FET: Small: Foundations of Quantum State Learning and Testing

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

The goal of this project is to advance our scientific understanding of how to most efficiently learn or test the state of a quantum particle system. Whenever researchers/engineers build a quantum device -- say, for a quantum teleportation experiment, or as a component of a quantum computer -- they need to check whether it works as intended. That is, they need to be able to determine the quantum state of the particles it produces. Given enough samples from the device, it is possible to determine the quantum state to any desired accuracy. But due to the expensive nature of quantum experiments, it is important to seek highly efficient strategies for estimating the unknown state, ones that use as few samples as possible. The overarching goal of the project is to try to mathematically determine the optimal samples vs. accuracy tradeoff for problems such as: a) learning an unknown quantum state; b) testing whether an unknown state is equal to a desired target state, etc. Success in the project will tell us the efficiency benchmark against which all practical state-learning methodologies can be judged. Additionally, the project contains a significant educational component involving undergraduate and graduate students in the research effort, and the insights gained will also filter into a free online quantum computing course curriculum. At a more technical level, the project seeks to understand the optimal sample complexity of many basic problems in quantum state tomography and estimation. Proposed problems include: a) estimating a d-dimensional state's eigenvalues using o(d^2) samples; b) quantum tomography and state certification with respect to more stringent measures of accuracy, such as chi-squared divergence; c) testing whether a d x d quantum state is separable (unentangled) or far from separable using o(d^4) samples; d) performing 'm-measurement shadow tomography' with a number of samples linear in log(m) and polylogarithmic in the dimension; and e) improving the computational efficiency of known sample-optimal algorithms. For many of these problems, the investigator plans to use and extend the representation theory-based quantum learning framework developed recently with other researchers. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.