Quantum communication complexity of linear regression

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.