Abstract
We propose a novel platform for quantum many body simulations of dipolar spin models using current circuit QED technology. Our basic building blocks are 3D Transmon qubits where we use the naturally occurring dipolar interactions to realize interacting spin systems. This opens the way toward the realization of a broad class of tunable spin models in both two-and onedimensional geometries. We illustrate the potential offered by these systems in the context of dimerized Majumdar-Ghosh-type phases, archetypical examples of quantum magnetism, showing how such phases are robust against disorder and decoherence, and could be observed within stateof-the-art experiments.
Abstract
PACS numbers: 03.67. Ac, 42.50.Dv, 75.10.Pq Introduction. -In the present work we propose and analyze a novel setup for an analog quantum simulator of quantum magnetism using superconducting qubits. The scheme builds on the remarkable recent developments in Circuit QED [1] [2][3][4][5][6] in the context of quantum simulation [7][8][9] [10] [11] , and especially the 3D Transmon qubit [12, 13] . The scheme promises a faithful implementation of many-body spin-1/2 Hamiltonians involving tens of qubits using state-of-the-art experimental techniques. The central idea behind the present work is to exploit the naturally occurring dipolar interactions between qubits to engineer the desired spin-spin interactions. In combination with the flexibility offered by solid-state setups for realizing arbitrary geometry arrangements, this allows us to design general dipolar spin models in ladder and 2D geometries. As we will show, our scheme competes favorably with present and envisaged quantum simulation setups for magnetism with cold atoms and trapped ions [14] [15] [16] , and enables us to address some of the key challenges of quantum simulation including equilibrium and non-equilibrium (quench) dynamics [17] . Moreover, we note that exploiting dipolar interactions to design dipolar spin models is conceptually different, and complementary to the remarkable recent experiments with superconducting circuits toward realizing the superfluid-Mott insulator transition, based on wiring up increasingly complex circuits of superconducting stripline cavities [7].
Abstract
In our analysis we address two of the key aspects of the design of our proposed simulator for quantum magnetism. First, we present a feasibility study of state-ofthe-art experimental setups: this includes a discussion of the general mechanism to generate dipolar interactions between 3D Transmons, combined with ab initio simulations of the coupling strength in our spin model for various geometries. Second, we illustrate how state-ofthe-art setups, composed of up to a dozen qubits and characterized by typical disorder and decoherence rates, are already able to demonstrate paradigmatic signatures of quantum magnetism. In particular, we show how a dimerized phase [18] , a valence-bond-solid reminiscent of the Majumdar-Ghosh state widely discussed in the context of quantum spin chains [19] , can be realized and probed with current technologies. The system dynamics we are interested in, is described by a generalized XY Hamiltonian of the form
Abstract
arXiv:1501.03098v1 [quant-ph] 13 Jan 2015 2 where S α j are spin operators at the lattice site j, r ij is the distance vector between i and j, the last term describes a disordered transverse field, and the inter-qubit couplings
Abstract
are in frequency units. The key element of our implementation is the realization of different patterns of quantum frustration [18] by tuning the form of the interaction couplings (Fig. 2) . As discussed below, the latter display a rich dependence as a function of the dipole angles θ j (see Fig. 2b ): this dependence is essential to tune the coupling between different spins from positive to negative, or to (approximately) set it to 0. Even more crucially, the 1/|r ij | 3 dependence (see Fig. 2a ) can be exploited to further modify the magnitudes of the couplings. In the theory of 1D and ladder systems [20, 21] , where the dipolar interaction is effectively local, this allows independent tunability of nearest-neighbor (NN) and next-nearest-neighbor (NNN) exchange, a fundamental ingredient to realize bond-order solids [19, 22] . We focus specifically on this case below, showing how, within our proposal, such states of matter are robust against both disorder and qubit decoherence. The setup can be straightforwardly extended to 2D geometries, where qualitatively new features emerge due to the non-locality of the dipolar couplings: in particular, the dynamics of Eq. (1) can by-pass the Mermin-WagnerHohenberg theorem [23, 24] , and thus support phases of matter with true long-range order even at finite temperature [25] . We emphasize that all of these phenomena are directly accessible within our proposal thanks to the naturally occurring dipolar interactions, while such interactions would be challenging to be implemented via wiring.
Abstract
Model system. -We now describe how the manybody dynamics of Eq. (1) can be realized in our setup, using state-of-the-art circuit QED technology, with the design flexibility and long coherence times 1/κ (κ ≤ 2π × 100 kHz) of Transmon qubits [12, 13] . This is schematically illustrated in Fig. 1 . Several Transmon qubits (in red) fabricated on a piece of sapphire (light blue area) are mounted inside a waveguide cavity (grey box). The Transmon qubits can be fabricated in an essentially arbitrary lattice configuration with locally controllable orientation. The waveguide cavity around the qubits is used to readout the state of selected qubits and apply a drive to a subset of qubits, providing means for both adiabatic state preparation and probing (see next section). This selectivity can be again achieved by partially rotating the qubits inside the cavity by a few degrees, such that their dipole moment has a finite overlap with the electric field in the cavity. To add more flexibility and mitigate disorder, each qubit can be fabricated on an individual piece of sapphire, and can be probed at room temperature [26] . For simplicity these individual pieces are not shown in Fig. 1 . From fabrication one can expect a disorder of about 3-4% in the Josephson • for three different antenna length. The dashed line indicates Ji,j = 0. The inset displays a typical result from HFSS simulations [27] , with the dipoles surrounded by the computed electric field (intensity decreasing from red to blue). Panel (b): angular dependence illustrated for the case θi = θj = θ at distance r = 1.5 mm. The solid line plots Eq. 2 with J0, rm extracted from best fits in (a). energy which one should be able to reduced to less than 1% (around 2π × 30 MHz spreading in h j ) by preselecting the qubits according to their measured normal state resistance.
Abstract
Designing the interaction. -The unique feature of our approach is the design flexibility in the interqubit interaction which results from the dipole-antenna structure of the Transmon. As one can expect, two dipole antennas in the near field interact like two magnetic spins [28] . By designing the shape and size of the antenna we can realize large interaction strengths (J ij ≈ 2π × 100 MHz) at inter-qubit distances of about 1 mm. The interaction between the qubits ultimately comes from an effective capacitance between the antennas, similar to [8, 9], with an angle and distance dependence akin to magnetic dipoles.
Abstract
To confirm and quantitatively assess the inter-qubit dipolar interactions, we have performed a finite element study to determine the coupling strength and dependence on distance and angle. These simulations are carried out by ab initio numerically solving the 3D Maxwell equations on finite grids (HFSS software [27]), and as such provide an extremely accurate quantitative benchmark for our modeled inter-qubit couplings. The quantitative insights gathered from the simulations are of basic importance to underpin the effects of the specific configuration that would be accessible in realistic setups, where the influence of the surrounding waveguide cavity and finite size effects of the Transmon antenna have to be systematically understood. A set of results can be seen in Fig. 2 (see also [29] for additional results). Assuming a dipoledipole interaction with an additional component due to the dispersive coupling of both qubits to the cavity with strength g (determined from independent simulations), we expect the following spatial dependence:
