There are two schools of“measurement-only quantum computation”.The first(Phys.Rev.Lett.86(22),5188-5191(2001))using prepared entanglement(cluster states)and the second(Phys.Rev.Lett.101(1),010501(2008))using collect...There are two schools of“measurement-only quantum computation”.The first(Phys.Rev.Lett.86(22),5188-5191(2001))using prepared entanglement(cluster states)and the second(Phys.Rev.Lett.101(1),010501(2008))using collections of anyons which,according to how they were produced,also have an entanglement pat-tern.We abstract the common principle behind both approaches and find the notion of a graph or even continuous family of equiangular projections.This notion is the leading character in the paper.The largest continuous family,in a sense made pre-cise in Corollary 4.2,is associated with the octonions and this example leads to a universal computational scheme.Adiabatic quantum computation also fits into this rubric as a limiting case:nearby projections are nearly equiangular,so as a gapped ground state space is slowly varied,the corrections to unitarity are small.展开更多
We discuss a possible definition for“k-width”of a closed d-manifold Md,and on embedding Md e↪ℝn,n>d≥k,generalizing the classical notion of width of a knot.We show that for every 3-manifold 2-width(M3)≤2 but tha...We discuss a possible definition for“k-width”of a closed d-manifold Md,and on embedding Md e↪ℝn,n>d≥k,generalizing the classical notion of width of a knot.We show that for every 3-manifold 2-width(M3)≤2 but that there are embeddings ei∶T3↪ℝ4 with 2-width(ei)→∞.We explain how the divergence of 2-width of embeddings offers a tool to which might prove the Goeritz groups Gg infinitely generated for g≥4.Finally we construct a homomorphismg∶Gg→MCG(#g S2×S2),suggesting a potential application of 2-width to 4D mapping class groups.展开更多
文摘There are two schools of“measurement-only quantum computation”.The first(Phys.Rev.Lett.86(22),5188-5191(2001))using prepared entanglement(cluster states)and the second(Phys.Rev.Lett.101(1),010501(2008))using collections of anyons which,according to how they were produced,also have an entanglement pat-tern.We abstract the common principle behind both approaches and find the notion of a graph or even continuous family of equiangular projections.This notion is the leading character in the paper.The largest continuous family,in a sense made pre-cise in Corollary 4.2,is associated with the octonions and this example leads to a universal computational scheme.Adiabatic quantum computation also fits into this rubric as a limiting case:nearby projections are nearly equiangular,so as a gapped ground state space is slowly varied,the corrections to unitarity are small.
基金funded by the"Microsoft Research","Aspen Center for Physics"and"UCSB".
文摘We discuss a possible definition for“k-width”of a closed d-manifold Md,and on embedding Md e↪ℝn,n>d≥k,generalizing the classical notion of width of a knot.We show that for every 3-manifold 2-width(M3)≤2 but that there are embeddings ei∶T3↪ℝ4 with 2-width(ei)→∞.We explain how the divergence of 2-width of embeddings offers a tool to which might prove the Goeritz groups Gg infinitely generated for g≥4.Finally we construct a homomorphismg∶Gg→MCG(#g S2×S2),suggesting a potential application of 2-width to 4D mapping class groups.
