Fig. 1: Schematic of the working principle of MEADD.

a Standard gate repetition amplifies the matrix elements to estimate but also amplifies other coherent and incoherent (noise) processes. b Making use of single-qubit dynamical decoupling sequences, MEADD filters out specific matrix elements while additionally echoing out the noisy time-varying single-qubit phases. The parameter we are estimating—together with other matrix elements commuting with the DD gates—then builds up linearly with depth and can be fitted from the slope, with Heisenberg scaling in precision with circuit depth. All MEADD circuits possess this simple structure, with a preparation and pre-measurement layer containing at most a single entangling operation together with at most four single-qubit gates, and terminated by standard single-qubit Z-basis measurement. c Fluctuations in a noisy parameter γ (red), representing single-qubit phases, for example, can overshadow the estimation of a stable parameter ϕ (black) in a simple gate repetition scheme that provides estimates depending on both of these parameters (purple). Note that a separate and necessarily noisy estimation of γ is therefore required to find ϕ. In contrast, MEADD provides a direct estimate of the stable parameter ϕ as illustrated at the bottom, thus yielding more precise and accurate characterization information.