H0 = j ωjσj σj−, where indicates the periodic boundary conditions. Here, 8 different we assume that all frequency shifts discussed in the pre- transition frequencies are needed. vious section (see also appendix D) have been absorbed into a redefinition of the ωj. In an experimentally re- qubit trans. frequency qubit trans. frequency alistic scenario, the qubits will however be affected by ωj /2π ωj /2π dissipation. The full dynamics of the circuit can there- 1 13.8 GHz 5 13.1 GHz fore be described by the Markovian master equation, 2 13.0 GHz 6 13.7 GHz 3 3.5 GHz 7 4.2 GHz 1 ρ˙ = −i~− [H, ρ] + Dr[ρ] + Dd[ρ] (17) 4 4.1 GHz 8 3.4 GHz where ρ is the state of the qubit excitations, H is as in TABLE I. Possible transition frequencies for an eight-qubit κ P + + + Eq. (1), Dr[ρ] = 2 j(2σj−ρσj − σj σj−ρ − ρσj σj−) de- Toric Code in natural frequency units. The qubit numbers P z z correspond to the numbering chosen in Fig. 3. These fre- scribes relaxation at a rate κ and Dd[ρ] = κ0 j(σj ρσj − ρ) pure dephasing at a rate κ . Note that both, the relax- quencies are assumed to already include the frequency shifts 0 calculated in appendix D. ation and dephasing terms Dr[ρ] and Dd[ρ] are invariant under the transformation into the rotating frame. In the following we turn to discuss realistic experimental parameters that lead to an effective Toric Code not need to be realized exactly but only with moderate Hamiltonian, the preparation and verification of topolog- precision. ical order in experiments and protocols for probing the anyonic statistics of excitations in a realization of the Toric Code with our approach. In all these discussions, A. Parameters we fully take the relaxation and dephasing processes described by Eq. (17) into account. 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 III. TOWARDS AN EXPERIMENTAL realized in qubits with Josephson energies in the range REALISATION 4.9 GHz ≤ EJq/h ≤ 94 GHz (h is Planck’s constant) and capacitances in the range 52 fF ≤ Cq;j ≤ 77 fF. Note that An experimental implementation of our approach is the frequencies are chosen such that neighboring qubits feasible with existing technology as it is based on trans- are detuned from each other by several GHz whereas the mon qubits which are the current state of the art in many transition frequencies of next nearest neighbor qubits dif- laboratories. Moreover, periodic driving of a coupling el- fer by ∼ 100 MHz. A practical advantage of our approach ement at frequencies comparable to the qubit transition with respect to frequency crowding is that it only requires frequencies is a well established technique [41, 60]. The a small set of continuous single frequency driving fields, minimal lattice of a Toric Code with periodic boundary whereas the frequency width of the pulses in digitized conditions in both lattice directions is the eight-qubit schemes grows inversely proportional to their durations. lattice sketched in Fig. 3. An important benefit of our rotating frame approach is its inherent insensitivity to For the coupling SQUIDs, the Josephson energy of each disorder due to largely spaced qubit transition frequen- junction should be EJ /h ≈ 10 GHz and the associated cies. The parameters we discuss in the sequel do thus capacitance 3.47 fF. Since high transparency Josephson 8 junctions typically have capacitances of 1 fF per 50 GHz Fig. 3. Moreover, these eight physical qubits are arranged of EJ , each SQUID is shunted by a capacitance of a on a 4 × 2 lattice so that every qubit is located on its few fF. For these parameters the modes ϕ ;j oscillate boundary, facilitating the control access. at ω /2π = 8.45 GHz, whereas the oscillation− frequency Controlling single qubit states and two qubit corre- − of the modes ϕ+;j is much higher since Cg CJ . lation measurements would require a charge line and a Furthermore the inductances coupling the qubits to readout resonator for each qubit. Our scheme, where the SQUIDs need to be chosen large enough so that the a single cell features four qubits with different transition resulting coupling term remains weak compared to the frequencies, has the beneficial property that four readout qubit nonlinearities. This requires inductances of about resonators can be connected to a single transmission line 100 nH which can be built as superinductors in the form and the qubit states read out using frequency division of Josephson junction arrays [61, 62] or with high ki- multiplexing [73–75]. Such a joint readout is capable of netic inductance superconducting nanowires [63]. We measuring the state of all four qubits and qubit pairs even note that our concept is compatible with nonlinearities when their transition frequencies are separated by several Josephson junction arrays may have. The amplitude of GHz. An improvement over the readout through a com- the drive is ϕac = 0.1 rad corresponding to an oscillat- mon transmission line would be to employ a common ing flux bias of amplitude 0.1 × Φ0 at the SQUID. These Purcell filter [76, 77] with two-modes, one mode placed parameters give rise to a strength for star and plaque- at 3.5 GHz and the second at 12.5 GHz. Such scheme tte interactions of Js = Jp = h × 1 MHz, which exceeds would not only allow a joint readout of four qubits but currently achievable dissipation rates of transmon qubits also protect them against the Purcell decay. Frequency significantly [32, 64–67]. The strength of the stabilizer division multiplexing can also be used for qubit excita- interactions can be enhanced further by increasing the tion where a single charge line can be split or routed to Josephson energies of the SQUIDs EJ and/or the ampli- four qubits in the cell. If a common transmission line is tudes of the oscillating drives ϕac. used for the readout the same line can also be used as a For the above parameters, the total oscillation fre- feedline for qubit state preparation [18]. quency, i.e. the sum of the frequencies of the qubit operators and the frequency of the external flux, is for any remaining two-body interaction term at least 10-30 C. Scalability times larger than the corresponding coupling strength. To leading order, these terms give rise to frequency shifts Importantly, our scheme is scalable as it also allows of the qubits which can be compensated for by a modifi- to realize the Toric Code on larger lattices. As already cation of the drive frequencies. All interaction terms con- mentioned in sections I and II C, the requirement for the tained in higher order correction are negligible compared choice of the qubit transition frequencies is that no near- to the four-body interactions forming the stabilizers. We est neighbor qubits and no next nearest neighbor qubits note that the experimental verification of the topological have the same transition frequencies. The resulting pat- phase is further facilitated by its robustness against local tern of transition frequencies for a large lattice is sketched perturbations, such as residual two-body interactions or in Fig. 4. It involves 16 qubit transition frequencies and shifts of the qubit transition frequencies, c.f. [68, 69]. can be realized with circuit parameters similar to the As an alternative to our approach, one could aim at im- ones given above. These 16 qubit transition frequencies plementing the Honeycomb model [14, 70], which allows are periodically repeated in both lattice directions and to obtain the Toric Code via a 4th order perturbative ex- therefore, in principle, sufficient for building any lattice pansion in the limit where one of the spin-spin couplings size. strongly exceeds the two others [5]. This strategy would Larger and more realistic Toric Code realizations will however require twice the number of qubits as our ap- obviously suffer from increased complexity. This could be proach. For typical qubit-qubit couplings of ∼100 MHz, drastically simplified when adding on-chip RF switches it would moreover only lead to stabilizer interactions that [78] for routing microwave signals to different cells and are two orders of magnitude weaker than in our approach. using selective broadcasting technique which would also reduce the number of microwave devices such as signal generators and arbitrary waveform generators [79]. B. Control and read-out

