Why spectral norms are used for computing the complexity of adiabatic Hamiltonian?

Question

In the context of adiabatic quantum computation the spectral norm was first used in the first adiabatic paper by Farhi et. al. when he demonstrated the relation of it to the conventional quantum computation. He showed that $$\Delta || H_P - H_B || << 1 $$. Later on van Dam et. al. also used spectral norm of Hamiltonian for computing the complexity. Daniel Nagaj also commented in chapter 2 of his PhD thesis that

It is thus usual to think the required resources of an AQC algorithm as $T.||H|| ...$

Why do we need spectral norm for computing adiabatic Hamiltonian complexity? Why can't we just express it in terms of the eigenvalue of $\tilde{H}(S)$?

Answer 1

The spectral norm $||H||$ determines the maximum energy involved in driving the evolution of the quantum system and thus the quantum computation. Any quantum evolution could be sped-up by a factor of $\lambda$ by simply driving the system with Hamiltonian $\lambda H$ as a simple consequence of Schrödinger's equation. The energy involved is simply another physical resource invested into solving the problem, that needs to be considered in complexity considerations, besides time. For example, if your Hamiltonian has a tiny spectral gap, you could of course enlarge it by simply rescaling the strengths of all interactions in your system until the gap is large enough in absolute terms. Of course, after having solved the engineering challenges involved, this will linearly accelerate your computationion, similar to overclocking your classical PC. But the speed gained was only achieved by investing another resource that needs to be kept track of.