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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 $\tilde{H}(S)$?

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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.

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  • $\begingroup$ can we say then that the input for an adiabatic algorithm running on an Ising model is the number of spins and the list of pairwise interaction energies? $\endgroup$ – Omar Shehab May 4 '13 at 14:32
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    $\begingroup$ yes, that should be sufficient. $\endgroup$ – Martin Schwarz May 4 '13 at 15:12
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    $\begingroup$ I think I understand it now. The time is dependent on the energy and vice versa. That's why we can't independently calculate the time or space (energy) complexity. I am not sure though whether time is dependent on the energy or also on the configuration of the energy. $\endgroup$ – Omar Shehab May 4 '13 at 15:15

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