Suppose a finite group satisfies the following property: If you take two random elements, then with probability bigger than 5/8 they commute. Then this group is commutative. 

Starting from this well-known result, it is natural to ask: Do similar results hold for other laws (p-groups, nilpotent groups...)? Are there analogous results for infinite groups? Are there phenomena specific to the infinite setup? 

We will survey known and new results in this area. New results are joint with Gideon Amir, Maria Gerasimova and Gady Kozma.

Quantum computers are becoming a reality and current generations of machines are already well beyond the 50-qubit frontier. However, hardware imperfections still overwhelm these devices and it is generally believed the fault-tolerant, error-corrected systems will not be within reach in the near term: a single logical qubit needs to be encoded into potentially thousands of physical qubits which is prohibitive.
 
Due to limited resources, in the near term, hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage but these need to resort to quantum error mitigation techniques. I will explain the basic concepts and introduce hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage. These have the potential to solve real-world problems---including optimisation or ground-state search---but they suffer from a large number of circuit repetitions required to extract information from the quantum state. I will detail a range of application areas of randomised quantum circuits, such as quantum algorithms, classical shadows, and quantum error mitigation introducing recent results that help lower the barrier for practical quantum advantage.