Gaussian tomography for cold-atom simulators

TL;DR

This work tackles the limitation that cold-atom readouts are typically restricted to densities by introducing a Gaussian tomography approach that uses non-interacting quenches to randomize measurement bases and recover charge-off-diagonal correlations. It offers two schemes: a local variant with short evolution for bilinear observables and a global variant that expands a 1D chain into 2D to access general correlations, both relying on ensembles of non-interacting unitary evolutions and a classical post-processing step with an optimal inverse. Numerical results show that local currents in 1D can be estimated with a few thousand samples to percent-level accuracy, and all non-local correlations in moderate-size systems with around ten thousand samples, indicating practical feasibility. The method requires only turning off interactions, density measurements, and a controllable quasiperiodic potential, making it readily implementable on existing platforms and enabling precision measurements beyond particle-number densities.

Abstract

A limitation of analog quantum simulators based on cold atoms in optical lattices is that readout is typically limited to observables diagonal in the charge basis, i.e., densities and density correlation functions. To overcome this limitation, we propose experiment-friendly schemes to measure charge-off-diagonal correlations (such as currents). Our protocols use non-interacting dynamics for random times followed by standard quantum gas microscope measurements to effectively measure in random bases. The main requirement of our scheme is the ability to turn off interactions, which can be done in many atomic species using Feshbach resonances. Importantly, our scheme requires no local control and otherwise also exhibits modest requirements in terms of total evolution time and number of repetitions. We numerically demonstrate efficient estimation of bilinear correlation functions, requiring less than $4000$ samples to measure local currents to 5% error (system-size independent) and $\sim 10^4$ samples to simultaneously measure all non-local correlations in 70-site systems. Due to its simplicity, our protocol is implementable in existing platforms and thus paves the way to precision measurements beyond particle number measurements.

Figures (2)

• Figure 1: We display the sample complexity for various different cases (estimating worst, average and specific observables) and target accuracy of $\varepsilon=0.05$ (left y-axis) as well as the value of the variances (right y-axis). Left: We show the expected performance for estimating the worst-case local observable (blue) and the middle current (green (1d) / black (2d)) in 1d and 2d lattices. In both cases we set $h_{\rm max}=6$, $\theta_{\rm L}=2/3$ and singular value truncation at $\delta=10^{-3}$. In the 1d setting, we used the hyper-parameter settings of $t_{\rm max}=5$ and $S=400$. In the 2d setting, we estimated a single current observable with the hyper-parameter settings of $t_{\rm max}=1.5$ and $S=1000$. For building the inverse for systems with greater than $N=36$, we used the measurement-site truncation method described in \ref{['app:truncation']} with $\ell_{\rm out}=2$ to reduce the computational demand. Right: Similar to the left panel but targeting global observables using $N^2+10N$ ancilla sites. We plot the average case sample complexity (orange) as well as the current in the middle (green) and a long range current where the distance is half the linear length of the one dimensional string (light blue). The inset shows the expected sample complexity for the worst case global observable. We do not randomize the phase or the strength of the laser and fix them at $\phi=0$ and $h=1$, respectively. Similarly, the laser angle and the evolution time are fixed at $\theta_{\rm L}\approx0.798$ and $t=0.75N$. The fit lines for the average- and worst-case follow asymptotic power law scaling with exponents $\alpha=1.88$ and $\alpha=2.88$, respectively.
• Figure 2: This shows the maximum eigenvalue of $G^{\rm opt}F_{\rm err}-\mathbb{I}$ for various different values of error $\nu$ for a perturbed forward map $F$ to get an error map $F_{\rm err}$ whereby the Hamiltonian has some random potential that has values sampled according to a standard distribution centered at zero with variance $\nu$. In both cases, we show the maximum across 50 random simulated error perturbations. Left: Local correlations estimation in a 1d chain without ancillas. Right: Global correlations estimation in a 1d chain with ancillas. Here, various combinations of the strength $h\in [0,5]$ and phase $\phi\in[0,\pi/2]$ of the Hamiltonian were tested with the best performing parameter combination selected for display.