Residual-Guided Global-Drive Design in Trapped-Ion Quantum Simulators

The gist: A residual-guided optimization protocol is introduced for systematically designing multi-tone global drive spectra to enhance programmability in trapped-ion quantum simulators, converging on theoretical limits using a number of laser tones that scales nearly linearly with system size.

The Model

In ion traps, the native interaction under the Mølmer-Sørensen protocol is approximated by long-range Ising models (Page 2). For N ions in the trap, a single pair of oppositely detuned lasers (bichromatic tone) labeled by m generates the interaction matrix Jij (omegam, µm) = R squared / X N k=1 B ikBjk µ 2m − ω 2k (Page 2). With M bichromatic tones simultaneously on, the effective interactions add up to Jeij = X M m=1 Jij (omegam, µm) = R X M m=1 omega 2m / X N k=1 B ikBjk µ 2m − ω 2k (Page 2). The search space is restricted to 2N tones, formed by placing a pair of red- and blue-detuned tones in the vicinity of each vibrational mode (Page 3). The ultimate task is to select the minimal number of tones out of the pool and to assign the Rabi and the beatnote frequencies of these tones, so that the synthesized interactions can approximate target interaction patterns to a certain accuracy (Page 3).

Residual-Guided Drive Design (RGDD)

The residual-guided drive design (RGDD) strategy avoids enumerating combinations of candidate tones by incrementally adding global tones guided by the current residual between the synthesized and target interactions (Page 3). The method operates in the far-detuned regime, and it incurs little additional computational cost to incorporate a penalty on motional excitation, which efficiently mitigates errors arising from incomplete closure of phase-space trajectories (Page 3). RGDD consists of two stages: initialization and residual-guided growth (Page 3). Initialization involves ranking each mode’s contribution to the target interaction matrix, selecting a small set of k0 modes with the largest c tar k, and greedily assigning either a red- or blue-detuned tone in its vicinity by minimizing the infidelity (9) (Page 3). After initialization, RGDD incrementally adds tones through a residual-guided growth process where the contribution of different modes to reducing the residual is measured by the coefficient c q k = ⟨R(q),J(k)⟩ (Page 3). A subset of k q modes with the largest c q k serves as the pool for a new tone to add, and each candidate in this pool can be added only once (Page 3).

Optimization Objective

The objective takes a scale-invariant format as cosine infidelity I(DM) = 1/2 (1 - D Je(DM), Jetar F Je(DM)F JetarF) (Page 4). To maintain the fidelity of Je(DM), the term Aα must be kept low, where Aα = X N k=1 αk(tg) squared (Page 4). The design objective incorporates a closure penalty, L(DM) = I(DM) + λ a max 0, ln Aα Atar squared (Page 4). This penalty vanishes for closure-compliant drives (Aα ≤ Atar) and grows quadratically in ln(Aα/Atar) otherwise (Page 4). The algorithm uses the L-BFGS-B optimizer to find the optimal parameters Dq+1 after selecting the next tone (Page 3).

Results on Interaction Patterns

The results in this section demonstrate that RGDD facilitates the construction of high-fidelity Hamiltonians with roughly O(N) tones, and nearly guarantees reaching the theoretical limit (Page 5). For power-law interactions J tar ij ∝ i − j −α, RGDD sharply outperforms the naive NNLS method (Page 5). The scaling results for ion numbers N = 14–110 across two interaction ranges are summarized in Fig. 2(a), which plots the number of tones per ion Mmin/N (Page 5). For long-range interaction example (α = 1.5), RGDD exhibits a decreasing trend with increasing ion number, whereas the NNLS results show the opposite, increasing trend (Page 5). The inclusion of the closure penalty in the objective proves essential during optimization to maintain low excitations, as penalized solutions have Aα < 1.0 at the largest system size considered (Page 5).

Extension to 2D Settings

The RGDD method extends naturally to two-dimensional (2D) ion crystals, where only the out-of-plane (drumhead) modes are included in the model due to their spectral isolation from the remaining modes (Page 7). Figure 4(c) demonstrates that RGDD drives the infidelity down to the theoretical lower bound Ilb (Page 7). With M/N ≈ 0.5, the optimized infidelity falls within a factor of two of Ilb, and with M ≈ N, it is indistinguishable from Ilb at the 10−3 level for all tested N (Page 7). The closure error from the unpenalized optimization in 2D settings is smaller than that in 1D settings, and introducing the penalty into the objective can control this error at a very low level while maintaining infidelity minimization (Page 7).

