Depth analysis of variational quantum algorithms for the heat equation

Variational quantum algorithms are a promising tool for solving partial differential equations. The standard approach for its numerical solution are finite difference schemes, which can be reduced to the linear algebra problem. We consider three approaches to solve the heat equation on a quantum computer. Using the direct variational method we minimize the expectation value of a Hamiltonian with its ground state being the solution of the problem under study. Typically, an exponential number of Pauli products in the Hamiltonian decomposition does not allow for the quantum speed up to be achieved. The Hadamard test based approach solves this problem, however, the performed simulations do not evidently prove that the ansatz circuit has a polynomial depth with respect to the number of qubits. The ansatz tree approach exploits an explicit form of the matrix what makes it possible to achieve an advantage over classical algorithms. In our numerical simulations with up to $n=11$ qubits, this method reveals the exponential speed up.