Bounded-Error Quantum Simulation via Hamiltonian and Lindbladian Learning

TL;DR

Bounded-error quantum simulation provides a scalable foundation for trusted analog quantum computation, bridging the gap between experimental platforms and predictive many-body physics, and the techniques presented here directly extend to digital quantum simulation.

Abstract

Analog Quantum Simulators offer a route to exploring strongly correlated many-body dynamics beyond classical computation, but their predictive power remains limited by the absence of quantitative error estimation. Establishing rigorous uncertainty bounds is essential for elevating such devices from qualitative demonstrations to quantitative scientific tools. Here we introduce a general framework for bounded-error quantum simulation, which provides predictions for many-body observables with experimentally quantifiable uncertainties. The approach combines Hamiltonian and Lindbladian Learning--a statistically rigorous inference of the coherent and dissipative generators governing the dynamics--with the propagation of their uncertainties into the simulated observables, yielding confidence bounds directly derived from experimental data. We demonstrate this framework on trapped-ion quantum simulators implementing long-range Ising interactions with up to 51 ions, and validate it where classical comparison is possible. We analyze error bounds on two levels. First, we learn an open-system model from experimental data collected in an initial time window of quench dynamics, simulate the corresponding master equation, and quantitatively verify consistency between theoretical predictions and measured dynamics at long times. Second, we establish error bounds directly from experimental measurements alone, without relying on classical simulation--crucial for entering regimes of quantum advantage. The learned models reproduce the experimental evolution within the predicted bounds, demonstrating quantitative reliability and internal consistency. Bounded-error quantum simulation provides a scalable foundation for trusted analog quantum computation, bridging the gap between experimental platforms and predictive many-body physics. The techniques presented here directly extend to digital quantum simulation.

Figures (10)