Abstract
The first term takes into account the qubit-qubit interaction mediated via the fundamental cavity mode with φ 1 , φ 2 the orientation of the qubits relative to the orientation of the electric field in the cavity. Higher order cavity modes contribute with a similar term with a coupling g depending on the mode structure; as these are typically very far detuned, we neglect them in the following. The second term is the direct dipole-dipole interaction between the qubits with r = |r 12 | being the distance between their centers. Both, the qubit-cavity interaction strength (which depends on the location of the qubit in the cavity) and the dipole-dipole interaction J 0 get modified by the length of the antenna, giving rise to the term d m in the Eq. 2, with d m the normalized antenna length. The remaining variables in this equation are the qubit cavity detuning ∆ ≈ 2π × 1.5 GHz and the angles θ 1 , θ 2 . These are the angles formed by the two dipoles with respect to a line connecting their centers, as can be seen in Fig. 2(b) . r m corrects for finite size effects of the dipole antennas.
Abstract
One can see in Fig. 2(a) that the qubit-qubit interaction strength in our numerical simulations very closely follows the analytical expression Eq. 2. The only fitting parameters in this expression are J 0 , which is the same for all three datasets, and the r m , which is adjusted for each length. The dependence on the distance for different antenna sizes can be seen in Fig. 2(a) . The interaction strength between the qubits for a parallel orientation is negative for short distances and falls of as 1/r 3 . For larger distances it becomes positive due to the additional coupling mediated off-resonantly via the cavity. This leads to the effect that qubits at a given distance (in the case of our cavity about 3.5 mm) do not interact with each other as these terms exactly cancel out. [44] By rotating one qubit around the other, as shown in Fig. 2(b) , we can see that the interaction strength goes from a negative value for a parallel orientation to a twice as large positive value for collinear qubits, as expected for a dipole-dipole interaction. The analytical curves agree well with the numerical simulations and demonstrate that the spatial dependence of the interaction behaves like a magnetic dipole-dipole interaction. Furthermore we note that the cavity mediated term can be fully suppressed by rotating the qubits perpendicular to the electric field of the cavity mode, as shown schematically in Fig. 1 . The finite-element simulations are in very good agreement with a simple analytical circuit model (see [29] ), which already captures the important interaction features.
Abstract
Frustrated spin ladders and dimer phases. -The setup discussed above paves the way to the realization of a broad class of lattice spin models [18] . The key challenge on the theory side is to identify many-body phenomena which can be realized within this toolbox and, crucially, are stable against the most common imperfection sources such as disorder and decoherence. Here, we show how a archetypical state of matter, a valence bond solid akin to the Majumdar-Ghosh (MG) state in frustrated spin chains [18, 19, 22, 30] , can be realized and probed with currently available technology. Specifically, we are interested in the model in Eq. (1) on a triangular ladder geometry (see Fig. 1b ). To simplify our discussion, since dipolar interactions are effectively local in 1D, we consider only NN and NNN couplings, that is:
Abstract
This model [22, 31] is known to exhibit a phase transition from a superfluid phase (J 1 J 2 ) to a dimer phase (DP) around J 2 /J 1 0.33, which is stable up to relatively large couplings [30] . This DP can be easily understood at the MG point J 2 /J 1 = 0.5, where the ground state factorizes onto a product of local triplets [22] :
Abstract
as illustrated in Fig. 1(b) . Beyond being a textbook example of a valence bond solid in 1D, the DP has triggered further interest in connection to the physics of the Peierls instability in Hubbard models [32] . In general, the DP is indicated by a bond-order parameter (BOP):
Abstract
which indicates the spontaneous formation of triplets along the odd-even bonds on an open chain. Deep into the DP, the phase can be identified in an even simpler way by monitoring the bond correlation of a single pair of spins in the middle of the chain, Fig. 3(a) , we compare the behavior of the dimer correlation B z (red line) against the BOP (dashed line) as a function of the coupling ratio J 2 /J 1 for a chain of L = 192 sites, obtained using densitymatrix-renormalization group simulations [33] [46]. Both quantities are maximized in the DP, as expected; in addition, close to the MG point, the difference between the BOP and B z is only at the few percent level even for very small system sizes of L = 8 spins, providing an ideal observable to detect the DP even in relatively small samples. Measurement techniques to access these observables exploiting current technology are discussed in Ref. [29] .
Abstract
A crucial point is, to which extent the DP and the corresponding signatures on small size systems are stable against disorder. Qualitatively, we expect that, close to the MG point, the phase is stable against weak disorder for h j < J 2 , as breaking a single triplet will cost energy J 2 . We have performed an exact diagonalization study of the resilience against disorder, which is included as a Gaussian distribution of h j around 0 with spread δh (J 1 setting the energy unit). In Fig. 3b , we show the behavior of the bond correlation as a function of J 2 /J 1 for a system of L = 10 sites and different disorder strength and average values. Up to a spread of around 0.45, the signatures of the dimer phase are very stable. Given that typical disorder strengths would be δh 0.2/0.3, this makes the observation of the DP realistic with current fabrication technology.
Abstract
Adiabatic state preparation and effect of dissipation. -We finally discuss how the DP can be realized in realistic settings via adiabatic state preparation. The system is initially in the vacuum state, where no excitations are present, that is, |Ψ in = j ⊗ |↓ j [47]. An external drive is used to inject excitations, and can be modeled by a time-dependent Hamiltonian
Abstract
The dynamics is well described by a master equation in Lindblad form [34] :
Abstract
where H T (t) = H + H d (t), the jump operators C j = √ κ j S + j and C j+L = √ γ j S z j describe single-spin decay and dephasing, respectively, with constants κ, γ 100 kHz. In our ramps, we employed a staggered driving field, Ω j = ((1 + (−1) j ) + δ j,L/2 )Ω, which is provided by the local spin addressability using angle tuning (see Fig. 1(a) ), and performs significantly better than other choices. The specific shape of the sweep parameters we choose is illustrated in the inset of Fig. 4a : first, the detunings ∆ j are slowly switched off while ramping up Ω. Subsequently, the driving is switched off in order to reach the desired final state [48].
Abstract
We quantitatively address the full system dynamics including disorder and decoherence by numerically solv- ing Eq. 6 for systems up to L = 10 spins. In Fig. 4 (a), we plot the average magnetization per site (blue dashed) and the bond correlation (both in the x-and z-basis, black and red line, respectively) as a function of time, with a system including decoherence rates with κ = γ = 2π × 10, 100, 500 kHz, and a disorder strength δh = 2π × 25 MHz. The effect of decoherence along the ramp is minimal: the final order parameters deviate only a few percent from the expected value even for κ = 2π × 100 kHz, making the phase detectable for typical experimental conditions. For values of the ratio J 2 /J 1 outside of the DP, dimerization does not occur (Fig. 4  (b) ). Quantum optimal control techniques [35, 36] could eventually be employed to further improve the ground state preparation.
Abstract
Conclusions and outlook. -We have shown how to use the naturally occurring dipolar interactions in 3D superconducting circuits to realize a platform for analogue quantum simulation of XY spin models. The possibility of realizing arbitrary lattice geometries with locallytunable dipole moments, in combination with their large interaction strength, opens the door to the investigation of a series of phenomena in quantum magnetism in both 1D [37, 38] 

Introduction. -In the present work we propose and analyze a novel setup for an analog quantum simulator of quantum magnetism using superconducting qubits. The scheme builds on the remarkable recent developments in Circuit QED [1] [2] [3] [4] [5] [6] in the context of quantum simulation [7] [8] [9] [10] [11] , and especially the 3D Transmon qubit [12, 13] . The scheme promises a faithful implementation of many-body spin-1/2 Hamiltonians involving tens of qubits using state-of-the-art experimental techniques. The central idea behind the present work is to exploit the naturally occurring dipolar interactions between qubits to engineer the desired spin-spin interactions. In combination with the flexibility offered by solid-state setups for realizing arbitrary geometry arrangements, this allows us to design general dipolar spin models in ladder and 2D geometries. As we will show, our scheme competes favorably with present and envisaged quantum simulation setups for magnetism with cold atoms and trapped ions [14] [15] [16] , and enables us to address some of the key challenges of quantum simulation including equilibrium and non-equilibrium (quench) dynamics [17] . Moreover, we note that exploiting dipolar interactions to design dipolar spin models is conceptually different, and complementary to the remarkable recent experiments with superconducting circuits toward realizing the superfluid-Mott insulator transition, based on wiring up increasingly complex circuits of superconducting stripline cavities [7] .

In our analysis we address two of the key aspects of the design of our proposed simulator for quantum magnetism. First, we present a feasibility study of state-ofthe-art experimental setups: this includes a discussion of the general mechanism to generate dipolar interactions between 3D Transmons, combined with ab initio simulations of the coupling strength in our spin model for various geometries. Second, we illustrate how state-ofthe-art setups, composed of up to a dozen qubits and characterized by typical disorder and decoherence rates, where S α j are spin operators at the lattice site j, r ij is the distance vector between i and j, the last term describes a disordered transverse field, and the inter-qubit couplings

