Abstract
Controlling the interaction graph between spins or qubits in a quantum simulator allows usercontrolled tailoring of native interactions to achieve a target Hamiltonian. Engineering longranged phononmediated spin–spin interactions in a trapped ion quantum simulator offers such a possibility. Trapped ions, a leading candidate for quantum simulation, are most readily trapped in a linear 1D chain, limiting their utility for readily simulating higher dimensional spin models. In this work, we introduce a hybrid method of analogdigital simulation for simulating 2D spin models which allows for the dynamic changing of interactions to achieve a new graph using a linear 1D chain. We focus this numerical work on engineering 2D rectangular nearestneighbor spin lattices, demonstrating that the required control parameters scale linearly with ion number. This hybrid approach offers compelling possibilities for the use of 1D chains in the study of Hamiltonian quenches, dynamical phase transitions, and quantum transport in 2D and 3D.
Introduction
Dynamical evolution of interacting quantum manybody systems is often intractable with classical computation. Controlled studies are best done in quantum simulators^{1,2,3,4} wherein the essential manybody dynamics is manifest but resides in an experimentally manageable configuration. Trapped ions^{3} owing to their inherent longrange interactions offer the ability to manipulate individual spin–spin interactions, in principle, arbitrarily.^{5} Longrange spin–spin interactions are straightforward to generate in ion trap quantum simulators, and additionally, can be controlled in their range, magnitude, and sign.^{6,7,8,9,10,11,12,13,14} Leveraging phonon modes to build interspin interactions is what makes a trapped ion system fully connected and thereby inherently higher dimensional. The advantages of fullconnectivity between qubits have recently been harnessed in quantum computing experiments.^{15} Importantly, the fullconnectivity potentially allows the ability to probe a rich variety of physical phenomena, such as quantum transport and localization, topological insulators,^{16} the Haldane model,^{17} as well as in topological quantum computation following the Kitaev honeycomb model.^{18}
However, despite a few notable experiments and proposals,^{10,14,19,20,21,22} most quantum simulations have been limited to onedimensional (1D) chain of ions due to the constraints of radiofrequency ion traps.^{23} While experimental efforts to broaden the number of ion traps with higher dimensional ion arrays are underway, significant experimental simplification is offered by leveraging existing 1D ion chains, especially considering remarkable progress, where N > 100 ions have been trapped in a linear geometry,^{24,25} and prospects for still larger system sizes looking optimistic in the future. In addition, existing experimental approaches can be experimentally resourceintensive, operating on either analog^{9,10,11,12,13,14,24,26,27,28,29,30} or digital^{31,32,33,34} quantum simulation protocols. Analogdigital hybrid protocols can incorporate the benefits offered by both approaches.^{35,36,37} In this work, we propose a hybrid quantum simulation that enables the dynamical engineering of a fully connected 1D ion chain to, in principle, an arbitrary 2D lattice (see Fig. 1). When the target lattice contains certain symmetries, for instance in the case of engineering rectangular lattices, our numerical results indicate that the quantum control required scales exceedingly favorably (\({\cal{O}}(N)\)) compared with other methods.
Results
We propose a protocol for timedomain engineering of the interaction graph between trapped ion spins in a quantum simulator. The method relies on a repeated and stroboscopic application^{38,39,40} of the full interaction Hamiltonians \(\hat{H}_{{\mathrm{int}}}\) and \(\hat H_{{\mathrm{int}}}\), and laser driven Stark shift gradients. A quantum simulator can be tuned such that the interspin interactions in a onedimensional chain of trapped ions has the form,
where \(\hat S_i^ \pm = \hat S_{x_i} \pm i\hat S_{y_i}\) are the raising and lowering spin operators acting on spin i. The longrange couplings are characterized by
where 0 < α < 3 sets the range of interactions.^{7,41} Here, J_{0} is the nearestneighbor coupling strength. Without the loss of generality, we assume J_{0} > 0. Our goal is to modify the interaction profile in Eq. (2) such that the 1D spinchain can be mapped onto rectangular lattices. We achieve this with the help of an external gradient field, represented by the Hamiltonian,
where \(\hat S_z\) is the z component of the spin1/2 operator. If the external Hamiltonian, \(\hat H_{{\mathrm{ext}}}\), is applied for a duration τ, the Hamiltonian in Eq. (1) is transformed to \(\hat{\tilde H}_{{\mathrm{int}}} = e^{i\hat H_{{\mathrm{ext}}}\tau }\hat H_{{\mathrm{int}}}e^{  i\hat H_{{\mathrm{ext}}}\tau } = \mathop {\sum}\nolimits_{i < j} {J_{ij}} \hat S_i^ + \hat S_j^  e^{i\omega _{ij}\tau } + h.c.\), where ω_{ij} = ω_{i} − ω_{j} (note \(\omega _{ij} \gg J_{ij}\)). Thus the phase tags ϕ_{ij} = ω_{ij}τ appear in \(\hat{\tilde H}_{{\mathrm{int}}}\). These phase tags give us a handle to modify the couplings J_{ij}. Figure 1 shows a schematic of the Hamiltonian engineering scheme. It works by removing (decoupling) interactions (forthwith “class B” interactions) that are absent in the target Hamiltonian graph, while appropriately weighting (engineering) the other interactions (“class A”), all by the global manipulation of all spins in the linear ion chain. Thus, the experimental implementation is considerably simpler than a fully digital simulation model, which requires individual single and two qubit gates on the ion chain.