In common with all two-dimensional lattices, the ar- IV. ADIABATIC PREPARATION OF chitecture we envision requires an increasing number of TOPOLOGICAL ORDER control lines for the qubits, SQUIDs and the read-out as one enhances the lattice size. Whereas solutions to this On the surface of a torus, i.e. for a two dimensional complexity challenge are already being developed, e.g. by lattice with periodic boundary conditions in both lattice placing the control lines on a second layer [35, 71, 72], a directions, the Hamiltonian (1) has four topologically or- minimal realization of the Hamiltonian (1) requires only dered, degenerate eigenstates |ψai (a = 1, 2, 3, 4), which eight qubits that form four stars and four plaquettes, see are joint eigenstates of the stabilizers As and Bp with 9 MHz

FIG. 5. The energy eigenvalues of H(λ) as a function of λ for the eight-qubit lattice with ∆ = 2π × 5MHz and Js = Jp = 2π × 2MHz. The initial ground state is the unique vacuum Q8 |ψ0i = j=1 |0j i. At λ ∼ 0.6 the ground state starts to become degenerate but the degenerate ground state manifold Thursday, June 25, 15 correlations are built up is always separated by a gap ∼ Js ∼ Jp from higher excited FIG. 4. Pattern of transition frequencies for an implemen- states. tation of a large lattice. The qubits are indicated by small colored squares, where the coloring encodes their transition frequencies. Here four principal transition frequencies are Hamiltonian is H(λ = 0) = P ∆σz/2 and the unique needed. These differ by a few GHz and are indicated by j j initial ground state is the vacuum with no qubit exci- the filling colors brown, blue, red and purple. To suppress Q two-body interactions between next nearest neighbor qubits, tations (all spins down), i.e. |ψ0i = j |0ji. The fi- each pair of these needs to be detuned from each other by nal Hamiltonian is the targeted Toric Code Hamiltonian ∼ 100 MHz. The resulting frequency differences are indicated given in Eq. (1). Whereas the initial detuning ensures by the colored frames around the respective squares. The that the state |ψ0i is also an eigenstate in the rotating gray diamond highlights the frequency set which is periodi- frame for Js = Jp = 0, the sweep will drive the system cally repeated in both lattice directions. adiabatically into the manifold of protected states, provided the parameters are tuned sufficiently slow. The spectrum of the minimal eight-qubit lattice, c.f. eigenvalue +1, see appendix E for a possible choice of Fig. 3, is shown in Fig. 5 as a function of the tuning pa- the |ψ i for an eight-qubit lattice. A state in this man- a rameter λ. There is one unique ground state at λ = 0 ifold of topologically ordered states can either be pre- which eventually becomes degenerate with three other pared via measurements of the stabilizer operators or via states to form the degenerate four-dimensional manifold an adiabatic sweep that starts from a product state [80]. of topologically ordered ground states at λ = 1. In fi- Measurements of stabilizers can in our architecture be nite size lattices, there is however, for any value of λ, performed via a joint dispersive readout of all qubits in a finite energy separation of the order of J or J be- a star or plaquette, see Sec. V. s p tween the ground state and all states which do not end For the adiabatic sweep, one starts in our realisation up in the topologically ordered manifold at λ = 1. We with the oscillating driving fields turned off, ϕ = 0 ac have numerically confirmed this for 8, 12 and 16 qubits. and the qubit transition frequencies blue-detuned by a This spectral gap makes the sweep robust because it is detuning ∆ with respect to their final values ω . In the j not important which state in this manifold is prepared. cryogenic environment of an experiment, this initializes All states in the manifold and any linear combination the system in a product state, where all qubits are in of them are equivalent. Therefore we use the probabil- their ground state, |ψ i = Q |0 i. Then the amplitudes 0 j j ity of ending up in the topologically ordered manifold, of the external oscillating fluxes, ϕ , are gradually in- 4 ac F = P hψ |ρ(T )|ψ i, as a criterion for a success- creased from zero to their final values ϕ , i.e. ϕ (t) = a=1 a S a ac,f ac ful preparation. Moreover, in the absence of dissipation λ(t) ϕ , where λ(t = 0) = 0, λ(T ) = 1, and T is the ac,f S S one will end up in a pure state since a unitary evolu- duration of the sweep. The qubit transition frequencies tion cannot change the purity of a state. To quantify are decreased to ω , i.e. ω (t) − ω = ∆[1 − λ(t)] > 0, j j0 j how unavoidable dissipative processes in a real experi- with ω (0) − ω = ∆ and ω (T ) − ω = 0. For this j0 j j0 S j ment would affect the preparation we also compute the sweep, the Hamiltonian of the circuit can be written in purity of the final state P = Tr(ρ2). the rotating frame as, Figure 6a shows the fidelity F for being in the topolog- X ∆ z X X ically ordered qubit subspace as a function of the sweep H(λ) = (1 − λ) σj − λJs As − λJp Bp, (18) 2 time TS and the initial detuning ∆. Here ρ(t) is the ac- j s p tual state of the system in the rotating frame and the where As and Bp are as in Eq. (1). Thus the initial |ψai (a = 1,..., 4) form the subspace of degenerate pro- 10 tected ground states. To test that the prepared state is spin correlations, but not by single-spin quantities, is not pure, we also plot the purity P as a function of TS and ∆ unique to topologically ordered states but also holds for in Fig. 6b. We choose Js = Jp = h×2 MHz, see Sec. II D, Bell states. In contrast, a property that is unique to topo- relaxation and dephasing times T1 = T2 = 50 µs, and a logically ordered states is that local perturbations can- 6 6 ramp profile λ(t) = (1 + ζ)(t/TS) /[1 + ζ(t/TS) ] with not transform one topologically ordered state |ψai into ζ = 100. For these values, a protected state can success- another one |ψbi. This crucial property can be experi- fully be prepared. mentally tested with our approach. For larger lattices one may also follow the scheme pro- In the rotating frame we consider, single-spin pertur- posed in [80] which does not rely on a finite energy sepa- bations on lattice site j can be taken to read, ration between the ground states and the higher excited α states that do not become topologically ordered in the Hpert,j(t) = g(t)σj for α = +, − or z, (19) sweep. In our case, the scheme of [80] proceeds as follows. where oscillating prefactors of σα have been included into | i j All qubits, which are initially in their states 0j , are first the time dependence of g(t). In this rotating frame, the | i x | i | i transformed into states +j with σj +j = +1 +j via excited states of the Toric Code Hamiltonian (1) are sep- a π/2-pulse. Since the resulting state Q |+ i is an eigen- j j arated from the degenerate topologically ordered states state of all star stabilizers As, these may then be switched P x by an energy 4J (assuming Js = Jp = J), irrespective of on abruptly together with a term −Bx j σj which sta- Q the lattice size. Therefore, the perturbations Hpert,j(t) bilizes the state j |+ji. Subsequently the plaquette cannot transform a topologically ordered state into a P x interactions are turned on and the term −Bx j σj is state outside the topologically ordered manifold provided turned off in a sweep that is adiabatic since the relevant |g(t)| 4J. Yet because of the topological order, no transition matrix elements of the instantaneous Hamilto- local perturbation is capable of transforming a topologi- nian always vanish [80]. cally ordered state |ψai into another one |ψbi. As a con- Once they are prepared, it is of course desirable to sta- sequence, provided their strength fulfills the condition bilize the topologically ordered states as long as possible. |g(t)| 4J, no local perturbation of the form (19) can Several approaches to this task have been put forward, affect a topologically ordered state |ψai. This robustness see [81, 82] and references therein, which are compati- can be tested experimentally in our approach by verifying ble with the rotating frame implementation we consider. that the prepared topologically ordered states are insen- Most approaches consider a shadow lattice of auxiliary sitive to any local operation, provided it is weak enough. qubits that are coupled to the primary lattice. In our However, as indicated above, the topologically ordered architecture, such auxiliary systems may not be required states are in our rotating frame implementation not ro- as the coupling SQUIDs could be employed for this task. bust against local dissipation because the coupling to an environment can change the energy in the Toric Code lattice by an amount that exceeds 4J. V. MEASURING TOPOLOGICAL This experimental verification of topological correla- CORRELATIONS tions can even be performed for the minimal lattice con- sisting of eight qubits, where it only requires measuring Our scheme allows to experimentally explore and ver- individual qubits and two-qubit correlations. An exam- ify topological order. To this end one prepares the lattice ple of possible outcomes for these measurements are given in one of the topologically ordered states |ψai and mea- in table III and discussed in appendix E. sures the state of individual physical qubits as well as a In recent years several strategies for state tomography Q x correlation such as h l v σl i for a non-contractible path in superconducting circuits have been demonstrated ex- v along one of the lattice∈ directions, which is closed in a perimentally [18, 39, 40, 83]. These either proceed by loop to implement the periodic boundary conditions. For reading out the qubit excitations via an auxiliary phase- the minimal eight qubit realization this is merely a mea- qubit or resonator. Both concepts are compatible with x x surement of a two-spin correlation, i.e. hσmσni. Non- our circuit design. In particular correlations between y Q y contractible loops of σ operations, such as l v0 σl , multiple qubits, including the stabilizers As and Bp can ∈ transform the topologically ordered states |ψai into one be measured in a joint dispersive read-out via a common another [5]. Here, these can be realized via strings of coplanar waveguide resonator [18, 30, 39, 40] or several π-pulses applied to the qubits along the loop. After ap- readout resonators connected to a single transmission line plying a string of π-pulses along one lattice direction, using frequency multiplexing [73–75]. which transforms |ψai into a different topologically correlated state |ψbi, one again measures individual spins as Q x VI. BRAIDING AND DETECTION OF ANYONS well as the correlation h l v σl i. Whereas the states of ∈ the individual spins should be the same for |ψai and |ψbi because the topologically correlated states are locally in- Our approach is also ideally suited for probing the frac- distinguishable, the results for the correlations should be tional statistics of excitations in the Toric Code. Once distinct. the lattice has been prepared in a topologically ordered The property of only being distinguishable via spin- state, for example via the sweep described in Sec. IV, the 11

