Analytically Continuing the Randomized Measurement Toolbox

Abstract

We develop a framework for extracting non-polynomial analytic functions of density matrices in randomized measurement experiments by a method of analytical continuation. A central advantage of this approach, dubbed stabilized analytic continuation (SAC), is its robustness to statistical noise arising from finite repetitions of a quantum experiment, making it well-suited to realistic quantum hardware. As a demonstration, we use SAC to estimate the von Neumann entanglement entropy of a numerically simulated quenched Néel state from Rényi entropies estimated via the randomized measurement protocol. We then apply the method to experimental Rényi data from a trapped-ion quantum simulator, extracting subsystem von Neumann entropies at different evolution times. Finally, we briefly note that the SAC framework is readily generalizable to obtain other nonlinear diagnostics, such as the logarithmic negativity and Rényi relative entropies.

Figures (3)

• Figure 1: SAC and traditional fitting methods vermersch2024enhancedChebyshevBook estimating the half-chain von Neumann entropy in the ground state of the transverse-field Ising model $H=-J\sum_i\sigma_i^z\sigma_{i+1}^z-h\sum_i\sigma_i^x$ away from criticality ($J=1$, $h=0.5$, 15 sites). Error percentage of different methods are shown as a function of the largest Rényi index in the dataset. Left: for noiseless Rényi inputs. Right: for inputs with 10% independent Gaussian noise, averaged over 200 realizations.
• Figure 2: Numerical benchmarks using simulated data for a 10-qubit Néel state quenched under Hamiltonian \ref{['eq:XY Hamiltonian']} with decoherence included. Left: von Neumann entropy estimates from different methods at fixed time, $t=5$ms, and subsystem size, $L=5$ sites. Center: von Neumann entropy versus time for $L= 5$. Right: vN entropy versus subsystem size at $t = 5$ms.
• Figure 3: (a)(b): $S_{vN}$ as a function of time $t$ and subsystem size $L$ obtained via SAC in a trapped-ion experiment with ten $^{40}\text{Ca}^+$ ions, initialized in an approximate Néel state and evolved under Hamiltonian \ref{['eq:XY Hamiltonian']}. For comparison, $S_{vN}$ obtained via SAC using simulation data, as well as Rényi data from simulated randomized measurements, are plotted in (c)(d). In (a)(c), $L = 5$. In (b)(d), $t = 5$ms.