Our scheme consists of a periodic application (with a cycle time of T_{cyc}) of a multipulse sequence on all the qubits simultaneously, as shown in Fig. 1b. Each cycle consists of two multipulse blocks with alternating pulses of \(\pm \hat H_{{\mathrm{int}}}\) (with duration {t_{k}}) and \(\hat H_{{\mathrm{ext}}}\) (with duration {τ_{k}}). \(\pm \hat H_{{\mathrm{int}}}\) in the first block are substituted by \(\mp \hat H_{{\mathrm{int}}}\) in the second block. As explained in the Supplementary Materials, the average Hamiltonian at the end of each cycle retains the form of Eq. (1), with modified couplings \(J_{ij}^\prime\) that depend on \(\{ t_k,\tau _k,\phi _{ij}^{(k)}\}\), where \(\phi _{ij}^{(k)} = \omega _{ij}\tau _k\). \(J_{ij}^\prime\) is related to J_{ij} by,
where
with \(\phi _{ij}^{({\mathrm{tot}})} = \mathop {\sum}\nolimits_{k = 1}^l {\phi _{ij}^{(k)}}\). Our choice of the pulse sequence in each block results in realvalued β_{ij}. The total phase accumulated in one block, \(\phi _{ij}^{({\mathrm{tot}})}\) has to be an integer multiple of π for \(J_{ij}^\prime\) to be realvalued (Eq. (4)). For the most efficient use of this protocol, we use a labeling scheme for mapping the ions in the 1D chain to the target lattice and the external field gradient {ω_{ij}} such that \(\phi _{ij}^{({\mathrm{tot}})} = 2n\pi\) (n = 0, 1, 2, ⋯), and hence \(J_{ij}^\prime = 0\) for as many class B couplings as possible. This is achieved by choosing a semilinear gradient field, which can be engineered in experiments (see Methods section). The remaining couplings in class B are set to zero, versus the class A couplings which are scaled to their target values by appropriately choosing \(\{ t_k,\tau _k = \phi _{ij}^{(k)}/\omega _{ij}\}\) in Eq. (5). In a target square lattice, we require equal horizontal and vertical nearestneighbor bonds, \(J_H^\prime = J_V^\prime\) as seen in Fig. 1a.
In an analogy to holography, the target Hamiltonian can be engineered by a Fourier expansion of the target couplings in the domain of phases imparted by the external field over the duration T = T_{cyc}/2 of a multipulse block. The Fourier ‘filtering’ function
is chosen such that \(F(\phi _{ij}^{({\mathrm{tot}})}) = \beta _{ij}\), where \(\phi _{ij}^{({\mathrm{tot}})} = (2n  1)\pi\), and W is a fitting parameter. The coefficients {t_{k}} in Eq. (5) are simply obtained from the Fourier coefficients {a_{i}}, as explained in Methods. The Fourier engineering allows a powerful means to achieve the target Hamiltonian efficiently while exploiting its inherent symmetries. Most importantly, by dynamically modifying the Fourier filtering function, the engineered Hamiltonians can be dynamically modified. This opens several possibilities for studying quantum transport, dynamical phase transitions under a Hamiltonian quench,^{24,30,42,43} and thermalization^{44} and manybody localization^{45,46,47,48,49} in high dimensions.