are in frequency units. The key element of our implementation is the realization of different patterns of quantum frustration [18] by tuning the form of the interaction couplings (Fig. 2) . As discussed below, the latter display a rich dependence as a function of the dipole angles θ j (see Fig. 2b ): this dependence is essential to tune the coupling between different spins from positive to negative, or to (approximately) set it to 0. Even more crucially, the 1/|r ij | 3 dependence (see Fig. 2a ) can be exploited to further modify the magnitudes of the couplings. In the theory of 1D and ladder systems [20, 21] , where the dipolar interaction is effectively local, this allows independent tunability of nearest-neighbor (NN) and next-nearest-neighbor (NNN) exchange, a fundamental ingredient to realize bond-order solids [19, 22] . We focus specifically on this case below, showing how, within our proposal, such states of matter are robust against both disorder and qubit decoherence. The setup can be straightforwardly extended to 2D geometries, where qualitatively new features emerge due to the non-locality of the dipolar couplings: in particular, the dynamics of Eq. (1) can by-pass the Mermin-WagnerHohenberg theorem [23, 24] , and thus support phases of matter with true long-range order even at finite temperature [25] . We emphasize that all of these phenomena are directly accessible within our proposal thanks to the naturally occurring dipolar interactions, while such interactions would be challenging to be implemented via wiring.

Model system. -We now describe how the manybody dynamics of Eq. (1) can be realized in our setup, using state-of-the-art circuit QED technology, with the design flexibility and long coherence times 1/κ (κ ≤ 2π × 100 kHz) of Transmon qubits [12, 13] . This is schematically illustrated in Fig. 1 . Several Transmon qubits (in red) fabricated on a piece of sapphire (light blue area) are mounted inside a waveguide cavity (grey box). The Transmon qubits can be fabricated in an essentially arbitrary lattice configuration with locally controllable orientation. The waveguide cavity around the qubits is used to readout the state of selected qubits and apply a drive to a subset of qubits, providing means for both adiabatic state preparation and probing (see next section). This selectivity can be again achieved by partially rotating the qubits inside the cavity by a few degrees, such that their dipole moment has a finite overlap with the electric field in the cavity. To add more flexibility and mitigate disorder, each qubit can be fabricated on an individual piece of sapphire, and can be probed at room temperature [26] . For simplicity these individual pieces are not shown in Fig. 1 . From fabrication one can expect a disorder of about 3-4% in the Josephson • for three different antenna length. The dashed line indicates Ji,j = 0. The inset displays a typical result from HFSS simulations [27] , with the dipoles surrounded by the computed electric field (intensity decreasing from red to blue). Panel (b): angular dependence illustrated for the case θi = θj = θ at distance r = 1.5 mm. The solid line plots Eq. 2 with J0, rm extracted from best fits in (a). energy which one should be able to reduced to less than 1% (around 2π × 30 MHz spreading in h j ) by preselecting the qubits according to their measured normal state resistance.