(a) (b)

0.95 0.95

0.9 0.9 0.85 F P 0.8 0.85 0.75

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

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. anyonic character of the excitations can be verified in an scheme that allows to realize a large variety of many-body experiment using single qubit rotations and the measure- interactions by coupling multiple qubits via dc SQUIDs ment of one stabilizer only. We here discuss a protocol that are driven by a suitably oscillating flux bias. The that has originally been proposed for cold atoms [84] but advanced status of experimental technology for supercon- is very well suited for superconducting circuits. ducting circuits makes available an ample toolbox for ex- For the prepared topologically ordered state |ψi, the ploring the physics of topological order with our scheme. protocol considers applying a single qubit rotation σy All physical qubits and coupling SQUIDs can be indi- to one of the qubits. This operation, which in our ap- vidually controlled with high precision, topologically or- proach can be implemented via a microwave pulse in a dered states can be prepared via an adiabatic ramp of the charge line to the qubit, creates a pair of so called e- stabilizer interactions, multi-qubit correlations including particles [5] on the neighboring vertices and puts the sys- stabilizers and correlations along non-contractible loops y tem into a state |ψ, ei. Alternatively,√ half a σ -rotation can be measured and anyons can be braided and their creates the state (|ψi + |ψ, ei)/ 2. Similarly, with a fractional statistics can be verified. Moreover, supercon- σx-rotation on another qubit, one creates a pair of m- ducting circuit technology allows for realizing periodic particles. Then one can move one of these m-particles boundary conditions, a feature that not easily available around one of the e-particles via successive σx-rotations in other platforms but crucial for exploring topological on the qubits along the chosen path and fuse both m- order. We thus expect our work to open up a realistic particles again. After this√ sequence, the system is in path towards an experimental investigations of this in- the state (|ψi − |ψ, ei)/ 2 due to the fractional phase triguing area of quantum many-body physics that at the π acquired from braiding the e- and m-particles. This same time will enable access for detailed measurements. √phase flip can for instance be detected√ with another Our scheme is highly versatile and not restricted to im- y σ -operation√ that maps (|ψi − |ψ, ei)/ 2 → |ψi and plementations of the Toric Code only. Several directions (|ψi + |ψ, ei)/ 2 → |ψ, ei and a subsequent measure- for generalizations appear very intriguing at this stage. ment of the corresponding stabilizer A . We note that s (i) By tuning the time-independent flux bias of the in our approach all single qubit operations can be imple- coupling SQUIDs away from the operating point for the mented via pulses applied through charge lines and the Toric Code, one can generate additional two-body in- stabilizer measurement can be realized via the technique teractions of the form σzσz between neighboring qubits. described in Sec. V. j l These can become stronger than the stabilizer interactions. The ability to tune the transition frequencies of the qubits makes our approach a very versatile quan- VII. CONCLUSION AND OUTLOOK tum simulator for exploring quantum phase transitions between topologically ordered and other phases of two- We have introduced a scheme and architecture for a dimensional spin lattices. quantum simulator of topological order that implements (ii) In our concept, the many-body interactions me- the central model of this class of quantum many-body diated by SQUIDs are not limited to 4-body interac- systems, the Toric Code, on a two dimensional super- tions. By connecting one port of a coupling SQUID to conducting circuit lattice. Our approach is based on a kl qubits and the other to kr qubits, k-body interactions 12 for k = kl + kr can be implemented in kth order of the quantum simulations of dynamical gauge fields theories, Schrieffer-Wolff expansion (7) together with a suitable where the coupling SQUID would represent the dynami- choice of frequencies for the flux bias ϕext. Thus 3-body cal degrees of freedom of the gauge field. interactions, for example, that are an order of magnitude larger than the stabilizer interactions of equation (16) can be implemented. For the parameters considered in section III, also 6-body interactions that are stronger VIII. ACKNOWLEDGEMENTS than dissipation would be feasible and even interactions among a larger number of qubits are within reach with M. S. and M. J. H. are grateful for discussions some improvement of circuit parameters. These capa- with Martin Leib at an early stage of this project. bilities of our scheme open the door to realizations of A. P. and A. W. acknowledge support by ETH Zürich. higher dimensional Toric Codes [7, 8] and other topolog- M. J. H. was supported by the EPSRC under grant No. ical codes such as color codes [85]. A useful property EP/N009428/1. M. S. and M. J. H. were supported in here is that the implementation of the sum of a star and part by the National Science Foundation under Grant plaquette interaction in one cell would only require half No. NSF PHY11-25915 for a visit to the program “Many- the frequency components for the driving fields. Body Physics with Light” at KITP Santa Barbara where (iii) If two qubits that are connected to the same port a part of this work has been done. of a coupling SQUID are identified with one lattice site of the simulated model, our approach can straight for- wardly be generalized to the analogue quantum simulation of non-Abelian lattice gauge theories, where the Appendix A: Lagrangian and Hamiltonian of entire engineered 4-body interactions play the role of the cou- circuit pling matrix between the color degrees of freedom of the adjacent lattice sites. In this appendix we present a detailed derivation of the (iv) Considering regimes, where the degrees of free- Hamiltonian in Eq. (5) that describes the considered cir- dom of the coupling SQUID can no longer be adiabati- cuit lattice. We start with the full Lagrangian describing cally eliminated, one can modify our approach towards the lattice depicted in Fig. 1, which reads,