Conclusion

The residual-guided optimization may extend to broader interaction engineering in trapped ions and beyond, such as Dicke interactions or faster gates near motional resonances (Page 9). The total laser power required for the RGDD-optimized parameters grows polynomially with system size, which is an acceptable scaling for experimental feasibility (Page 15). The optimized Rabi frequencies exhibit no extreme values and are well balanced, with most magnitudes within the same order of magnitude, which helps keep the experimental implementation on waveform generators stable (Page 15). The total laser power required for the RGDD-optimized parameters grows polynomially with system size, which is an acceptable scaling for experimental feasibility (Page 15). The amplitudes of the optimized drives for all tested interactions span only two orders of magnitude, which is compatible with the tunability of available modulators in experiments (Page 15). The residual-guided optimization may extend to broader interaction engineering in trapped ions and beyond, such as Dicke interactions [57–59] and to faster gates near motional resonances (Page 9). The mode-space perspective in this work can be generalized beyond the far-detuned Ising regime, such that the residual-guided strategy could be adapted to Dicke interactions [57–59] and to faster gates near motional resonances (Page 9). Lastly, as indicated in Ref. [45], the ordering of ions in interaction graph engineering can be vital in practical applications, and future development of optimization methods should address this aspect (Page 9).

How it works

1. The algorithm incrementally adds global tones, guided by the current residual between the synthesized and target interactions (Page 3).

2. At each step, the next global tone is chosen to maximally reduce this residual by selecting a mode with the largest c q k (Page 3).

3. A subset of k q modes with the largest c q k serves as the pool for a new tone to add, and each candidate in this pool can be added only once (Page 3).

4. The next tone is selected from this pool according to the unweighted infidelity criterion (Page 13).

5. A joint refinement is performed to get the optimal parameter set Dq+1 after selecting the (q + 1)th tone (Page 3).

Edge-weight Guidance

For short-range interaction optimization, a weighted version of the objective is introduced where weights are assigned to nearest-neighbor and next-nearest-neighbor interactions (Page 14). The weights are only incorporated into the calculation of the residual (13), which ranks the importance of the unselected candidate tones (Page 14). The pool for the next tone to add is expanded to include k w q additional modes with weighted residual coefficients c q k,w = ⟨R(q),J(k)⟩ w (Page 13). The next tone is selected from this expanded pool according to the unweighted infidelity criterion (Page 13).

Closure Penalty and Gate Time

The objective incorporates a closure penalty, L(DM) = I(DM) + λ a max 0, ln Aα Atar squared (Page 4). The penalty vanishes for closure-compliant drives (Aα ≤ Atar) and grows quadratically in ln(Aα/Atar) otherwise (Page 4). The optimization scans tg = s tg0 with s ranging from 0.5 to 200 and chooses the smallest tg satisfying Aα ≤ Atar for all N with given targets and M (Page 4). Raising the penalty to λ a = 0.3 immediately confines the closure error to the preset target Atar = 1 (Page 12). The inclusion of the penalty term in the objective function proves essential during optimization to maintain low excitations (Page 5).

Nonadiabatic Protocol Baseline

The comparison with a simple equal-spacing placement [41, 45] shows that RGDD reaches I ≤ 1.

1. Bold header: Residual-Guided Global Drive Design for Arbitrary Interaction Graphs

This method enables the synthesis of both long-range and traditionally difficult short-range coupling topologies by incrementally adding global tones guided by the current residual, which allows for the creation of Hamiltonians that approximate a target interaction pattern to a satisfying accuracy.

2. Bold header: Scalable Hamiltonian Synthesis

The algorithm is designed to be efficient in both hardware and optimization, ensuring that the number of laser tones scales nearly linearly with system size while maintaining high fidelity, which is critical for scaling up trapped-ion architectures.

3. Bold header: High-Fidelity Simulation of Complex Many-Body Physics

The system can achieve a high degree of control over interaction patterns, as demonstrated by its ability to approximate representative interactions with a tone count that scales almost linearly with system size, thus enhancing the experimental feasibility for studying many-body physics.

4. Bold header: Robust Performance Across Interaction Regimes

The method shows sharp[ly] outperforms the naive NNLS method in synthesizing both long-range power-law interactions and short-range models like the SSH and dimerized chains, confirming its robustness across different interaction topologies.

5. Bold header: Mitigation of Motional Excitation Errors

By incorporating a penalty for residual motional excitation, the algorithm effectively mitigates errors associated with incomplete closure of phase-space trajectories, ensuring that the synthesized dynamics adhere to the desired Ising evolution despite practical constraints like finite gate times.

Trapped-ion quantum simulators are powerful platforms for exploring many-body physics, yet engineering of arbitrary interaction graphs remains a central challenge. To leverage the intrinsic structure of native motional modes and thereby enhance programmability, a residual-guided optimization protocol is introduced for the systematic design of multi-tone global drive spectra. While the achievable interaction fidelity is fundamentally bounded by the available motional modes, this adaptive strategy converges on the theoretical limit using a number of laser tones that scales nearly linearly with system size. The versatility of this method enables the synthesis of both long-range and traditionally difficult short-range coupling topologies. Additionally, by incorporating a penalty for residual motional excitation, the algorithm effectively mitigates errors associated with incomplete closure of phase-space trajectories. Overall, this scalable framework significantly expands the programmable capabilities of trapped-ion architectures while maintaining low computational overhead and high potential for experimental implementation.