How to learn a quantum state (and how not to)

How to learn a quantum state (and how not to)

Seminar author:Yihui Quek

Event date and time:11/29/2021 10:00:am

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Learning an unknown n-qubit quantum state is a fundamental challenge in quantum computing. Full tomography, however, requires exponential-in-n many copies of \rho in order to estimate it up to small trace distance. We consider two variants of this problem:

1. “Pretty-good tomography” (based on https://arxiv.org/abs/2102.07171, to appear at NeurIPS 2021 (Spotlight talk)): Motivated by computational learning theory, Aaronson and others introduced several “reduced” models of learning quantum states which impose weaker requirements on the learner: PAC-learning, shadow tomography for learning “shadows” of a quantum state, online learning, whose complexities scale only linearly in n. We show that many of these models imply each other and are characterised by a combinatorial parameter, the sequential fat-shattering dimension of quantum states. As an application, we improve shadow tomography (for classes of quantum states).
2. Probabilistic modelling (based on https://arxiv.org/abs/2110.05517): Consider now states generated at the output of quantum circuits of only local gates. By measuring such circuits in the computational basis, can we learn an algorithm that generates more such samples? (Notice this is a weaker requirement than learning the entire state.) More importantly, is there a quantum advantage for such a task? This question is at the heart of several near-term algorithms, such as Quantum Circuit Born Machines. We prove both a no-go result and a go result for this setting.