 
Summary: The Learnability of Quantum States
Scott Aaronson
MIT
Abstract
Traditional quantum state tomography requires a number of measurements that grows exponentially
with the number of qubits n. But using ideas from computational learning theory, we show that one
can do exponentially better in a statistical setting. In particular, to predict the outcomes of most
measurements drawn from an arbitrary probability distribution, one only needs a number of sample
measurements that grows linearly with n. This theorem has the conceptual implication that quantum
states, despite being exponentially long vectors, are nevertheless "reasonable" in a learning theory sense.
The theorem also has two applications to quantum computing: first, a new simulation of quantum one
way communication protocols, and second, the use of trusted classical advice to verify untrusted quantum
advice.
1 Introduction
Suppose we have a physical process that produces a quantum state. By applying the process repeatedly,
we can prepare as many copies of the state as we want, and can then measure each copy in a basis of our
choice. The goal is to learn an approximate description of the state by combining the various measurement
outcomes.
This problem is called quantum state tomography, and it is already an important task in experimental
physics. To give some examples, tomography has been used to obtain a detailed picture of a chemical
