13 research outputs found
Enumerative Combinatorics
Enumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds
Products of independent Gaussian random matrices
Ipsen JR. Products of independent Gaussian random matrices. Bielefeld: Bielefeld University; 2015
Optimal learning of quantum Hamiltonians from high-temperature Gibbs states
We study the problem of learning a Hamiltonian to precision
, supposing we are given copies of its Gibbs state
at a known inverse
temperature . Anshu, Arunachalam, Kuwahara, and Soleimanifar (Nature
Physics, 2021, arXiv:2004.07266) recently studied the sample complexity (number
of copies of needed) of this problem for geometrically local -qubit
Hamiltonians. In the high-temperature (low ) regime, their algorithm has
sample complexity poly and can be implemented with
polynomial, but suboptimal, time complexity.
In this paper, we study the same question for a more general class of
Hamiltonians. We show how to learn the coefficients of a Hamiltonian to error
with sample complexity and
time complexity linear in the sample size, . Furthermore, we prove a
matching lower bound showing that our algorithm's sample complexity is optimal,
and hence our time complexity is also optimal.
In the appendix, we show that virtually the same algorithm can be used to
learn from a real-time evolution unitary in a small regime
with similar sample and time complexity.Comment: 59 pages, v2: incorporated reviewer comments, improved exposition of
appendi