Designing the interaction. -The unique feature of our approach is the design flexibility in the interqubit interaction which results from the dipole-antenna structure of the Transmon. As one can expect, two dipole antennas in the near field interact like two magnetic spins [28] . By designing the shape and size of the antenna we can realize large interaction strengths (J ij ≈ 2π × 100 MHz) at inter-qubit distances of about 1 mm. The interaction between the qubits ultimately comes from an effective capacitance between the antennas, similar to [8, 9] , with an angle and distance dependence akin to magnetic dipoles.

To confirm and quantitatively assess the inter-qubit dipolar interactions, we have performed a finite element study to determine the coupling strength and dependence on distance and angle. These simulations are carried out by ab initio numerically solving the 3D Maxwell equations on finite grids (HFSS software [27] ), and as such provide an extremely accurate quantitative benchmark for our modeled inter-qubit couplings. The quantitative insights gathered from the simulations are of basic importance to underpin the effects of the specific configuration that would be accessible in realistic setups, where the influence of the surrounding waveguide cavity and finite size effects of the Transmon antenna have to be systematically understood. A set of results can be seen in Fig. 2 (see also [29] for additional results). Assuming a dipoledipole interaction with an additional component due to the dispersive coupling of both qubits to the cavity with strength g (determined from independent simulations), we expect the following spatial dependence:

The first term takes into account the qubit-qubit interaction mediated via the fundamental cavity mode with φ 1 , φ 2 the orientation of the qubits relative to the orientation of the electric field in the cavity. Higher order cavity modes contribute with a similar term with a coupling g depending on the mode structure; as these are typically very far detuned, we neglect them in the following. The second term is the direct dipole-dipole interaction between the qubits with r = |r 12 | being the distance between their centers. Both, the qubit-cavity interaction strength (which depends on the location of the qubit in the cavity) and the dipole-dipole interaction J 0 get modified by the length of the antenna, giving rise to the term d m in the Eq. 2, with d m the normalized antenna length. The remaining variables in this equation are the qubit cavity detuning ∆ ≈ 2π × 1.5 GHz and the angles θ 1 , θ 2 . These are the angles formed by the two dipoles with respect to a line connecting their centers, as can be seen in Fig. 2 (b). r m corrects for finite size effects of the dipole antennas.

One can see in Fig. 2 (a) that the qubit-qubit interaction strength in our numerical simulations very closely follows the analytical expression Eq. 2. The only fitting parameters in this expression are J 0 , which is the same for all three datasets, and the r m , which is adjusted for each length. The dependence on the distance for different antenna sizes can be seen in Fig. 2(a) . The interaction strength between the qubits for a parallel orientation is negative for short distances and falls of as 1/r 3 . For larger distances it becomes positive due to the additional coupling mediated off-resonantly via the cavity. This leads to the effect that qubits at a given distance (in the case of our cavity about 3.5 mm) do not interact with each other as these terms exactly cancel out. [44] By rotating one qubit around the other, as shown in Fig. 2 (b), we can see that the interaction strength goes from a negative value for a parallel orientation to a twice as large positive value for collinear qubits, as expected for a dipole-dipole interaction. The analytical curves agree well with the numerical simulations and demonstrate that the spatial dependence of the interaction behaves like a magnetic dipole-dipole interaction. Furthermore we note that the cavity mediated term can be fully suppressed by rotating the qubits perpendicular to the electric field of the cavity mode, as shown schematically in Fig. 1 . The finite-element simulations are in very good agreement with a simple analytical circuit model (see [29] ), which already captures the important interaction features.

Frustrated spin ladders and dimer phases. -The setup discussed above paves the way to the realization of a broad class of lattice spin models [18] . The key challenge on the theory side is to identify many-body phenomena which can be realized within this toolbox and, crucially, are stable against the most common imperfection sources such as disorder and decoherence. Here, we show how a archetypical state of matter, a valence bond solid akin to the Majumdar-Ghosh (MG) state in frustrated spin chains [18, 19, 22, 30] , can be realized and probed with currently available technology. Specifically, we are interested in the model in Eq. (1) on a triangular ladder geometry (see Fig. 1b ). To simplify our discussion, since dipolar interactions are effectively local in 1D, we consider only NN and NNN couplings, that is:

