Optimization of algorithmic errors in analog quantum simulations

Analog quantum simulation is emerging as a powerful tool for uncovering classically unreachable physics such as many-body real-time dynamics. A complete quantification of uncertainties is necessary in order to make precise predictions using simulations on modern-day devices. Therefore, the inherent physical limitations of the device on the parameters of the simulation must be understood. This paper examines the interplay of errors arising from simulation of approximate time evolution with those due to practical, real-world device constraints. These errors are studied in Heisenberg-type systems on analog quantum devices described by the Ising Hamiltonian. A general framework for quantifying these errors is introduced and applied to several proposed time evolution methods, including Trotter-like methods and Floquet-engineered constant-field approaches. The limitations placed on the accuracy of time evolution methods by current devices are discussed. Characterization of the scaling of coherent effects of different error sources provides a way to extend the presented Hamiltonian engineering methods to take advantage of forthcoming device capabilities.

Figure 1

The device Hamiltonian $H_{\text{dev}}$ can be used to emulate a target system of interest with Hamiltonian $H_{\text{tar}}$ using Trotter-like methods (top path) or continuous driving methods (bottom path). In the limit of perfect pulses (equivalently, infinite magnetic field), the dashed curves implement the target time evolution exactly. The solid curves represent the mappings that are approximate due to algorithmic errors. These errors increase the overall error when mapping between Hamiltonians due to idle errors and when implementing the time evolution operator $U\left(t\right)$ (Trotter errors). Algorithmic errors affect constant-field time evolution engineering methods as well. The constant-field method ${\mathcal{C}}_1$ and one of the Trotter methods ${\mathcal{S}}_1$ are labeled as resulting time evolution methods within the space of possible $U\left(t\right)$ operators.

Figure 2

Pulse sequences for generating Heisenberg evolution from $ZZ$ interactions, corresponding to Eqs. (6) and (8, 9, 10, 11). For each pulse sequence, the magnetic-field strength (in units of $\mathrm{\Omega }$) is given as a function of time (in units of $\epsilon =\frac{\pi }{2\mathrm{\Omega }}$). The length of each sequence $U$ is denoted by ${\tau }_U$. The idling times $t_i$ are shown in the graphics for each of the Trotter sequences.

Figure 3

Different leading-order error terms contributing to the error rate analysis of the four Trotter methods considered in this paper. The optimal scaling is given by the negative slope of each line at $k_1=1$ and the corresponding error rate scaling is given by the $k_1=1$ intercept.

Figure 4

Comparison of analytic bounds for the error rate for a single Trotter step from Eqs. (22) and (23) to numerical results as a function of the device parameter $\epsilon$ for $2\times 2$ systems. The analytic bounds consistently exceed the numerical results by factors of $O\left(1\right)–O\left(10\right)$, which is an expected consequence of using the triangle inequality to evaluate the bounds [63].

Figure 5

The effect of multiple Trotter steps on errors for $2\times 2$ systems with $\epsilon \approx {10}^{-5}$ and pulse sequences ${\mathcal{S}}_{1/2}$ and ${\mathcal{S}}_1$. The quantity ${\delta }_1-{\delta }_n=\left|\right|U_{\text{trot}}\left(T\right)-e^{-i{H}_{XXX}T}\left|\right|-\left|\right|{\prod }_n{U}_{\text{trot}}(T/n)-e^{-i{H}_{XXX}T}\left|\right|$ is positive when taking many small steps of $U_{\text{trot}}(T/n)$ replicates $e^{-i{H}_{XXX}T}$ better than taking one large step $U_{\text{trot}}\left(T\right)$. Here, $T=n{\tau }_U$ and ${\tau }_U$ is chosen optimally according to Eq. (21). Taking many steps to reach a total time $T$ is beneficial for ${\mathcal{S}}_{1/2}$ and ${\mathcal{S}}_1$ compared to taking a single large step when the step size is chosen optimally. This demonstrates the importance of choosing the step size correctly to take advantage of the Trotter error scaling.

Figure 6

Numerical error rates corresponding to Eq. (14) for a single Trotter step (Floquet period for ${\mathcal{C}}_1$) for $2\times 2$ systems. The slopes of lines on these log-log plots indicate the scaling of the error rate for each method (see Table 1) and the $y$ intercepts give the scaling prefactor. The lines for ${\mathcal{S}}_{1/2},{\mathcal{S}}_1$, and ${\mathcal{C}}_1$ are the same as the numerical ones in Fig. 4. Current device capabilities report pulse control to the ns level [76], meaning pulse sequences with $\epsilon \approx O\left({10}^{-5}\right)$ can be applied, indicated by the dashed vertical line.

Figure 7

Scaling of the contributions $\mathcal{E}$ to the error from different sources in a single Trotter step of the pulse sequences ${\mathcal{S}}_{1/2}$ and ${\mathcal{S}}_1$ as a function of system geometry. $N_y$ is varied on the $x$ axis and each line represents the contribution to the error with a different value of $N_x$. Here the analytic bounds of Eqs. (C19, C20, C21, C22) are compared to values obtained by numerically evaluating the norm of Eq. (15) for each type of contribution. Note that in the latter three panels, the $N_x=1,N_y=1$ points are much lower than the rest of the $N_x=1$ lines because of the fact that there are three-body contributions to the error for these terms; when the system has less than three sites, these contributions are absent, leading to a much smaller error.

Figure 8

$J_1,J_2,J_3$ sums evaluated for explicit array configurations with fits. The extent of the array in the $x$ direction is $N_x=20$ atoms. While there is a deviation for smaller array sizes, a linear scaling is still seen.