Practically, the global spin–spin interactions (\(\pm \hat H_{{\mathrm{int}}}\)) are realized by laser driven Mølmer–Sørensen couplings^{6} and the single qubit phase gates (by \(\hat H_{{\mathrm{ext}}}\)) are realized by imprinting light shift (AC Stark shift) in the qubit frequency by an additional laser beam with an intensity gradient. The sign of the internal Hamiltonian (\(\pm \hat H_{{\mathrm{int}}}\)) can be flipped by changing the frequencies of global laser beams.^{50} The scheme can be extended to other 2D lattice geometries, 3D lattices, and can potentially be adapted to other systems with longrange interactions and control over individual spins. Our approach therefore offers both a simplification of control parameters and a favorable linear scaling with ion number, and offers compelling possibilities for exploiting the remarkable versatility of longrange coupled linear chain of ions for the generation of exotic engineered Hamiltonians.
Numerical simulations
We successfully reproduce the spin–spin interaction graph of target 2D square lattices at discrete evolution times t = nT_{cyc} (n = 1, 2, ⋯). We demonstrate control over dynamically changing the vertical and horizontal nearest neighbour bonds (\(J_{\mathrm{V}}^\prime\) and \(J_{\mathrm{H}}^\prime\) in Fig. 1a).
Figure 2a, b illustrates the results for a 2 × 3 and a 3 × 3 square lattice, respectively. These results are obtained using the labeling and field gradient schemes presented in Fig. 5 in Methods section. The semilinear field gradient in Fig. 5 assigns \(\phi _{ij}^{({\mathrm{tot}})} = 2n\pi\) to most coupling in Class B and \(\phi _{ij}^{({\mathrm{tot}})} = (2n  1)\pi\) to all Class A couplings. Decoupling interactions in Class B with an \(\phi _{ij}^{({\mathrm{tot}})} = (2n  1)\pi\) and rescaling Class A interactions (e.g., \(J_{\mathrm{H}}^\prime = J_{\mathrm{V}}^\prime\) in Fig. 1a) are accomplished through the Fourier filtering functions F(ϕ) specified by coefficients {a_{i}} (see Eq. (6) and Table 1). The first and second row of the table correspond, respectively, to {a_{i}} for engineering a 2 × 3 and a 3 × 3 square lattice. The target lattices are engineered through application of multipulse sequences constructed from the given Fourier coefficients taking the steps discussed in Methods section. The Fourier expansion coefficients for engineering an m × m square lattice (\(m = \sqrt N\)) up to m = 7 are given in Supplementary Materials. In Fig. 3, we show that the number of pulses in each cycle scales linearly with the number of ions.
On the target square lattices, spins interact with strength \(J_{ij}^\prime\), which is nonzero only when the ionpairs (i, j) are nearest neighbors in the target lattice. Since the interactions are decaying with distance in the original 1D chain, for α > 0 in Eq. (2), \(J_{{\mathrm{ij}}}^\prime\) can be at most J_{0}/m^{α} for an m′ × m square lattice.
The engineered interaction matrix formed by the couplings \(\{ J_{ij}^\prime \}\) matches the target interaction matrix of the 2D square lattice with an RMS error of <0.1%. Here we define the RMS error as \(\sqrt {\mathop {\sum}\nolimits_{ij} {(J_{ij}^\prime  J_{ij}^\prime ({\mathrm{Target}}))^2} } /\mathop {\sum}\nolimits_{ij} {J_{ij}^\prime ({\mathrm{Target}})}\). In Fig. 2, we also compare spin dynamics under the engineered lattice (red dots) with that of the ideal target (green curve), and find excellent agreement. Here the systems are initially prepared in ψ_{0}〉 = ↑_{1}↓_{2}↓_{3}↓_{4}↓_{5}↓_{6}〉 for N = 6 and ψ_{0}〉 = ↑_{1}↓_{2}↓_{3}↓_{4}↓_{5}↓_{6}↓_{7}↓_{8}↓_{9}〉 for N = 9 at t = 0, when the pulse sequence is turned on. Here, ↑_{i}〉 and ↓_{i}〉 are the eigenstates of \(\hat S_z\) for the ith spin. The probability of the system being in ψ_{0}〉 (Fig. 2a(iii), b(iii)) follows the expected dynamics of the ideal 2D square lattice. These numerical simulations were performed using the time dependent master equation solver based on the QuTip python package.^{51,52}
The nearperfect matching of the target and engineered spin dynamics in Fig. 2 indicates small intrinsic errors in the dynamical Hamiltonian engineering protocol. However, additional errors may creep into an experimental realization due to imperfect single qubit gates. In our numerical simulation, we find that RMS error between the target and engineered interaction matrices scales linearly with single qubit phase error, with 1% error in single qubit phase (in each pulse of \(\hat H_{{\mathrm{ext}}}\)) contributing to ~1.2% error in \(J_{ij}^\prime\).
