# Download Communications In Mathematical Physics - Volume 287 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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**Extra resources for Communications In Mathematical Physics - Volume 287**

**Example text**

By construction, a cycle of bubble rays Bθ1 , . . , Bθq with the same angles lands at the fixed point αa . 5. We have φ1 (z) = φ1 (w) if and only if z ∼r w. Proof. 7, only uni-accessible points can be identified. 4 the lemma now follows if at least one of z and w is bi-accessible. Hence we can assume that both z and w are either landing points of infinite bubble rays B1 , B2 ⊂ K ◦ ◦ , or that one of z and w or both lies on a uni-accessible point on the boundary of a bubble. Denote −β1 , −β2 the angles of the external rays landing at z and w respectively.

More concretely, it has long been known that it is NP-hard to find the ground state energy of some classical spin systems, such as a three-dimensional Ising model. Kitaev [KSV02] extended these results to the quantum realm, by defining the quantum analogue of NP, called QMA (for “quantum Merlin-Arthur”). QMA is thus, roughly speaking, the class of problems that can be efficiently checked on a quantum computer, provided we are given a “witness” quantum state related to the answer to the problem.

The gate performed at any given time cannot depend explicitly on the time (the number of gates already performed), but can depend on location. We will define the gates in terms of a variety of possible transition rules, and to remove any ambiguity, we will ensure that for a legal state of the system, only one transition rule will apply at any given time. For QMA-completeness, we need some additional properties ensuring that enough of the constraints are locally checkable. The problem of moving to one dimension is somewhat similar to a quantum cellular automaton in that the transition rules need to depend only on the local environment and not on some external time coordinate, but differs from a cellular automaton in a number of ways.