N,M X Cq;n,m L = φ2 ϕ˙ 2 + E cos(ϕ ) (A1) 0 2 n,m Jq;n,m n,m n,m=1 M,N X φ2 − 0 (ϕ − ϕ )2 + (ϕ − ϕ )2 + (ϕ − ϕ )2 + (ϕ − ϕ )2 2L a;n,m n,m a;n,m n,m+1 b;n,m n+1,m b;n,m n+1,m+1 n,m=1 M,N M,N X Cg Cg X ϕext + φ2 ϕ˙ 2 + ϕ˙ 2 + C (ϕ ˙ − ϕ˙ )2 + 2E cos( ) cos(ϕ − ϕ ). 0 2 a;n,m 2 b;n,m J a;n,m b;n,m J 2 a;n,m b;n,m n,m=1 n,m=1

As explained in the main text, ϕn,m is the phase at node ϕ ;n,m from now on. We now turn to derive the Hamil- − (n, m), Cq;n,m is the capacitance and EJq;n,m the Joseph- tonian associated to the Lagrangian in Eq. (A1). son energy of the qubit at that node. ϕa;n,m and ϕb;n,m As there is neither capacitive coupling between the are the phases at the auxiliary nodes next to the SQUID qubits in the lattice nor between the qubits and the cou- located between nodes (n, m), (n, m + 1), (n + 1, m), and pling SQUIDs, the kinetic energy can be decomposed into (n + 1, m + 1). The Cg are the capacitances of these the contribution of individual qubits and that of the in- auxiliary nodes with respect to ground and CJ is the dividual coupling SQUIDs, capacitance of each junction in the SQUID including a N,M shunt capacitance parallel to the junctions. EJ is the X φ2 ˙ ˙ K = 0 C ϕ˙ 2 + Φ~ T C Φ~ , (A3) Josephson energy of each junction in the SQUID. 2 q;n,m n,m s;n,m s s;n,m As the Josephson junctions of the dc-SQUIDs only cou- n,m=1 ple to the diﬀerence between the phases of the two adja- ~ T cent auxiliary nodes, we deﬁne the modes, where Φs;n,m = (ϕ+;n,m, ϕ ;n,m) and −

ϕ ;n,m = ϕa;n,m ± ϕb;n,m, (A2) Cg ± 2CJ + 2 0 Cs = Cg (A4) and describe the dc-SQUIDs in terms of ϕ+;n,m and 0 2 13 is the capacitive matrix associated with one SQUID. where we have used that Cg CJ in inverting Cs. We The Hamiltonian of the lattice is obtained in the stan- thus arrive at the Hamiltonian, dard way via a Legendre transform of the Lagrangian L in Eq. (A1) by deﬁning the qubit and SQUID momenta,

∂L ∂L πn,m = , and π ;n,m = . (A5) ∂ϕ˙ n,m ± ∂ϕ˙ ;n,m ± The kinetic energy thus reads,

N,M " 2 2 2 # X πn,m π ;n,m π+;n,m K = + − + (A6) 2 φ2C 4 φ2C φ2C n,m=1 ~ 0 q;n,m ~ 0 J ~ 0 g

N,M X ECq;n,m H 2 2 − = 4 2 πn,m + 4ELϕn,m EJq;n,m cos(ϕn,m) (A7) n,m=1 ~ N,M X EC+ 2 EC 2 2 2 ϕext + 4 π+;n,m + 4 − π ;n,m + ELϕ+;n,m + ELϕ ;n,m − 2EJ cos cos(ϕ ;n,m) 2 2 − − 2 − n,m=1 ~ ~ M,N X − EL [ϕq+;n,mϕ+;n,m + ϕq ;n,mϕ ;n,m] , − − n,m=1

where we have introduced the notation Appendix B: Eﬀective time-dependent Hamiltonian

ϕq ;n,m = ϕn,m + ϕn,m+1 ± ϕn+1,m ± ϕn+1,m+1. (A8) ± 2 In this appendix we present details of the derivation EL = φ0/(2L) is the inductive energy associated with an of the effective Hamiltonian for the qubits only, which inductor L and the charging energies of the qubits and will be dominated by the four-body interactions corre- SQUID modes are sponding to the stabilizer operators in the Toric Code Hamiltonian, see Eq. (1). e2 e2 e2 ECq;n,m = ,EC+ = and EC = , 2Cq;n,m Cg − 4CJ As the modes ϕ ;n,m of the SQUIDs oscillate at frequencies that strongly± differ from the transition frequen- where e is the elementary charge. The first line in cies of the qubits and the oscillation frequencies of the Eq. (A7) describes the qubits, the second line the applied flux biases, they remain in their ground states SQUIDs and the third line the interactions between to a high degree of precision and can be adiabatically qubits and SQUIDs via the inductances L. eliminated from the description. To this end we apply The Hamiltonian (A7) is quantized in the standard the Schrieffer-Wolf transformation introduced in Eq. (7) way by imposing the canonical commutation relations with the generator

[ϕj, πl] = i~δj,l and [ϕ ;j, π ;l] = i~δj,l. (A9) ± ± The external ﬂux applied through the SQUID loops and N,M i X hence the associated phase is composed of a constant dc- S = (ϕq+;n,mπ+;n,m + ϕq ;n,mπ ;n,m), (B1) 2 − − part and a time dependent ac-part as given in Eq. (6). ~ n,m=1 We choose ϕdc = π and ϕac ϕdc = π, and obtain to leading order in ϕac, ϕ ϕ and expand the transformed Hamiltonian up to fourth cos ext ≈ − ac F (t). (A10) 2 2 order in S,