A crucial feature of the protocol presented here is the ability to dynamically change the Hamiltonian within the same symmetry class, by changing the pulse sequence obtained from the Fourier decomposition of the target interaction profile. This enables simulation of manybody dynamics, such as quantum quench experiments that are hard to simulate numerically. As an example, we show a quench from two decoupled chains of three spins each into a 2 × 3 square lattice in Fig. 4. To simulate the decoupled chains, N_{3} couplings are set to zero (see Fig. 6 in Methods section) in estimating the Fourier filtering function and hence the multipulse sequence. The spin–spin correlations between the previously uncoupled chains start to build up after the quench. We show the engineered dynamics (red dots) of the twopoint correlation functions C_{12}(t) between spins 1 and 2 and C_{14}(t) between spins 1 and 4. They follow closely to the ideal target dynamics (green line).
Proposal for experimental implementation
The experimental implementation of the multipulse scheme can be achieved in a trapped ion system such as ^{171}Yb^{+} trapped in a radiofrequency (Paul) trap. When the confining potential is sufficiently anisotropic, lasercooled ions will form a linear chain.^{53} The hyperfine states ↓〉 ≡ ^{2}S_{1/2}, F = 0, m_{F} = 0〉 and ↑〉 ≡ ^{2}S_{1/2}, F = 1, m_{F} = 0〉 form the spin (qubit) states for this ion.^{54}
The interspin interactions in Eq. (1) can be simulated^{12,13} by shining the ions with a laser beam that imparts optical dipole forces, using the Mølmer–Sørensen scheme.^{6} When the frequencies of the laser beams (referred to as the Mølmer–Sørensen detuning in this manuscript) are appropriately chosen to offresonantly excite the centerofmass collective vibrational (phonon) modes of the ion chain, a longrange interaction in the form of Eq. (2) can be obtained. The global sign of \(\hat H_{{\mathrm{int}}}\) can be flipped by changing the Mølmer–Sørensen detuning, with additional laser beams improving the accuracy as discussed in Supplementary Materials.
The field gradient in \(\hat H_{{\mathrm{ext}}}\) can be implemented with laser beams imprinting AC Stark shifts (ω_{i} in Eq. (3)), and by spatially modulating the laser intensity using a spatial light modulator (SLM) or an acousto optic deflector (AOD). Another potentially easier experimental implementation will be to combine a global tightly focused laser beam with additional relatively low power beams created by an AOD. The global beam propagating along the axis of the ion chain can be focused before hitting the ions, such that its intensity varies linearly on the ion chain. The jumps in the gradient field can be added by beams created by an SLM or AOD and shining on the ions from the transverse direction. A 100 mW laser beam propagating along the ion chain, detuned from the ^{171}Yb^{+ 2}S_{1/2} − ^{2}P_{1/2} resonance by 10^{5} natural linewidths, and focused to ~2 microns will create a twophoton differential AC Stark shift gradient ω_{i} on the order of ~1 MHz. Thus, the total time (τ_{tot}) that the Stark shifting beam is shining on the ions in a time cycle can be limited to a few microseconds, minimizing spontaneous emission errors.
Discussion
In summary, we have proposed an analogdigital hybrid quantum simulation protocol to dynamically engineer a 2D lattice in a linear chain of ions. For an arbitrary target lattice, the Hamiltonian engineering protocol presented here requires \({\cal{O}}(N^2)\) Fourier coefficients, hence the number of pulses. However, in presence of common symmetries between the target lattice and the external field gradient, as is the case for a square lattice, the number of pulses in the engineering timesequence is drastically reduced to \({\cal{O}}(N)\). Note that we can also engineer the target lattice using \(\hat H_{{\mathrm{int}}}\) only instead of \(\pm \hat H_{{\mathrm{int}}}\) where all the interactions are scaled or set to zero by Fourier filtering function (Eq. (6)) alone. However, for small system sizes (N < 36), using \(\pm \hat H_{{\mathrm{int}}}\) instead of \(\hat H_{{\mathrm{int}}}\) reduces the number of pulses approximately by up to a factor of 2 (see Table 2 in Supplementary Materials).