This model [22, 31] is known to exhibit a phase transition from a superfluid phase (J 1 J 2 ) to a dimer phase (DP) around J 2 /J 1 0.33, which is stable up to relatively large couplings [30] . This DP can be easily understood at the MG point J 2 /J 1 = 0.5, where the ground state factorizes onto a product of local triplets [22] :

as illustrated in Fig. 1(b) . Beyond being a textbook example of a valence bond solid in 1D, the DP has triggered further interest in connection to the physics of the Peierls instability in Hubbard models [32] . In general, the DP is indicated by a bond-order parameter (BOP):

which indicates the spontaneous formation of triplets along the odd-even bonds on an open chain. Deep into the DP, the phase can be identified in an even simpler way by monitoring the bond correlation of a single pair of spins in the middle of the chain, Fig. 3(a) , we compare the behavior of the dimer correlation B z (red line) against the BOP (dashed line) as a function of the coupling ratio J 2 /J 1 for a chain of L = 192 sites, obtained using densitymatrix-renormalization group simulations [33] [46]. Both quantities are maximized in the DP, as expected; in addition, close to the MG point, the difference between the BOP and B z is only at the few percent level even for very small system sizes of L = 8 spins, providing an ideal observable to detect the DP even in relatively small samples. Measurement techniques to access these observables exploiting current technology are discussed in Ref. [29] .

A crucial point is, to which extent the DP and the corresponding signatures on small size systems are stable against disorder. Qualitatively, we expect that, close to the MG point, the phase is stable against weak disorder for h j < J 2 , as breaking a single triplet will cost energy J 2 . We have performed an exact diagonalization study of the resilience against disorder, which is included as a Gaussian distribution of h j around 0 with spread δh (J 1 setting the energy unit). In Fig. 3b , we show the behavior of the bond correlation as a function of J 2 /J 1 for a system of L = 10 sites and different disorder strength and average values. Up to a spread of around 0.45, the signatures of the dimer phase are very stable. Given that typical disorder strengths would be δh 0.2/0.3, this makes the observation of the DP realistic with current fabrication technology.

Adiabatic state preparation and effect of dissipation. -We finally discuss how the DP can be realized in realistic settings via adiabatic state preparation. The system is initially in the vacuum state, where no excitations are present, that is, |Ψ in = j ⊗ |↓ j [47]. An external drive is used to inject excitations, and can be modeled by a time-dependent Hamiltonian

The dynamics is well described by a master equation in Lindblad form [34] :

where H T (t) = H + H d (t), the jump operators C j = √ κ j S + j and C j+L = √ γ j S z j describe single-spin decay and dephasing, respectively, with constants κ, γ 100 kHz. In our ramps, we employed a staggered driving field, Ω j = ((1 + (−1) j ) + δ j,L/2 )Ω, which is provided by the local spin addressability using angle tuning (see Fig. 1(a) ), and performs significantly better than other choices. The specific shape of the sweep parameters we choose is illustrated in the inset of Fig. 4a : first, the detunings ∆ j are slowly switched off while ramping up Ω. Subsequently, the driving is switched off in order to reach the desired final state [48] .

We quantitatively address the full system dynamics including disorder and decoherence by numerically solv- ing Eq. 6 for systems up to L = 10 spins. In Fig. 4  (a) , we plot the average magnetization per site (blue dashed) and the bond correlation (both in the x-and z-basis, black and red line, respectively) as a function of time, with a system including decoherence rates with κ = γ = 2π × 10, 100, 500 kHz, and a disorder strength δh = 2π × 25 MHz. The effect of decoherence along the ramp is minimal: the final order parameters deviate only a few percent from the expected value even for κ = 2π × 100 kHz, making the phase detectable for typical experimental conditions. For values of the ratio J 2 /J 1 outside of the DP, dimerization does not occur (Fig. 4  (b) ). Quantum optimal control techniques [35, 36] could eventually be employed to further improve the ground state preparation.

