



01044nas a2200133   4500008004100000245005300041210005000094260001500144520066000159100001700819700001800836700001900854856003700873       2016                        eng d00aA finite presentation of CNOT-dihedral operators0 afinite presentation of CNOTdihedral operators  c2016/12/313 a<p>We give a finite presentation by generators and relations of unitary operators expressible over the {CNOT, T, X} gate set, also known as CNOT-dihedral operators. To this end, we introduce a notion of normal form for CNOT-dihedral circuits and prove that every CNOT-dihedral operator admits a unique normal form. Moreover, we show that in the presence of certain structural rules only finitely many circuit identities are required to reduce an arbitrary CNOT-dihedral circuit to its normal form. By appropriately restricting our relations, we obtain a finite presentation of unitary operators expressible over the {CNOT, T } gate set as a corollary.</p>1 aAmy, Matthew1 aChen, Jianxin1 aRoss, Neil, J.  uhttps://arxiv.org/abs/1701.0014001702nas a2200157   4500008004100000245004500041210004500086260001400131490000700145520128800152100001801440700001801458700001401476700001701490856003701507       2012                        eng d00aFrom Ground States to Local Hamiltonians0 aFrom Ground States to Local Hamiltonians  c2012/8/300 v863 a  Traditional quantum physics solves ground states for a given Hamiltonian,
while quantum information science asks for the existence and construction of
certain Hamiltonians for given ground states. In practical situations, one
would be mainly interested in local Hamiltonians with certain interaction
patterns, such as nearest neighbour interactions on some type of lattices. A
necessary condition for a space $V$ to be the ground-state space of some local
Hamiltonian with a given interaction pattern, is that the maximally mixed state
supported on $V$ is uniquely determined by its reduced density matrices
associated with the given pattern, based on the principle of maximum entropy.
However, it is unclear whether this condition is in general also sufficient. We
examine the situations for the existence of such a local Hamiltonian to have
$V$ satisfying the necessary condition mentioned above as its ground-state
space, by linking to faces of the convex body of the local reduced states. We
further discuss some methods for constructing the corresponding local
Hamiltonians with given interaction patterns, mainly from physical points of
view, including constructions related to perturbation methods, local
frustration-free Hamiltonians, as well as thermodynamical ensembles.
1 aChen, Jianxin1 aJi, Zhengfeng1 aZeng, Bei1 aZhou, D., L.  uhttp://arxiv.org/abs/1110.6583v4