The engineered interactions in the target 2D lattice will become weaker with increasing system size N, approximately as 1/N^{2} for small α in Eq. (2). This scaling in the interaction strength is inherited from the 1D chain for α ≈ 0. Since the separation between the vibrational normal modes in the ion chain will decrease with increasing N, the coupling strength J_{0} will have to scale down accordingly in order to avoid direct excitation of phonons that limit the validity of a spinonly Hamiltonian (see Supplementary Materials for more details). To engineer an m × m square lattice with equal nearestneighbor couplings, nearestneighbor couplings N_{1} of the 1D chain have to be scaled down to the mth neighbor coupling, N_{m} of the 1D chain (see Fig. 6 in Methods section). This further scales the target interaction in the 2D lattice down by 1/m^{α}, making the overall scaling 1/N^{(2+α/2)} for N = m^{2}. However, for small α, the effect of 1/m^{α} factor is small compared with 1/N^{2} scaling inherited from the original 1D chain. For the results presented here with N = 6 and N = 9 ions, we have chosen α = 0.2, which is experimentally realizable in current systems, and provides sufficiently strong target interactions. When α = 0.2, the target coupling of a 2 × 3 square lattice is estimated to be 2π × 300 Hz, while for a 3 × 3 square lattice, the target coupling is estimated to be 2π × 102 Hz. It should be noted that the average Hamiltonian theory employed here works when \(J_0T_{{\mathrm{cyc}}} \ll 1\). Due to the linear scaling of the number of pulses in a cycle with N (Fig. 3), T_{cyc} is expected to scale linearly. That is, longer simulation times are necessary as the system size increases. Since the initial coupling J_{0} scales as 1/N^{2} while T_{cyc} grows with N, the average Hamiltonian validity condition is readily satisfied for any system size.
While trapped ion qubits have long single qubit coherence time,^{9} scaling the simulation to a large number of spins where classical computation of dynamics may be intractable will require isolating experimental noise sources, such as intensity fluctuations of the global Mølmer–Sørensen and Stark shifting laser beams, and drifts in the collective phonon mode frequencies.
In a small chain of ions, the couplings are inhomogeneous. The errors due to the inhomogeneity can be mitigated by using an anharmonic trapping potential for the ions^{55} and spatially modulating the global Mølmer–Sørensen laser beams to increase the homogeneity of the couplings. The errors can also be reduced at the expense of increasing the number of pulses within a cycle and using a field gradient that breaks the symmetry between interactions belonging to an interaction class N_{d} where d = j − i is the distance between the spin pairs (i, j) in the 1D chain (see Methods section).
As the number of ions N increases, the spacing between the vibrational modes decreases for a given frequency bandwidth of the phonon spectrum, constrained by experimental resources (such as the radiofrequency power for a Paul trap). This will make it harder to engineer \( \hat H_{{\mathrm{int}}}\) with a single global Mølmer–Sørensen beam that is detuned close to the centerofmass mode. The accuracy of engineering \( \hat H_{{\mathrm{int}}}\) is enhanced by introducing additional global beams with Mølmer–Sørensen detuning near the neighboring phonon modes as demonstrated in Supplementary Materials. Engineering \( \hat H_{{\mathrm{int}}}\) for very large N will require either a large number of global Mølmer–Sørensen beams to cancel the effect of multiple modes, or reducing the overall intensity of laser beams resulting in a reduction in the strength of interactions (J_{0}) in the 1D chain.
Methods
Labeling and field gradient scheme
To map the interactions J_{ij} in the 1D spinchain onto the target 2D rectangular lattice, we have categorized them into classes N_{d} (d = j − i), with N_{1} denoting the nearestneighbor couplings in the 1D chain, N_{2} denoting the next nearestneighbor couplings, and so on. Here, we ignore inhomogeneities due to the finite size effect. As seen in Fig. 5 top left corner, we employ a labeling scheme in which the N_{1} and N_{m} couplings form the horizontal and vertical bonds of an m′ × m target square lattice. That is class A = {N_{1}, N_{m}}, with the exception of couplings J_{km,km+1}, k = 1, …, m′ − 1, that form a toroidal linkage between the edges of the square lattice and must be excluded from class A.
A semilinear field gradient as illustrated in Fig. 5 assigns proper phase tags to the couplings in the original 1D chain. That is ϕ = (2n − 1)π to class A couplings with unique phase tags to J_{km,km+1} and ϕ = 2nπ to most class B couplings, n being an integer. The external field profile is assumed to increase linearly with a constant slope ω_{0} and added jumps of 2ω_{0} (for even m) or 3ω_{0} (for odd m) between the kmth and (km + 1)th spins to address the toroidal linkages J_{km,km+1}. Hence,
Given \(\phi = \phi _{ij}^{({\mathrm{tot}})} = \omega _{ij}\tau _{{\mathrm{tot}}}\), we can achieve desired phase tags upon adjusting ω_{0}τ_{tot} = π.