• Figure 1: Overview of the BEQS protocol: The analog quantum simulator is treated as a "black-box" device. Product input states are prepared, evolved for variable times under the unknown dynamics, and local observables are measured (e.g., via randomized measurements) to yield time-resolved traces of few-body correlation functions, containing information about the implemented dynamics. This information is compiled, in the first step of the protocol, to determine the dynamics using Hamiltonian and Lindbladian Learning. This yields a set of "optimal" Hamiltonian and Lindbladian parameters (optimality specified below), along with their associated uncertainties. Using the quantum simulator as a computational device to compute observable expectation values, $\expval{O_t}$, we need to bound the uncertainty in $\expval{O_t}$ resulting from our uncertainty in the Hamiltonian and Lindbladian. Equivalently, we must ensure that the error incurred by assuming the device implements the learned Hamiltonian and Lindbladian does not exceed a specified threshold, with high confidence constituting step 2 of our protocol.
• Figure 2: Step 1 of the BEQS protocol: Hamiltonian and Lindbladian Learning. ( a) In an experiment with $N=10$ ions, we monitor the residual norm, $\Vert \boldsymbol{r}\Vert_2$, representing the learning error, as a function of $N_{\rm shots}$ for different ansätze $H_{XY}^{(k)}$ (Eq. \ref{['eq:ansatzK']}) with increasing maximum interaction distance $k$, with and without dissipation (thick lines). The ansatz closest to the anticipated shot-noise scaling consists of a long-range XY model, Eq. \ref{['eq:XYmodel']}, with additional transverse field, $H=H_{XY}+\sum_i B_i\sigma^z_i$, and a Lindbladian dominated by collective dephasing, $L_{\rm coll}=\sum_i\sigma_i^z$. The remaining deviation from shot-noise scaling is attributed to measurement errors (see App. \ref{['app:exp']}). To verify that, we re-simulate the experiment with and without measurement errors (thin solid/dashed lines), and reproduce the observed and ideal (shot-noise) scaling, as well as a deviation from shot-noise scaling if measurement errors are included. The learned Hamiltonian parameters including $1\sigma$ error-bars are shown in ( b) and ( c). In ( b) we show the interaction matrices, $J_{ij}^{(x,x)}$ and $J_{ij}^{(y,y)}$. Each interaction value is plotted at the midpoint $|i+j|/2$ between involved sites. This collapses the $(i,j)$ plane while preserving physically relevant structure: translation invariance appears as horizontal arrangements and spatial decay of interactions manifests as a systematic decrease in magnitude as $|i-j|$ increases. In this representation, points at the same horizontal location correspond to interactions centered around the same center point, $|i+j|/2$ , while their vertical spread reflects how the interaction strength varies with separation. This enables quick visual identification of whether the system is homogeneous, long-ranged, or exhibits spatial modulation of couplings. In ( c) we show an effective site-dependent magnetic fields $B_i$, and $({\bf d})$ shows the learned dissipation rates for local and collective dephasing, as well as spontaneous emission. Error bars are obtained via $N_b=300$ bootstrap resamples of the measurement outcomes, which also yields the error covariance matrix $\Sigma$ (Eq. \ref{['eq:HDistribution']}), represented here as a correlation matrix$\hat{\Sigma}=\mathrm{diag}(\Sigma)^{-1/2}\,\Sigma\,\mathrm{diag}(\Sigma)^{-1/2}$ in ( e), showing only weak off-diagonal correlations. For a detailed discussion of these results see Sec. \ref{['subsec:ExpResultsIntegral']}.
• Figure 3: Step 2 of the BEQS protocol: Error bounds. ( a) Following Hamiltonian and Lindbladian Learning, our knowledge of the realized dynamics is described by an ensemble of Hamiltonians \ref{['eq:HDistribution']} (and Lindbladians \ref{['eq:statistical_model']}), parametrized by a random Gaussian variable, $\boldsymbol{g}$. While the mean learned Hamiltonian $H_{\rm learned}$ would realize the expected computation of an observable expectation value, $\expval{O_t}$, the uncertainty in $H'(\boldsymbol{g})$ ultimately propagates non-linearly into observable expectation values. ( b) In a BEQS we need to ensure that these uncertainties do not lead to large errors relative to the expected computation, $\expval{O_t}$. Here, two errors are of interest (see Sec. \ref{['sec:ErrorBoundsShort']}): the expected error, as the difference between $\expval{O_t}$, and the ensemble mean $\mathds{E}[\expval{O'_t(\boldsymbol{g})}]$, and the concentration of $\expval{O'_t(\boldsymbol{g})}$ around the ensemble mean, which can be controlled by a concentration inequality, effectively bounding the probability of large deviations. ( c) For the learned Hamiltonian and Lindbladian ensemble in Fig. \ref{['fig:Hlearning10']}, we can still classically simulate the dynamics of a state under the ensemble. Solid lines represent trajectories predicted by the mean Hamiltonian and Lindbladian. Shaded areas represent the $95\%$ prediction intervals estimated from empirical percentiles using $50$ samples from the Hamiltonian and Lindbaldian ensemble. Dashed lines represent the respective estimates of the distribution means. From these simulations one could estimate the expected error, Eq. \ref{['eq:ExpectedError']}, and a concentration bound, Eq. \ref{['eq:TailBound']}, including corresponding uncertainties though bootstrapping. For more details see Secs. \ref{['sec:ErrorBoundsShort']} and Appendix \ref{['sec:boundedError']}.
• Figure 4: Decay of interactions for $N=10$ ions. Average interaction $\bar{J}_{i,i+d}$ as a function of the inter-ion distance, $d$. The decay is expected to follow a power-law. However, one observes that the decay of interactions is much faster than predicted by the ratio of nearest-neighbor to next-nearest-neighbor interactions. A better model for the learned Hamiltonian seems to be an exponential decay for which this ratio better predicts the long-range couplings.
• Figure 5: Dephasing Lindbladian for $10$ ions. ( a) Choosing a general dephasing Lindbladian as an ansatz, we can resolve the structure of the dephasing matrix $\Gamma$ in Eq. \ref{['eq:general_dephasing']}. While perfect collective dephasing leads to a constant dephasing matrix of the form $\Gamma_{ij}=\Gamma_{\rm col}$, independent single-qubit dephasing yields a diagonal dephasing matrix $\Gamma_{ij}=\Gamma_{\rm loc}\delta_{ij}$. In the experiment we can recover the structure of the dephasing matrix and transform the resulting master equation to Lindblad form by diagonalizing the dephasing matrix $\Gamma$. ( b) We find a dominant dephasing process, with jump operator $L=\sum_{i}a_i\sigma^z_i\approx L_{\rm coll}/\sqrt{N}$, resembling a collective dephasing of the qubits. Other jump operators are close to single- or few-qubit dephasing at much smaller rates.