S S 1 1 1 H˜ = e He− = H + [S, H] + [S, [S, H]] + [S, [S, [S, H]]] + [S, [S, [S, [S, H]]]] + ... (B2) 2! 3! 4! 14

Using the canonical commutation relations (A9), the describing the individual qubits, a part HIq describing commutators in the expansion (B2) read, pure qubit-qubit interactions, a part HS describing the coupling SQUIDs and a part HqS describing the residual [S, H] = interactions between qubits and SQUIDs, N,M X H˜ = H0q + HIq + HS + HqS. (B4) = ELϕq+;n,mϕ+;n,m + ELϕq ;n,mϕ ;n,m − − n,m=1 The Hamiltonian of the individual qubits, H0q, and the N,M X qubit-qubit interactions, HIq, are given in Eqs. (8) and − 4 ECq;n,mπn,m (πS ;n,m + πS+;n,m) − (9). We here repeat them for completeness, n,m=1 N,M N,M X EL X 2 2 H0q = Hq;n,m with (B5) − (ϕq+;n,m + ϕq ;n,m) 2 − n,m=1 n,m=1 E N,M Cq;n,m 2 2 − X ϕext Hq;n,m =4 2 πn,m + 2ELϕn,m EJq;n,m cos(ϕn,m) + EJ cos sin(ϕ ;n,m)ϕq ;n,m ~ 2 − − n,m=1 and [S, [S, H]] = N,M X N,M − X EL 2 2 HIq = 2EL ϕn,mϕn,m+1 (B6) = (ϕq+;n,m + ϕq ;n,m) n,m=1 2 − n,m=1 N,M The term describing the SQUIDs reads, X 2 + 2 ECq;n,m (πS ;n,m + πS+;n,m) − M,N " ˜ # n,m=1 X EC+;n,m 2 2 HS = 4 π+;n,m + ELϕ+;n,m (B7) N,M 2 n,m=1 ~ X EJ ϕext 2 + cos cos(ϕ ;n,m)ϕq ;n,m M,N " # 2 2 − − ˜ n,m=1 X EC ;n,m 2 2 + 4 − π ;n,m + ELϕ ;n,m 2 − − [S, [S, [S, H]]] = n,m=1 ~ N,M M,N X EJ ϕext 3 ϕext X = − cos sin(ϕ ;n,m)ϕq ;n,m − 2EJ cos cos(ϕ ;n,m), 4 2 − − 2 − n,m=1 n,m=1 [S, [S, [S, [S, H]]]] = where E˜C+;n,m and E˜C ;n,m are the renormalized charg- N,M − X EJ ϕext 4 ing energies of the + and − modes, E˜C ;n,m = EC ;n,m+ = − cos cos(ϕ ;n,m)ϕq ;n,m ± ± 8 2 − − (E + E + E + E )/4. The n,m=1 q;n,m q;n,m+1 q;n+1,m q;n+1,m+1 qubit-SQUID interactions read, where we have here introduced the abbreviations, N,M X ECq;n,m HqS = − 4 πn,mπS ;n,m (B8) πS ;n,m = π ;n,m + π ;n,m 1 ± π ;n 1,m ± π ;n 1,m 1. ~2 − ± ± ± − ± − ± − (B3)− n,m=1 H N,M We note that the terms [S, [S, [S, ]]] and X ϕext [S, [S, [S, [S, H]]]] do not contain any contributions + EJ cos sin(ϕ ;n,m)ϕq ;n,m 2 − − from the modes ϕ+;n,m which are just harmonic os- n,m=1 cillators. They would vanish for an entirely harmonic N,M 1 EJ X ϕext 2 coupling circuit and only the Josephson terms of the + cos cos(ϕ ;n,m)ϕq ;n,m 2! 2 2 − − coupling SQUIDs that are nonlinear in ϕ ;n,m lead to n,m=1 contributions. − N,M Truncating the expansion (B2) at fourth order is a 1 EJ X ϕext 3 − cos sin(ϕ ;n,m)ϕq ;n,m 3! 4 2 − − good approximation since the amplitudes of the terms n,m=1 ϕn,mπ ;n,m/~ are much smaller than unity, see Eq. (B16) ± N,M for details. Moreover, as we explain in the sequel, the fre- 1 EJ X ϕext 4 − cos cos(ϕ ;n,m)ϕq ;n,m, quencies contained in the ﬂux bias φ that is applied 4! 8 2 − − ext n,m=1 to the SQUIDs can be chosen such that interactions contained in the fourth order terms [S, [S, [S, [S, H]]]] become where again we have neglected terms ∝ πn,mπS+;n,m be- the dominant terms of the expansion (B2). cause |ω+ − ωn,m| |ω − ωn,m|, see Eq. (B15), which As stated in Eq. (8) of Sec. II A, the transformed makes these terms highly− oﬀ-resonant so that we only Hamiltonian H˜ can be written as a sum of a part H0q keep the dominant contributions ∝ πn,mπS ;n,m. − 15

1. Second-quantized form 2. Accuracy of truncating the transformation

The free Hamiltonian of the qubits can be written in After having introduced the amplitudes ϕj and ϕ in second-quantized form by defining the ladder operators Eqs. (B9) and (B13), we can now estimate the amplitude± of the Schrieffer-Wolff generator S, see Eqs. (7) or (B1), qi and qi† via to check whether a truncation of the expansion (B2) at fourth order provides a good approximation. For the ϕj = ϕ (qj + q†), (B9) j j scenario with at most a single excitation in each qubit i~ and vanishingly small excitation numbers in the SQUID πj = − (qj − qj†), with 2ϕj modes, which is of interest here, the generator S has a 1/4 magnitude given by 2ECq;j ϕj = . E + 4E 1 ϕj Jq;j L (B16) 4 ϕ ± Here, qj (qj†) is the annihilation (creation) operator for for each site. Typical parameters of our scheme lead to qubit j [j = (1, 1),..., (N,M)] and the effective Joseph- ϕj . 0.7 and ϕ & 1.65 so that ϕj/4ϕ ≈ 0.1 and the son energy of the qubits is EJq;j +4EL due to the contri- truncation of the± expansion in Eq. (B2)± is well justified. butions of the four adjacent inductors. In terms of these creation and annihilation operators, H0q reads, Appendix C: Effective Hamiltonian in the rotating (N,M) frame X h i H0q = ~ωjqj†qj − ~Ujqj†qj†qjqj (B10) j=(1,1) To analyze which terms of the transformed Hamilto- nian (8) respectively (B4) provide the dominant contri- where we have approximated cos(ϕj) by its expansion butions, we now move to a rotating frame, where all up to fourth order in ϕj and the transition frequencies qubits rotate at their transition frequencies ωn,m and the ωj and nonlinearities Uj are given by SQUID modes rotate at their oscillation frequencies ω . In transforming to this rotating frame, the raising and± 1q ωj =~− 8ECq;j(EJq;j + 4EL), and (B11) lowering operators transform as, iωj t iω±t ECq;j EJq;j qj → e− qj, and s ;j → e s ;j (C1) Uj = . (B12) ± ± 2~ EJq;j + 4EL for j = (1, 1),... (N,M). For the SQUID modes we introduce the annihilation (creation) operators s (s† ) according to, 1. Adiabatic elimination of the SQUID modes ± ±