Fourier filtering of interactions
The interactions that were not automatically be canceled by our chosen field gradient can be rescaled to their target value by designing a Fourier filter function F(ϕ) (see Eq. (6)). The N_{1} couplings in class A have to be rescaled to match the N_{m} couplings for α ≠ 0 in Eq. (2). In addition, the couplings in class B for which ϕ = (2n − 1)π have to be rescaled to zero. These scalings are performed through F(ϕ).
In Fig. 6, we show the Fourier filter function fit for engineering a 2 × 3 square lattice when α = 0.2. The nearestneighbor couplings N_{1}, except the toroidal linkage (J_{34}) have a phase of ϕ = π. The N_{3} couplings accumulate a phase of 5π. Hence we require \(F(\pi )/F(5\pi ) = \frac{1}{{3^\alpha }} = 0.80\) such that N_{1} couplings (except J_{34}) are equal to N_{3}. The toroidal linkage J_{34} (ϕ = 3π) and N_{5} couplings (ϕ = 7π) are scaled to zero. We have introduced a global rescaling factor to all the target couplings, by setting F(5π) = 0.7. The global rescaling of all the target couplings ensures an efficient Fourier fit with minimum number of parameters.
In general, the Fourier filter function should satisfy \(F(\phi = \phi _{ij}^{({\mathrm{tot}})}) = \beta _{ij}\) for all couplings with a phase ϕ = (2n − 1)π to engineer target Hamiltonian graph. By comparing Eq. (5) with Eq. (6), one can find the proper multipulse sequence for implementing the Fourier filtering function F(ϕ) by taking the following steps:

1.
The number of singlequbit phase gates (by \(\hat H_{{\mathrm{ext}}}\)) l in each block is l = 2i + 1, where the number of Fourier terms in Eq. (6) is i + 1.

2.
The pulse sequence in each block is antisymmetric about the central \(\hat H_{{\mathrm{ext}}}\) pulse. That is t_{k} = t_{l−k} and \(\pm \hat H_{{\mathrm{int}}_k} = \mp \hat H_{{\mathrm{int}}_{l  k}}\) for k = 1, …, l − 1. This leads to the cancellation of all even order correction terms to the average Hamiltonian.

3.
The time intervals {t_{k}} are proportional to the coefficients in the Fourier filter function: t_{k} = Ta_{k}/2 for k = 1, …, l − 1 and t_{l} = Ta_{0}, with the constraint that \(\mathop {\sum}\nolimits_{j = 0}^l {a_j = 1}\). This constraint can be relaxed for an efficient search for the Fourier coefficients at the expense of reducing all the couplings in the target lattice by a global rescaling factor. Numerically, we find that β_{i,i+m} = 0.7 allows us to find efficient solutions for up to N = 100.

4.
A negative coefficient (a_{j} < 0) in Eq. (6) is implemented by an \(\hat H_{{\mathrm{ext}}}\) pulse followed by \( \hat H_{{\mathrm{int}}}\). If we want to use \(\hat H_{{\mathrm{int}}}\) only instead of \(\pm \hat H_{{\mathrm{int}}}\) the Fourier search must be performed under the constraint that a_{j} > 0 for all j.

5.
We choose the phase gates to be of equal duration τ, except the central phase gate in each block which has to have a duration of τ′ = τ_{tot} − (l − 1)τ. τ can be readoff from W = τ/τ_{tot}.
Data availability
The data and numerical codes that support the findings of this study are available from the corresponding author upon reasonable request.
References
 1.
AspuruGuzik, A. & Walther, P. Photonic quantum simulators. Nat. Phys. 8, 285 (2012).
 2.
Bloch, I., Dalibard, J. & Nascimbene, S. Quantum simulations with ultracold quantum gases. Nat. Phys. 8, 267 (2012).
 3.
Blatt, R. & Roos, C. F. Quantum simulations with trapped ions. Nat. Phys. 8, 277 (2012).
 4.
Cirac, J. I. & Zoller, P. Goals and opportunities in quantum simulation. Nat. Phys. 8, 264 (2012).
 5.
Korenblit, S. et al. Quantum simulation of spin models on an arbitrary lattice with trapped ions. New J. Phys. 14, 095024 (2012).
 6.
Mølmer, K. & Sørensen, A. Multiparticle entanglement of hot trapped ions. Phys. Rev. Lett. 82, 1835 (1999).
 7.
Deng, X.L., Porras, D. & Cirac, J. I. Effective spin quantum phases in systems of trapped ions. Phys. Rev. A 72, 063407 (2005).
