A polynomial-time dissipation-based quantum algorithm for solving the ground states of a class of classically hard Hamiltonians

In this work, we give a polynomial-time quantum algorithm for solving the ground states of a class of classically hard Hamiltonians. The mechanism of the exponential speedup that appeared in our algorithm comes from dissipation in open quantum systems. To utilize the dissipation, we introduce a new idea of treating vectorized density matrices as pure states, which we call the vectorization picture. By doing so, the Lindblad master equation (LME) becomes a Schr\"odinger equation with non-Hermitian Hamiltonian. The steady state of the LME, therefore, corresponds to the ground states of a special class of Hamiltonians. The runtime of the LME has no dependence on the overlap between the initial state and the ground state. For the input part, given a Hamiltonian, under plausible assumptions, we give a polynomial-time classical procedure to judge and solve whether there exists LME with the desired steady state. For the output part, we propose a novel measurement strategy to extract information about the ground state from the original steady density matrix. We show that the Hamiltonians that can be efficiently solved by our algorithms contain classically hard instances assuming $\text{P}\neq \text{BQP}$. We also discuss possible exponential complexity separations between our algorithm and previous quantum algorithms without using the vectorization picture.