Fig. 1: Demonstration of our protocol.

The connected blue balls are for the n-qubit quantum state, and the boxes are for the random local unitary. The diagrams represent the order of operations. A Demonstration of few-shot randomized measurements (FSRM). In FSRM, the protocol consists of N_U measurement rounds. In the i-th round, the measurement basis is rotated by a randomly chosen unitary, resulting in \({U}_{i}^{\dagger }{\mathbb{M}}{U}_{i}\), where \({\mathbb{M}}\) is a fixed measurement. Unlike standard RM or CS methods, only N_M = k outcomes are collected per round, denoted by b_i,1,...,b_i,k, which are then used to get a real number O(b_i,1,...,b_i,k) from some function O. The average of the real numbers is the estimation of the target quantity \({{\mathcal{T}}}_{k}(\rho )\). In particular, choosing k = 2 or 3 allows estimation of the mixed-state entanglement moments p₂ and p₃. Importantly, FSRM does not require recording the applied unitaries, relying solely on the few collected outcomes to obtain an unbiased estimation. B Demonstration of BM-enhanced FSRM. We slice the state into pairs of qubits. After the random unitary rotation shown in (A), all pairs undergo a Bell-basis measurement (BM) or a computational-basis measurement (CM) with some probability, i.e., \({\mathbb{M}}\in \{{\rm{CM,\; BM}}\}\). For BM, we record bases with labels 0, 1, 2, 3, and for CM, we record bases with labels 00, 01, 10, 11 for post-processing. The overall measurement basis can be represented by an n-bit string \(\overrightarrow{s}\). These recorded bases, with the measurement shot outcomes, can give an unbiased estimation following the process in (A).