  A large number of problems in science can be solved by preparing a specific
eigenstate of some Hamiltonian H. The generic cost of quantum algorithms for
these problems is determined by the inverse spectral gap of H for that
eigenstate and the cost of evolving with H for some fixed time. The goal of
spectral gap amplification is to construct a Hamiltonian H' with the same
eigenstate as H but a bigger spectral gap, requiring that constant-time
evolutions with H' and H are implemented with nearly the same cost. We show
that a quadratic spectral gap amplification is possible when H satisfies a
frustration-free property and give H' for these cases. This results in quantum
speedups for optimization problems. It also yields improved constructions for
adiabatic simulations of quantum circuits and for the preparation of projected
entangled pair states (PEPS), which play an important role in quantum many-body
physics. Defining a suitable black-box model, we establish that the quadratic
amplification is optimal for frustration-free Hamiltonians and that no spectral
gap amplification is possible, in general, if the frustration-free property is
removed. A corollary is that finding a similarity transformation between a
stoquastic Hamiltonian and the corresponding stochastic matrix is hard in the
black-box model, setting limits to the power of some classical methods that
simulate quantum adiabatic evolutions.