 8.
Kim, K. et al. Entanglement and tunable spinspin couplings between trapped ions using multiple transverse modes. Phys. Rev. Lett. 103, 120502 (2009).
 9.
Kim, K. et al. Quantum simulation of frustrated ising spins with trapped ions. Nature 465, 590 (2010).
 10.
Britton, J. W. et al. Engineered twodimensional ising interactions in a trappedion quantum simulator with hundreds of spins. Nature 484, 489 (2012).
 11.
Islam, R. et al. Emergence and frustration of magnetism with variablerange interactions in a quantum simulator. Science 340, 583–587 (2013).
 12.
Richerme, P. et al. Nonlocal propagation of correlations in quantum systems with longrange interactions. Nature 511, 198 (2014).
 13.
Jurcevic, P. et al. Quasiparticle engineering and entanglement propagation in a quantum manybody system. Nature 511, 202 (2014).
 14.
Bohnet, J. G. et al. Quantum spin dynamics and entanglement generation with hundreds of trapped ions. Science 352, 1297–1301 (2016).
 15.
Linke, N. M. et al. Experimental comparison of two quantum computing architectures. Proc. Natl Acad. Sci. USA 114, 3305–3310 (2017).
 16.
Qi, X.L. & Zhang, S.C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057 (2011).
 17.
Haldane, F. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken timereversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).
 18.
Kitaev, A. Y. Faulttolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).
 19.
Sawyer, B. C. et al. Spectroscopy and thermometry of drumhead modes in a mesoscopic trappedion crystal using entanglement. Phys. Rev. Lett. 108, 213003 (2012).
 20.
Yoshimura, B., Stork, M., Dadic, D., Campbell, W. C. & Freericks, J. K. Creation of twodimensional coulomb crystals of ions in oblate paul traps for quantum simulations. EPJ Quantum Technol. 2, 2 (2015).
 21.
Richerme, P. Twodimensional ion crystals in radiofrequency traps for quantum simulation. Phys. Rev. A 94, 032320 (2016).
 22.
Li, H.K. et al. Realization of translational symmetry in trapped cold ion rings. Phys. Rev. Lett. 118, 053001 (2017).
 23.
Wineland, D. J. et al. Experimental issues in coherent quantumstate manipulation of trapped atomic ions. J. Res. Natl. Inst. Stand. Technol. 103, 259 (1998).
 24.
Zhang, J. et al. Observation of a manybody dynamical phase transition with a 53qubit quantum simulator. Nature 551, 601 (2017).
 25.
Pagano, G. et al. Cryogenic trappedion system for large scale quantum simulation. Quantum Sci. Technol. 4, 014004 (2019).
 26.
Friedenauer, A., Schmitz, H., Glueckert, J. T., Porras, D. & Schätz, T. Simulating a quantum magnet with trapped ions. Nat. Phys. 4, 757 (2008).
 27.
Gerritsma, R. et al. Quantum simulation of the klein paradox with trapped ions. Phys. Rev. Lett. 106, 060503 (2011).
 28.
Islam, R. et al. Onset of a quantum phase transition with a trapped ion quantum simulator. Nat. Commun. 2, 377 (2011).
 29.
Senko, C. et al. Realization of a quantum integerspin chain with controllable interactions. Phys. Rev. X 5, 021026 (2015).
 30.
Jurcevic, P. et al. Direct observation of dynamical quantum phase transitions in an interacting manybody system. Phys. Rev. Lett. 119, 080501 (2017).
 31.
Lanyon, B. P. et al. Universal digital quantum simulation with trapped ions. Science 334, 57–61 (2011).
 32.
Barreiro, J. T. et al. An opensystem quantum simulator with trapped ions. Nature 470, 486 (2011).
 33.
Linke, N. M. et al. Measuring the renyi entropy of a twosite fermihubbard model on a trapped ion quantum computer. Phys. Rev. A 98, 052334 (2018).
 34.
Hempel, C. et al. Quantum chemistry calculations on a trappedion quantum simulator. Phys. Rev. X 8, 031022 (2018).
 35.
Hayes, D., Flammia, S. T. & Biercuk, M. J. Programmable quantum simulation by dynamic hamiltonian engineering. New J. Phys. 16, 083027 (2014).
 36.
Arrazola, I., Pedernales, J. S., Lamata, L. & Solano, E. Digitalanalog quantum simulation of spin models in trapped ions. Sci. Rep. 6, 30534 (2016).
 37.
Lamata, L., ParraRodriguez, A., Sanz, M. & Solano, E. Digitalanalog quantum simulations with superconducting circuits. Adv. Phys.: X 3, 1457981 (2018).