Conclusions and outlook. -We have shown how to use the naturally occurring dipolar interactions in 3D superconducting circuits to realize a platform for analogue quantum simulation of XY spin models. The possibility of realizing arbitrary lattice geometries with locallytunable dipole moments, in combination with their large interaction strength, opens the door to the investigation of a series of phenomena in quantum magnetism in both 1D [37, 38] and 2D [25] , complementing the remarkable developments in cold atom and trapped ion systems [14] [15] [16] . Our ideas are not limited to Transmon qubits, but could be implemented with, e.g., Xmon qubits [6] or Fluxonium qubits coupled to an antenna [39] . It would be interesting to explore these developments in view of realizing Hamiltonian dynamics for surface code architectures [40] or as a building block for coupled cavity array experiments [41, 42] . 
Supplemental Material to 'Dipolar Spin Models with Arrays of Superconducting Qubits'
In the first part and second part of this appendix the details of numerical finite element simulations and the lumped element circuit model are explained. In the third part the Hamiltonian for two capacitively coupled Transmon qubits inside a cavity is derived. More details on the qubit state measurement and flux tuning are given in part four. In part five, we present an extended discussion of the dimer order parameters utilized in the main text to characterize the dimer phase.
HFSS SIMULATIONS
In this section, finite element electromagnetic simulations that were performed using the ANSYS HFSS [1] software are explained in more detail. Qubits are modeled as linear LC resonators (see also [2] ). The capacitance of the resonator originates from the capacitance between two rectangular films that are separated by distance d. These pads couple to electromagnetic waves like a dipole antenna. The Josephson junction of the qubit is modeled by a lumped element inductance connecting the two antenna pads. Fig. 5 shows the electric field distribution on a sapphire substrate that holds the qubit structure together with structure variable names used in this note.
HFSS SIMULATIONS
It should be noted that finite-element calculations of a structure with objects that are orders of magnitude different in dimensions requires plenty of computational resources. In this work, extremely fine meshing is needed in areas where accurate understanding of the physics of small objects (like qubits) is needed. Fig. 6 shows how different mesh sizes are used for different structure size to accurately calculate the eigenmodes of the electromag- netic fields of two qubits in a cavity.
HFSS SIMULATIONS
In this study, the coupling strength is defined as the minimum difference in the mode frequencies of two qubits when the inductance of one qubit is swept in order to tune its mode frequency in and out of resonances with the other(see Fig. 14) .
Single qubit inside cavity
In order to better understand the effect of the antenna geometry on the qubit properties, a few cases have been studied. First, the width of the antenna pads w was changed and the corresponding capacitance was extracted as shown in Fig. 7 . It can be concluded that the capacitance increases linearly as a function of the pad width.
Single qubit inside cavity
Then, the coupling strength of the qubit to the cavity g was calculated as a function of position of the qubit with respect to the side wall of the cavity. The coupling strength has a cosine form as expected from a standing wave in the cavity. The results are shown in Fig. 8 . The coupling strength g was also studied as a function of the pad height h as shown in Fig. 9 where the qubit was located at the center of the cavity. It can be seen from the figure that the coupling strength increases linearly as a function of the antenna height. This is in agreement with dipole antenna physics.
Single qubit inside cavity
Two qubits in a cavity Fig. 10 shows how the coupling strength varies when one of the qubits is kept at a constant distance (centerto-center) of 1.5 mm from the other while the angle of the vector that connects the centres of two qubits is changed. In our simulations, the coupling strength is defined to be a positive number. However, in this particular case, the sign of the coupling strength has been changed to negative for values below ≈ 40 degrees. This is to highlight the fact that the sign of the coupling is different in the two regions. This sign change can be seen from HFSS simulations which show parallel/anti-parallel field vectors for the lowest mode, indicating a positive/negative coupling. As explained in section , the results are in strong agreement with the proposed circuit model.
Single qubit inside cavity
In another study, the coupling strength was evaluated when moving one qubit with respect to the other along a straight line. In Fig. 11 , the coupling strength is shown for a qubit moving along the x axis while the distance along y is kept at 1 mm. This geometry is inspired by the circuit model prediction that there should be two zeros along this line. A fit to the circuit model is also shown in the graph.
CIRCUIT MODEL
In this section, details of a circuit model that represents two-qubits inside a cavity are explained. The model is used to evaluate the coupling strength between the elements.
Circuit diagram
Dissipation-less LC resonators are used to model the cavity and qubits (see also [2] ). The nonlinearity of the qubits is not taken into account as it does not affect the coupling strength. Capacitors are used in order to account for the coupling between elements. The coupling between each qubit and the cavity is represented by a coupling capacitor C cav . As qubits are assumed to couple to each other through their dipole antenna, a total of four capacitors are used to represent coupling of each antenna pad to the two pads of the other qubit (see Fig.  12 ). 2 where r is the center to center distance between pads of the antenna. This models the dipole-dipole coupling of two qubits. The two purple structures on Q1 and Q2 represent the physical shapes of the qubits.
