We present a general procedure for constructing lattices of qubits with a Hamiltonian composed of nearest-neighbor two-body interactions such that the ground state encodes a cluster state. We give specific details for lattices in one, two, and three dimensions, investigating both periodic and fixed boundary conditions, as well as present a proof for the applicability of this procedure to any graph. We determine the energy gap of these systems, which is shown to be independent of the size of the lattice but dependent on the type of lattice (in particular, the coordination number), and investigate the scaling of this gap in terms of the coupling constants of the Hamiltonian. We provide a comparative analysis of the different lattice types with respect to their usefulness for measurement-based quantum computation.
Motzkin chain is a model of nearest-neighbor interacting quantum s = 1 spins with open boundary conditions. It is known that it has a unique ground state which can be viewed as a sum of Motzkin paths. We consider the case of periodic boundary conditions and provide several conjectures about structure of the ground state space and symmetries of the Hamiltonian. We conjecture that the ground state is degenerate and independent states are distinguished by eigenvalues of the third component of total spin operator. Each of these states can be described as a sum of paths, similar to the Motzkin paths. Moreover, there exist two operators commuting with the Hamiltonian, which play the roles of lowering and raising operators when acting at these states. We conjecture also that these operators generate a C -type Lie algebra, with rank equal to the number of sites. The symmetry algebra of the Hamiltonian is actually wider, and extended, besides the cyclic shift operator, by a central element contained in the third component of total spin operator.
It is known that integrable models associated to rational R matrices give rise to certain non-abelian symmetries known as Yangians. Analogously ‘boundary ’ symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay and Short from the field theory point of view, in the context of the principal chiral model on the half line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton non-preserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary non-local charges directly at the quantum level. The non-local charges are simply coproduct realizations of particular boundary quantum algebras called ‘boundary ’ or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the symmetry of the open spin chain, namely we show that a number of the boundary non-local charges are in fact conserved quantities.
It is well known that integrable models associated to rational R matrices give rise to certain non-Abelian symmetries known as Yangians. Analogously boundary symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay, and Short from the field theory point of view, in the context of the principal chiral model on the half-line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton nonpreserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary nonlocal charges directly at the quantum level. The nonlocal charges are simply coproduct realizations of particular boundary quantum algebras called boundary or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the exact symmetry of the open spin chain, namely we show that a number of the boundary nonlocal charges are in fact conserved quantities.
Searching for simple models that possess nontrivial controlling properties is one of the central tasks in the field of quantum technologies. In this work, we construct a quantum spin-
1/2 chain of finite size, termed as controllable spin wire (CSW), in which we have ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{z}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{z}$ (Ising) interactions with a transverse field in the bulk and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{x}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{z}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{z}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{z}$ couplings with a canted field on the boundaries. The Hamiltonians on the boundaries, dubbed as tuning Hamiltonians (TH's), bear the same form as the effective Hamiltonians emerging in the so-called ``quantum entanglement simulator'' that is originally proposed for mimicking infinite models. We show that tuning the TH's (parametrized by $\ensuremath{\alpha}$) can trigger nontrivial controlling of the bulk properties, including the degeneracy of energy/entanglement spectra, and the response to the magnetic field
hbulk in the bulk. A universal point dubbed as ${\ensuremath{\alpha}}^{s}$ emerges. For $\ensuremath{\alpha}>{\ensuremath{\alpha}}^{s}$, the ground-state diagram versus
hbulk consists of three ``phases,'' which are Ne\'eL and polarized phases, and an emergent pseudomagnet phase, distinguished by entanglement and magnetization. For $\ensuremath{\alpha}<{\ensuremath{\alpha}}^{s}$, the phase diagram changes completely, with no steplike behaviors to distinguish phases. Due to its controlling properties and simplicity, the CSW could potentially serve in future the experiments for developing quantum devices.