Dequantized algorithms show that quantum computers do not have exponential
speedups for many linear algebra problems in terms of time and query
complexity. In this work, we show that quantum computers can have exponential
speedups in terms of communication complexity for some fundamental linear
algebra problems. We mainly focus on solving linear regression and Hamiltonian
simulation. In the quantum case, the task is to prepare the quantum state of
the result. To allow for a fair comparison, in the classical case the task is
to sample from the result. We investigate these two problems in two-party and
multiparty models, propose near-optimal quantum protocols and prove
quantum/classical lower bounds. In this process, we propose an efficient
quantum protocol for quantum singular value transformation, which is a powerful
technique for designing quantum algorithms. As a result, for many linear
algebra problems where quantum computers lose exponential speedups in terms of
time and query complexity, it is possible to have exponential speedups in terms
of communication complexity.Comment: 28 page