Local minima in quantum systems

Finding ground states of quantum many-body systems is known to be hard for both classical and quantum computers. Consequently, when a quantum system is cooled in a low-temperature thermal bath, the ground state cannot always be found efficiently. Instead, the system may become trapped in a local minimum of the energy. In this work, we study the problem of finding local minima in quantum systems under thermal perturbations. Although local minima are much easier to find than ground states, we show that finding a local minimum is hard on classical computers, even when the task is merely to output a single-qubit observable at any local minimum. By contrast, we prove that a quantum computer can always find a local minimum efficiently using a thermal gradient descent algorithm that mimics natural cooling processes. To establish the classical hardness of finding local minima, we construct a family of two-dimensional Hamiltonians such that any problem solvable by polynomial-time quantum algorithms can be reduced to finding local minima of these Hamiltonians. Therefore, cooling systems to local minima is universal for quantum computation and, assuming that quantum computation is more powerful than classical computation, finding local minima is classically hard but quantumly easy.