Predicting properties of large-scale quantum systems is crucial for the development of quantum science and technology. Shadow estimation is an efficient method for this task based on randomized measurements, where many-qubit random Clifford circuits are used for estimating global properties like quantum fidelity. Here we introduce the minimal Clifford measurement (MCM) to reduce the number of possible random circuits to the minimum, while keeping the effective post-processing channel in shadow estimation. In particular, we show that MCM requires 2^n+1 distinct Clifford circuits, and it can be realized by Mutually Unbiased Bases (MUB), with n as the total qubit number. By applying the Z-Tableau formalism, this ensemble of circuits can be synthesized to the -S-CZ-H- structure, which can be decomposed to 2n-1 fixed circuit modules, and the total circuit depth is at most n+1. Compared to the original Clifford measurements, our MCM significantly reduces the circuit complexity and the compilation costs. In addition, we find the sampling advantage of MCM on estimating off-diagonal operators, and extend this observation to the biased-MCM scheme to enhance the sampling improvement further.

Quantum cryptography is one of the most widely applied research fields in quantum information science. It includes two main branches, quantum key distribution and quantum random number generation. The central task in the studies of quantum cryptography is to prove the security regardless of the eavesdropper's computational power. In literature, the security analysis has been done both analytically and numerically. Compared to analytical methods which tend to require techniques specific to quantum cryptography protocols, numerical ones are more general since they can be directly applied to general protocols with little adaptation. However, current numerical methods are carried out based on some assumptions such as working in the asymptotic limit and collective attacks from eavesdroppers. In this talk, I will introduce how to remove these assumptions and develop a numerical finite-size security analysis framework against general attacks.