 38.
Warren, W., Sinton, S., Weitekamp, D. & Pines, A. Selective excitation of multiplequantum coherence in nuclear magnetic resonance. Phys. Rev. Lett. 43, 1791 (1979).
 39.
Baum, J., Munowitz, M., Garroway, A. & Pines, A. Multiplequantum dynamics in solid state nmr. J. Chem. Phys. 83, 2015–2025 (1985).
 40.
Ajoy, A. & Cappellaro, P. Quantum simulation via filtered hamiltonian engineering: application to perfect quantum transport in spin networks. Phys. Rev. Lett. 110, 220503 (2013).
 41.
Porras, D. & Cirac, J. I. Effective quantum spin systems with trapped ions. Phys. Rev. Lett. 92, 207901 (2004).
 42.
Heyl, M., Polkovnikov, A. & Kehrein, S. Dynamical quantum phase transitions in the transversefield ising model. Phys. Rev. Lett. 110, 135704 (2013).
 43.
Vosk, R. & Altman, E. Dynamical quantum phase transitions in random spin chains. Phys. Rev. Lett. 112, 217204 (2014).
 44.
Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854 (2008).
 45.
Schreiber, M. et al. Observation of manybody localization of interacting fermions in a quasirandom optical lattice. Science 349, 842–845 (2015).
 46.
Smith, J. et al. Manybody localization in a quantum simulator with programmable random disorder. Nat. Phys. 12, 907–911 (2016).
 47.
Bordia, P. et al. Coupling identical onedimensional manybody localized systems. Phys. Rev. Lett. 116, 140401 (2016).
 48.
Choi, J.y et al. Exploring the manybody localization transition in two dimensions. Science 352, 1547–1552 (2016).
 49.
Lüschen, H. P. et al. Signatures of manybody localization in a controlled open quantum system. Phys. Rev. X 7, 011034 (2017).
 50.
Gärttner, M. et al. Measuring outoftimeorder correlations and multiple quantum spectra in a trappedion quantum magnet. Nat. Phys. 13, 781 (2017).
 51.
Johansson, J., Nation, P. & Nori, F. Qutip: an opensource python framework for the dynamics of open quantum systems. Comput. Phys. Commun. 183, 1760–1772 (2012).
 52.
Johansson, J. R., Nation, P. D. & Nori, F. QuTiP 2: a Python framework for the dynamics of open quantum systems. Comput. Phys. Commun. 184, 1234–1240 (2013).
 53.
Schiffer, J. Phase transitions in anisotropically confined ionic crystals. Phys. Rev. Lett. 70, 818 (1993).
 54.
Olmschenk, S. et al. Manipulation and detection of a trapped yb+ hyperfine qubit. Phys. Rev. A 76, 052314 (2007).
 55.
Lin, G.D. et al. Largescale quantum computation in an anharmonic linear ion trap. EPL (Europhys. Lett.) 86, 60004 (2009).
Acknowledgements
We thank Industry Canada and University of Waterloo for financial assistance. F.R. and S.M. have been in part financially supported by Institute for Quantum Computing. This work is partially supported by a cooperative agreement with the Army Research Laboratory (W911NF1720117). A.A. would like to thank A. Pines and P. Cappellaro for insightful discussions. F.R., S.M., C.Y.S., N.K. and R.I. acknowledge valuable discussions with YiHong Teoh.
Author information
Affiliations
Contributions
F.R. and S.M. wrote the numerical codes and carried out simulations based on the initial idea by A.A. All authors took part in developing the content of the study and in writing the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Rajabi, F., Motlakunta, S., Shih, CY. et al. Dynamical Hamiltonian engineering of 2D rectangular lattices in a onedimensional ion chain. npj Quantum Inf 5, 32 (2019). https://doi.org/10.1038/s415340190147x
Received:
Accepted:
Published:
Further reading

Reprogrammable and highprecision holographic optical addressing of trapped ions for scalable quantum control
npj Quantum Information (2021)

Programmable quantum simulations of spin systems with trapped ions
Reviews of Modern Physics (2021)

Localized dynamics following a quantum quench in a nonintegrable system: an example on the sawtooth ladder
Journal of Physics B: Atomic, Molecular and Optical Physics (2021)

Quantum Simulations with Complex Geometries and Synthetic Gauge Fields in a Trapped Ion Chain
PRX Quantum (2020)

Machine learning design of a trappedion quantum spin simulator
Quantum Science and Technology (2020)