Quantum Physics

We give the first instance-optimal sample complexity bounds for shadow tomography using few-copy measurements in the high-precision regime. More concretely, we study the problem of learning expectation values of a given set of observables of an unknown quantum state to precision $\epsilon$ in $L_p$-norm, using (possibly adaptive) measurements that act on one or a few copies at a time, and we are interested in the regime that $\epsilon$ is below some concrete and potentially dimension-dependent threshold. In this setup, we prove the necessary and sufficient number of copies, for any given set of observables, is characterized by a simple optimization formula involving a quadratic form of the inverse Fisher information matrix up to a logarithmic factor. Our results establish a rigorous correspondence between quantum learning and quantum metrology. 

I will discuss some surprising examples of quantum circuits that can be realized in constant T-depth. Some of these constructions, such as single-qubit rotation and its programmable variants, as well as  quantum part of Shor's factoring algorithm, require a catalyst state. But there are also other constructions that do not, such as reversible encoded addition.