On the average-case complexity of learning output distributions of quantum circuits

05/09/2023
by   Alexander Nietner, et al.
0

In this work, we show that learning the output distributions of brickwork random quantum circuits is average-case hard in the statistical query model. This learning model is widely used as an abstract computational model for most generic learning algorithms. In particular, for brickwork random quantum circuits on n qubits of depth d, we show three main results: - At super logarithmic circuit depth d=ω(log(n)), any learning algorithm requires super polynomially many queries to achieve a constant probability of success over the randomly drawn instance. - There exists a d=O(n), such that any learning algorithm requires Ω(2^n) queries to achieve a O(2^-n) probability of success over the randomly drawn instance. - At infinite circuit depth d→∞, any learning algorithm requires 2^2^Ω(n) many queries to achieve a 2^-2^Ω(n) probability of success over the randomly drawn instance. As an auxiliary result of independent interest, we show that the output distribution of a brickwork random quantum circuit is constantly far from any fixed distribution in total variation distance with probability 1-O(2^-n), which confirms a variant of a conjecture by Aaronson and Chen.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset