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2. However, it does not appear to be possible to make the translational invariant adiabatic construction nondegenerate. 2. Results related to QMA-completeness. Theorem 1 means that efficient simulation of general one-dimensional adiabatic quantum systems is probably impossible. One might expect that calculating, or at least approximating, some specific property of a system, such as its ground state energy, would be more straightforward, as this does not require complete knowledge of the system.
Gottesman, S. Irani, J. Kempe surprising, but in retrospect, we can provide an intuitive explanation. The reason is that the k-local Hamiltonian essentially allows us to encode an extra dimension, namely, time, by making the ground state a superposition of states corresponding to different times. In other words, the result implies that the correct analogue of one-dimensional local Hamiltonian is two-dimensional MAX-k-SAT, which is of course NP-complete. Indeed, there are many cases in physics where one-dimensional quantum systems turn out to be most closely analogous to two-dimensional classical systems.
The main result of [AvDK+ 04], namely the universality of adiabatic computation, was based on the observation that the circuit-to-Hamiltonian construction is useful also in the adiabatic context, where it is used The Power of Quantum Systems on a Line 45 to design the final Hamiltonian of the adiabatic evolution. It turns out that the same connection between adiabatic universality proofs and QMA-completeness proofs can also be made for many of the follow-up papers on the two topics. It is often the case that whenever adiabatic universality can be proven for some class of Hamiltonians, then the local Hamiltonian problem with the same class (roughly) can be shown to be QMA-complete — and vice versa.