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This talk presents a quantum algorithm for performing persistent homology, the identification of topological features of data sets such as connected components, holes and voids. Finding the full persistent homology of a data set over n points using classical algorithms takes time O(2^{2n}), while the quantum algorithm takes time O(n^2), an exponential improvement. The quantum algorithm does not require a quantum random access memory and is suitable for implementation on small quantum computers with a few hundred qubits.

In a fair comparison of the performance of a quantum algorithm to a classical one it is important to treat them on equal footing, both regarding resource usage and parallelism. We show how one may otherwise mistakenly attribute speedup due to parallelism as quantum speedup. As an illustration we will go through a few quantum machine learning algorithms, e.g. Quantum Page Rank, and show how a classical parallel computer can solve these problems faster with the same amount of resources. Our classical parallelism considerations are especially important for quantum machine learning algorithms, which either use QRAM, allow for unbounded fanout, or require an all-to-all communication network.

The introduction of neural networks with deep architecture has led to a revolution, giving rise to a new wave of technologies empowering our modern society. Although data science has been the main focus, the idea of generic algorithms which automatically extract features and representations from raw data is quite general and applicable in multiple scenarios. Motivated by the effectiveness of deep learning algorithms in revealing complex patterns and structures underlying data, we are interested in exploiting such tool in the context of many-body physics. I will first introduce the Boltzmann Machine, a stochastic neural network that has been extensively used in the layers of deep architectures. I will describe how such network can be used for modelling thermodynamic observables for physical systems in thermal equilibrium, and show that it can faithfully reproduce observables for the 2 dimensional Ising model. Finally, I will discuss how to adapt the same network for implementing the classical computation required to perform quantum error correction in the 2D toric code.

Density functional theory (DFT) is an extremely popular approach to electronic structure problems in both materials science and chemistry and many other fields. Over the past several years, often in collaboration with Klaus Mueller at TU Berlin, we have explored using machine-learning to find the density functionals that must be approximated in DFT calculations. I will summarize our results so far, and report on two new works.