Coupling strength
In order to calculate the coupling between elements, the following method is used: The impedance of the system (Z in ) seen through one of the qubits (here Q 1 ) is calculated. The poles of the impedance are the resonance modes of the total system, in this example three. By sweeping the inductance L 1 (of Q 1 ) we can see the avoided crossing between the modes (see Fig. 14) . Coupling is defined as the minimum distance between the two modes at the avoided crossing.
Coupling strength
A further step was taken by relating the inter-qubit coupling capacitances (as in Fig. 12 ) to the physical distances among the qubit antenna poles. This distance dependence is assumed to be of the following form [3] :
Coupling strength
where is the average over dielectric constants of the substrate and the air (or vacuum), w = w a = w b is the width of antenna paddles and s is the separation between closest edges as shown in the inset of Fig. 13 . The parameters a L , a R , b R , b L are defined in Fig. 13 . As this study concerns square shaped paddles (see top right inset of Fig. 13) , it is not possible to apply equation 7 to any arbitrary relative position of the two qubits without complicated corrections. Therefore, just to demonstrate the general behavior, a simplified relation (Eq. 8 follows Eq. 7 when the separation s becomes much larger than the paddle width w -this effect is shown in Fig. 13 for the coupling is chosen which assumes that the capacitance decays quadratically as a function of the distance:
Coupling strength
where a is a coefficient used to adjust the coupling strength. This means that the coupling strength J i,j can be calculated based on the physical position of the two qubits with respect to each other. In this study, Q 1 is assumed to be fixed in position and Q 2 changes its distance and angle with respect to Q 1 .
Coupling strength
Assuming that the two qubits are placed parallel with respect to each other, we have calculated the coupling J i,j as a function of the center-to-center distance. In Fig. 15 the results are shown for two qubits without a mediating cavity. As can be seen, the coupling is heavily suppressed along a straight line. This line can be thought of as when the dipole electric field from Q 1 is horizontal, that is perpendicular to the dipole antenna of Q 2 and hence, there will be no coupling. This is typical behavior of dipoles and is well expected. However, by adding a cavity to the model (see Fig. 16 ), one can observe that the straight line shape of suppression will no more hold and instead, a curved line emerges.
Coupling strength
Adding the cavity has certain consequences on how two qubits couple. For example, if one moves Q 2 along the x-axis while it is held at y = 1 mm (or y = −1 mm), the coupling might be suppressed twice. This is verified by HFSS simulations as seen in Fig. 11 . Our lumpedelement model shows behavior that is consistent with HFSS simulations as already demonstrated in Figs. 10 Example for three modes of a system with two qubits and a cavity.The minimum distance of two modes is defined as the coupling 2Ji,j if it is two qubits and as 2g if it is the qubit cavity coupling. Note that despite the fact that Q2 has 9 nH of inductance, the crossing happens at lower inductances due to the coupling to Q1. Dashed line shows how resonance frequency of a typical LC circuit would behave without coupling to the other resonators.
FIG. 15:
Contour plot showing the coupling strength between two qubits Ji,j as a function of the distance between them, in the x and y direction, in absence of the cavity. The color scale is logarithmic to highlight the rapid change in Ji,j. In such a case, the dipole physics dominates and the coupling will be suppressed along a straight line. The white area is not accessible due to the dimensions of the qubits. and 11.
HAMILTONIAN DERIVATION Interacting Transmons
Let us consider two interacting transmons as depicted in Fig. 17 . The Lagrangian describing this system is given by
HAMILTONIAN DERIVATION Interacting Transmons
with Φ T ≡ (Φ 1 , Φ 2 ). For small phase fluctuations, we can expand the cosine and get H = H 0 + H 1 , with
HAMILTONIAN DERIVATION Interacting Transmons
where
HAMILTONIAN DERIVATION Interacting Transmons
. We now quantize this Hamiltonian using the usual ladder operator replacement for Q j and Φ j which is
HAMILTONIAN DERIVATION Interacting Transmons
where 
HAMILTONIAN DERIVATION Interacting Transmons
Interacting transmons inside a cavity
HAMILTONIAN DERIVATION Interacting Transmons
Let us now consider the circuit depicted in Fig. 18 which is a simplified version of Fig. 12 with the capacitive coupling network between the qubits reduced to one capacitor. This lumped element model will provide the right Hamiltonian but does not capture the angle and distance dependence explicitly. We first write the Lagrangian of the system, which reads The associated Hamiltonian H = H 0 +H 1 can be written as equations (11) and (12), with
HAMILTONIAN DERIVATION Interacting Transmons
Here, L i ≡ different coils accordingly using e.g. an optimal control algorithm should in principle allow us to generate any desired flux at the qubit location even though there are interdependencies.
SCALING OF THE BOND ORDER PARAMETER AND BOND CORRELATIONS
We provide here some additional information on the bond order parameter and correlation used in the main text to characterize the dimer phase (DP). The main message here is that, while the DP is rigorously characterized by the true order parameters D α , simpler correlation functions, which can immediately be accessed in the experiments, already qualitatively demonstrate the existence of the DP.
SCALING OF THE BOND ORDER PARAMETER AND BOND CORRELATIONS
As a first figure of merit, we show in top two panels of 

{'paper_id': '117110618', 'title': 'Dipolar Spin Models with Arrays of Superconducting Qubits', 's2_url': 'https://api.semanticscholar.org/CorpusID:117110618', 'source': 'S2ORC'}