Everyone is welcome to attend the online QuSoft seminar with Aarthi Sundaram (Microsoft), with the title: Quantum learning algorithms imply circuit lower bounds.
Abstract: In this talk, I will discuss the first general connection established between the design of quantum algorithms and circuit lower bounds. Specifically, let C be a class of polynomial-size concepts, and suppose that C can be PAC-learned with membership queries under the uniform distribution with error 1/2−γ by a time T quantum algorithm. We prove that if γ^2⋅T≪ (2^n)/n, then BQE ⊈ C, where BQE is an exponential-time analogue of BQP. This result is optimal in both γ and T, since it is not hard to learn any class C of functions in (classical) time T=2^n (with no error), or in quantum time T=poly(n) with error at most 1/2−Ω(2{−n/2}) via Fourier sampling. In other words, even a marginal improvement on these generic learning algorithms would lead to major consequences in complexity theory.
Our proof builds on several works in learning theory, pseudorandomness, and computational complexity, and crucially, on a connection between non-trivial classical learning algorithms and circuit lower bounds established by Oliveira and Santhanam (CCC 2017). Extending their approach to quantum learning algorithms turns out to create significant challenges. To achieve that, we show among other results how pseudorandom generators imply learning-to-lower-bound connections in a generic fashion, construct the first conditional pseudorandom generator secure against uniform quantum computations, and extend the local list-decoding algorithm of Impagliazzo, Jaiswal, Kabanets, and Wigderson (SICOMP 2010) to quantum circuits via a delicate analysis. We believe that these contributions are of independent interest and might find other applications.
Reference: https://arxiv.org/abs/2012.01920
For the link, please contact: Subhasree.Patro@cwi.nl