ϕ ;j = ϕ (s ;j + s† ;j), (B13) For the parameters of our architecture, the SQUID fre- ± ± ± ± quencies ω differ strongly from the qubit transition fre- i~ ± π ;j = − (s ;j − s† ;j), with quencies ωj and the frequencies contained in the driving ± 2ϕ ± ± ± fields. The SQUID modes ϕ+;n,m and ϕ ;n,m therefore !1/4 remain to a very good approximation in− their ground E˜C ± ϕ = , states, |0+;n,mi and |0 ;n,mi. They can thus be adiabat- ± E L ically eliminated from− the description, where the dominant contributions come from the ϕ ;n,m modes because and get − ω+ ω . The leading− order of the adiabatic elimination is ob- (N,M) X h i tained by projecting HS and HqS onto the ground states HS = ~ω+s+;† js+;j + ~ω s† ;js ;j , (B14) QN,M − − − of the SQUIDs, |0SQUIDsi = |0 ;n,m, 0+;n,mi. j=(1,1) n,m=1 − Whereas HS only gives rise to an irrelevant constant, (0) where we get for HqS = h0SQUIDs|HqS|0SQUIDsi the expression given in Eq. (11), which we here repeat for completeness q 1 ˜ ω = 4~− EC EL (B15) N,M ± ± (0) 1 ˜ X 2 HqS = − EJ F (t) ϕq ;n,m 8 − and we have neglected the last term of Eq. (B7) since n,m=1 (C2) cos (ϕext/2) ≈ −(ϕac/2)F (t) and the frequencies con- N,M 1 1 X tained in F (t) strongly differ from ω+ and ω . Note ˜ 4 − + EJ F (t) ϕq ;n,m, 16 4! − that CJ Cg and thus E˜C+ E˜C , so that ω+ ω . n,m=1 − − 16 where, Hamiltonian H1 contains 16 terms of the form S1S2S3S4, iωj t iωj t + where Sj ∈ {e− σ−, e σ } (see C6). Ignoring the ˜ j j EJ = χ ϕac EJ with (C3) oscillating pre-factor F (t), these terms rotate at the an- 2 ! gular frequencies as listed in table II. Denoting the sum ϕ ;n,m χ = exp − − (C4) 2 term frequency term frequency + + + + + + + − σ1 σ2 σ3 σ4 ν1 σ1 σ2 σ3 σ4 ν5 − − − − + + − + In deriving Eq. (C2) we have used the ground state ex- σ1 σ2 σ3 σ4 −ν1 σ1 σ2 σ3 σ4 ν6 + + − − + − + + pectation values σ1 σ2 σ3 σ4 ν2 σ1 σ2 σ3 σ4 ν7 + − + − − + + + σ1 σ2 σ3 σ4 ν3 σ1 σ2 σ3 σ4 ν8 h | | i + − − + − − − + 0 ;n,m π ;n,m 0 ;n,m = 0, σ1 σ2 σ3 σ4 ν4 σ1 σ2 σ3 σ4 −ν5 − − − − + + − − − + − h0 ;n,m| cos(ϕ ;n,m) |0 ;n,mi = χ, (C5) σ1 σ2 σ3 σ4 −ν4 σ1 σ2 σ3 σ4 −ν6 − − − − + − + − + − − σ1 σ2 σ3 σ4 −ν3 σ1 σ2 σ3 σ4 −ν7 h0 ;n,m| sin(ϕ ;n,m) |0 ;n,mi = 0. − − + + + − − − − − − σ1 σ2 σ3 σ4 −ν2 σ1 σ2 σ3 σ4 −ν8 (0) The Hamiltonian HqS is a sum of contributions from each frequencies: cell formed by four qubits at its corners which interact ν1 = ω1 + ω2 + ω3 + ω4 via one coupling SQUID. ν2 = ω1 + ω2 − ω3 − ω4 When considering the terms of one cell only, as de- ν3 = ω1 − ω2 + ω3 − ω4 scribed in Sec. II A, and labelling its qubits by indices ν4 = ω1 − ω2 − ω3 + ω4 1,2,3 and 4, i.e. (n, m) → 1, (n + 1, m) → 2, (n + 1, m + ν5 = ω1 + ω2 + ω3 − ω4 1) → 3, (n, m + 1) → 4, one can divide the Hamiltonian ν6 = ω1 + ω2 − ω3 + ω4 ν7 = ω1 − ω2 + ω3 + ω4 H(0) for one cell into three parts as described in Eq. (12), qS ν8 = −ω1 + ω2 + ω3 + ω4

(0) TABLE II. The four-body terms contained in H1 of Eq. (C6) HqS = Hcell = H1 + H2 + H3, (C6) cell and their rotation frequencies. where of the terms in the first column of table II by V1 and 1 ˜ the sum of the terms in the second column by V2 it can H1 = EJ F (t)ϕ1ϕ2ϕ3ϕ4, x x x x 16 readily be verified that V1 + V2 = AS = σ1 σ2 σ3 σ4 and y y y y 1 2 V1 − V2 = BP = σ1 σ2 σ3 σ4 . Motivated by the specific H2 = − E˜J F (t)(ϕ1 + ϕ4 − ϕ2 − ϕ3) , 8 form of the Hamiltonian H1, see Eq. (C6), we thus define X two oscillating fields FX and FY , H3 = −E˜J F (t) Cijklϕiϕjϕkϕl (ijkl) FX (t) = f1,1,1,1 + f1,1, 1, 1 + f1, 1,1, 1 + f1, 1, 1,1 − − − − − − FY (t) = f1,1,1, 1 + f1,1, 1,1 + f1, 1,1,1 + f 1,1,1,1 Here, H1 contains the four-body interactions which will − − − − (C7) give rise to stabilizer interactions, H2 contains all two- body interactions and H3 all fourth-order interactions with fa,b,c,d = cos[(aω1 + bω2 + cω3 + dω4)t]. Each of beyond the four-body interaction, i.e. terms where at the terms in FX (FY ) brings two of the four-body pro- least two of the indices i, j, k and l are the same as we P cesses in the first (second) column of table II into res- indicate by the notation (ijkl). The coefficients Cijkl onance (makes them non-rotating). Next to resonant have the values Cijkl = ±1/32 for terms of the from terms, each field also generates rotating terms. The ro- 2 ϕj ϕkϕl with j =6 k, j =6 l and k =6 l, Cijkl = ±1/64 for tation frequency of these terms is the detuning between 2 2 terms of the from ϕj ϕk with j =6 k, Cijkl = ±1/96 for different components of FX and FY . As the least detun- 3 ∼ terms of the from ϕj ϕk with j =6 k and Cijkl = ±1/384 ing is in the GHz range ( 0.4 GHz) and the strength 4 of four-body interaction is in the MHz range (∼ 1 MHz), for terms of the from ϕj . Here, “+” signs apply whenever an even number of the indices i, j, k, l refer to the same the rotating terms can be neglected in a RWA. Hence the side of the SQUID (i.e. the same diagonal) and “−” signs external oscillating field Fs = FX + FY gives rise to the otherwise. Whereas we discuss perturbations originating star interaction As and the external field Fp = FX − FY gives rise to the plaquette interaction B , from H2 and H3 in appendix D, we now turn to discuss p H1 in more detail. ( x x x x Fs = FX + FY ⇒ H1 ∝ As = σ1 σ2 σ3 σ4 F (t) = y y y y Fp = FX − FY ⇒ H1 ∝ Bp = σ1 σ2 σ3 σ4 2. Four-body interactions (C8) and the strength of the stabilizer interaction is

Due to the nonlinearities of the qubits, we can here 4 1 Y restrict the description to the two lowest energy levels J = J = E˜ ϕ (C9) + s p 16 J i of each node by replacing qj → σj− (qj† → σj ). The i=1 17

Besides these stabilizer interactions, there are also two- X δ(ωα + ωβ) hh ˆ i ˆ i Heff,2 = 2 HTC, hα , hβ body interactions between the qubits which may lead to 2 ωαωβ α,β ~ corrections in the effective Hamiltonian (1). We now turn hh i i to estimate their effect. X δ(ωα + ωβ + ωγ ) ˆ ˆ ˆ + 2 hα, hβ , hγ 3 ωβωγ α,β,γ ~

X δ(ωα + ωβ + ωγ ) hhh i i i Appendix D: Perturbations ˆ ˆ ˆ Heff,3 = 3 HTC, hα , hβ , hγ 3 ωαωβωγ α,β,γ ~ In this section we estimate the effects of the remain- X δ(ωα + ωβ + ωγ + ωδ) hhhˆ ˆ i ˆ i ˆ i ˜ + 3 hα, hβ , hγ , hδ ing terms of the transformed Hamiltonian H, see Eq. (8) 4~ ωβωγ ωδ or (B4). These include contributions contained in the α,β,γ,δ Hamiltonians HIq, see Eq. (B6), and HqS, see Eq. (B8). and δ(ω) is the Dirac δ-function. In our approach, the transition frequencies of the qubits Since ||hˆ || ω , the first term of H and the ω and the frequency spectrum of the oscillating fluxes α ~ α eff,2 n,m first term of H are at least two orders of magnitude ν are chosen such the all undesired terms are highly off- eff,3 k smaller than H and we neglect them. We now turn to resonant and therefore strongly suppressed. They how- TC discuss the other perturbations to H one by one. ever lead to a frequency shift TC

(1) (2) (3) (4) ∆j = ∆j + ∆j + ∆j + ∆j (D1) 1. Corrections to the adiabatic elimination of the SQUIDs (1) (2) (3) (4) for each qubit j, where ∆j , ∆j , ∆j and ∆j are given (0) in Eqs. (D8), (D11),(D13) and (D18). These frequency Besides the terms contained in Hqs that are found shifts can easily be compensating for by modifying the by projecting the Hamiltonian H˜ onto the ground states frequency spectrum of the applied driving fields accord- of the SQUID modes, there are first-order corrections ingly, so that we did not include them in our preceding to the adiabatic elimination. These are processes medi- analysis. They can be calculated via the following time ated by the creation and annihilation of a virtual ex- dependent perturbation theory. citation in a SQUID mode. To estimate their effect, In the frame where all qubits rotate at their transi- we first note that terms proportional to cos(ϕ ;n,m) do ˜ − tion frequencies, the transformed Hamiltonian H can be not create single excitations in the SQUID modes since written as h0 ;n,m| cos(ϕ ;n,m) |1 ;n,mi = 0 and calculate the matrix− elements − − X iωαt H˜ = HTC + hˆαe , (D2) α h0 ;n,m| sin(ϕ ;n,m) |0 ;n,mi = 0, − − − h0 ;n,m| sin(ϕ ;n,m) |1 ;n,mi = χ, (D4) where HTC is the Toric Code Hamiltonian as introduced − − − h1 ;n,m| sin(ϕ ;n,m) |1 ;n,mi = 0. in Eq. (1) and the remaining terms have been written as − − − ˆ a sum of all their frequency components. Hence hα is the In the subspace of at most one excitation, we may thus iωαt sum of all terms that rotate as e , where ωα contains replace all contributing frequencies, i.e. the oscillation frequencies of the operators and, for terms with a prefactor F (t), χ sin(ϕ ;n,m) → χ(s ;n,m + s† ;n,m) = ϕ ;n,m, (D5) the frequency of the oscillating flux bias. Note that both, − − − ϕ − − ωα > 0 and ωα < 0 contribute to the sum in Eq. (D2) ˜ (0) since H is Hermitian. to approximate ∆HqS = HqS − HqS , c.f. Eq. (C6), as Following approaches outlined in [86–88], the dynamics N,M generated by a Hamiltonian of the form as in Eq. (D2) X ECq;n,m ≈ − ||ˆ || ∆HqS 4 2 πn,mπS ;n,m (D6) with hα ~ωα can be approximated by a time in- ~ − dependent effective Hamiltonian that is an expansion in n,m=1 powers of the interaction strength over oscillation fre- N,M E˜J X quency [87], − F (t) ϕ ;n,mϕq ;n,m 2ϕ − − − n,m=1 Heff = Heff,0 + Heff,1 + Heff,2 + Heff,3 + ..., (D3) where we have neglected the terms in Eq. (B8) that are 3 2 where the individual contributions read proportional to ϕq ;n,m as these are a factor ϕn,m/24 − 1 smaller than the terms that are linear in ϕq ;n,m. We − Heff,0 = HTC now consider corrections to HTC coming from second order processes of ∆HqS. X δ(ωα + ωβ) h i Heff,1 = hˆα, hˆβ Using the formulation in second quantization, see 2 ωβ α,β ~ Eqs. (B9) and (B13), one finds for the first term of ∆HqS 18 that the perturbation theory of Eq. (D3) is applicable if For the set of parameters as given in Sec. III, we find (2) |∆j | . 20 MHz. The next higher order non-vanishing ECq;j 1. (D7) term of the expansion (D3) is here the second term of ~(ω − ωj)ϕjϕ − − Heff,3 which is more than two orders of magnitude smaller (2) For this regime, the effective Hamiltonian Heff,1 accord- than ∆j . ing to Eq. (D3) for the first term of ∆HqS just contains a frequency shift 2. Perturbations contained in HIq 4E2 ∆(1) = − Cq;j . (D8) j 2 2 2 ˜ ~ (ω − ωj)ϕj ϕ The transformed Hamiltonian H, see Eq. (8) or (B4) − − contains residual interactions between neighboring qubits for each qubit j = (n, m). Here, the prefactor 4 ac- on each diagonal n as described in Eq. (B6). Their counts for the fact that each qubit has 4 neighboring strength is 2ELϕn,mϕn,m+1 for the interaction between SQUIDs. For the set of parameters discussed in Sec. III, qubits (n, m) and (n, m + 1) on diagonal n. Hence pro- we find for the qubits with high transition frequencies vided that (1) ECq;j/(hϕjϕ ) ∼ 500 MHz leading to ∆j ∼ 200 MHz, − 2ELϕn,mϕn,m+1 i.e. a shift of 50 MHz from each coupling to a SQUID. 1 (D12) For qubits with low transition frequencies the shifts are ~(ωn,m − ωn,m+1) ∆(1) ∼ 100 MHz, i.e. a shift of 25 MHz from each cou- j the perturbation theory of Eq. (D3) is applicable and pling to a SQUID. Besides frequency shifts, the first term leads to the frequency shift of ∆HqS can also lead to interactions between two qubits of one cell. These are ineffective provided any pair of 2 2 2 2 2 2 (3) 4ELϕn,mϕn,m 1 4ELϕn,mϕn,m+1 qubits in a cell is sufficiently detuned from each other, − ∆n,m = 2 + 2 , ~ (ωn,m − ωn,m 1) ~ (ωn,m − ωn,m+1) − E E (D13) Cq;j Cq;l 1, (D9) 2 2 for qubit (n, m). For the parameters discussed in Sec. III, ~ (ωl − ωj)(ω − ωj)ϕjϕlϕ − − we find 2ELϕn,mϕn,m+1/h ≈ 300 MHz whereas neighboring qubits on a diagonal are detuned by almost 10 GHz so which holds since any pair of qubits in one cell is at least (3) detuned by 700 MHz from each other, see table I. Note that |∆n,m| ∼ 18 MHz. In a higher order of the pertur- that the detuning between two neighboring qubits on a bation theory of Eq. (D3), each qubit (n, m) can also me- diagonal n that would couple via two SQUIDs is signif- diate an interaction between its two neighboring qubits on the diagonal n due to the couplings 2ELϕn,m 1ϕn,m icantly higher (∼ 10 GHz). The next higher order non- − vanishing term of the expansion (D3) is here the second and 2ELϕn,mϕn,m+1. To suppress these interactions of strength 10 MHz, next-nearest neighbor qubits need to term of Heff,3 which is more than two orders of magnitude (1) be sufficiently detuned from one another, smaller than ∆j . In a similar way as second-order terms can lead to 2 2 4ELϕn,m 1ϕn,mϕn,m+1 qubit-qubit interactions in one cell, fourth-order terms − 1, (D14) 2 ~ (ωn,m − ωn,m+1)(ωn,m 1 − ωn,m+1) could mediate interactions between two qubits in neigh- − boring cells. For our parameters, these interactions would have a strength of ∼ 0.5 MHz but are ineffective which requires |ωn,m 1 − ωn,m+1| & 2π × 100 MHz as − as qubits in neighboring cells are always detuned by at fulfilled by the frequencies in table I. least 100 MHz. Furthermore, there could be interactions between Via an analogous calculation, one finds for the second qubits in neighboring cells, say qubit (n, m) and qubit term of ∆HqS that the perturbation theory of Eq. (D3) (n, m + 2), since qubits (n, m) and qubit (n, m + 1) is applicable if couple via 2ELϕn,mϕn,m+1 and qubits (n, m + 1) and qubit (n, m + 2) couple in a second order process via E˜ ϕ their interaction with a common SQUID as discussed in J j 1 (D10) Sec. D 1. This third-order interactions are strongly sup- 2~[(ω ± ωj) ± νk] − pressed since their strength is ∼ 3 MHz and thus 30 times and leads to the frequency shifts (taking into account smaller than the detuning between the respective qubits. that each qubit couples to 4 SQUIDs)

