I'm standing in front of a quantum computer built out of atoms and light at the UK's National Quantum Computing Centre on the outskirts of Oxford. On a laboratory table, a complex matrix of mirrors and lenses surrounds a Rubik's Cube-size cell where 100 cesium atoms are suspended in grid formation by a carefully manipulated laser beam. The cesium atom setup is so compact that I could pick it up, carry it out of the lab, and put it on the backseat of my car to take home. I'd be unlikely to get very far, though.

We consider the problem of minimizing a continuous function given given access to a natural quantum generalization of a stochastic gradient oracle. We provide two new methods for the special case of minimizing a Lipschitz convex function. Each method obtains a dimension versus accuracy trade-off which is provably unachievable classically and we prove that one method is asymptotically optimal in low-dimensional settings. Additionally, we provide quantum algorithms for computing a critical point of a smooth non-convex function at rates not known to be achievable classically. To obtain these results we build upon the quantum multivariate mean estimation result of Cornelissen et al. [25] and provide a general quantum variance reduction technique of independent interest.

Among them, quantum machine learning is one of the most exciting applications of quantum computers.

Our code is available at https://github.com/martaskrt/qdeq.

See Figure 1 for an illustration of the different SNR thresholds and Section 2.1 for more

Given a convex function f: Rd R, the problem of sampling from a distribution e f(x) is called log-concave sampling. This task has wide applications in machine learning, physics, statistics, etc.