˜2 2 8 x 3. Residual two-body interactions in H(0) (2) EJ ϕj X X ω + (−1) ωj qS − − ∆j = 2 2 x 2 2 (D11) ~ [ω + (−1) ωj] − νk k=1 x=0,1 − When projected onto the ground states of the SQUID for each qubit j, where the frequencies νk that are con- modes in the zeroth order of adiabatic elimination, the (0) tained in the driving field F (t) are defined in table II. Hamiltonian HqS = h0SQUIDs|HqS|0SQUIDsi contains 19 terms that are quadratic in ϕq ;n,m and thus contain two- The two commutators in the last two lines of Eq. (D17) body interactions, see Eq. (C2).− In the rotating frame, would lead to transitions out of the qubit subspace and the two-body interaction Hamiltonian of one cell as given can thus be ignored because of the nonlinearity of the by H2 in Eq. (C6) can be written as, qubits. Hence, to leading order the Hamiltonian H2 will lead 4 8 nh i to shifts of the transition frequencies of the qubits. Tak- X X i(ωj ωl+νk)t i(ωj ωl νk)t H2 = Bj,l e − + e − − qj†ql ing into account that each cell is embedded into the to- j,l=1 k=1 tal lattice and hence each qubit experiences frequency h i o i(ωj +ωl+νk)t i(ωj +ωl νk)t + e + e − q†q† + H.c. , shifts from the two-body interactions in all four cells it is j l part of, we find that the qubit transition frequencies are (D15) shifted by ˜ where the coupling coefficients read Bj,l = ∓EJ ϕjϕl/8, 2 8 E˜ X X 2ωj where the “−” sign applies for j = l and (j, l) = (1, 4) or ∆(4) = − J ϕ4 (D18) j 8 j 4ω2 − ν2 (2, 3), and the “+” sign applies otherwise. The pertur- l: j l =1 k=1 j k;j,l { | − | } bation theory of Eq. (D3) applies to H2 provided that 2 8 2 2 E˜ X X ϕj ϕ [ωj + ωl] − J l 2 2 , E˜J ϕ ϕ 32 [ωj + ωl] − ν j l l: j l =1 k=1 k;j,l 1 (D16) { | − | } 8~(ωj ± ωl ± νk) ˜2 8 2 2 − EJ X X ϕj ϕl [ωj ωl] + 2 2 , 32 [ωj − ωl] − ν for all j, l and k, c.f. Eq. (D15). Hence the spectrum l: j l =1 k=1 k;j,l of the driving fields must be chosen such that conditions { | − | } (D10) and (D16) are met. P where the sum l: j l =1 runs over all qubits l that are To estimate the corrections due to H2 as written in { | − | } nearest neighbors to qubit j and νk;j,l denotes the drive Eq. (D15), one thus needs to consider the commutation frequencies at the SQUID that connects qubits j and l. relations, For the set of parameters as given in Sec. III, we find (4) |∆j | . 30 MHz. [qjql†, qj†ql] = (1 − δj,l) ql†ql − qj†qj , (D17) [qjql, qj†ql†] = (1 + δj,l) qj†qj + ql†ql + 1 , Appendix E: Expected Measurement Outcomes [qjql, qjql†] = (1 + δj,l) qjqj, In this section we discuss a set of possible measurement [qjql†, qj†ql†] = (1 + δj,l) ql†ql†, out comes for the topologically ordered states on a eight- as these are the only ones that, together with their os- qubit lattice, c.f. Fig. 3. One ground state of the Toric cillating prefactors Fs or Fp, lead to non-rotating terms. Code Hamiltonian on this lattice can be written as

8 1 1 y y y y 1 y y y y 1 y y y y 1 y y y y Y |ψ1i = √ ( + σ1 σ4 σ5 σ3 )( + σ2 σ3 σ6 σ4 )( + σ5 σ8 σ1 σ7 )( + σ6 σ7 σ2 σ8 ) |+ji , (E1) 4 2 j=1

x y y where |+ji is the eigenstate of σj with eigenvalue 1, |ψ3i = σ4 σ8 |ψ1i x y y y y σj |+ji = |+ji. The other three ground states can be |ψ i = σ σ σ σ |ψ i . y y y y 4 4 8 5 6 1 found by applying the loop operators σ5 σ6 , σ4 σ8 and y y y y σ4 σ8 σ5 σ6 , Whereas the reduced density matrices of individual spins are all completely mixed (i.e. proportional to an identity matrix) for all these four states, they can be distinguished y y |ψ2i = σ5 σ6 |ψ1i (E2) via their two spin correlations, see table III.

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x x x x x x x x x x x x x x x x hσ1 σ2 i hσ3 σ4 i hσ5 σ6 i hσ7 σ8 i hσ3 σ7 i hσ1 σ5 i hσ4 σ8 i hσ2 σ6 i |ψ1i 0 1 0 1 0 1 0 1 |ψ2i 0 1 0 1 0 -1 0 -1 |ψ3i 0 -1 0 -1 0 1 0 1 |ψ4i 0 -1 0 -1 0 -1 0 -1

TABLE III. Two-qubit correlations along non-contractible loops for the 4 ground states of the Toric Code on an 8-qubit lattice.

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