Anisotropic Ferromagnetic Quantum Domains

Alcaraz, Francisco C.; Salinas, S. R.; Wreszinski, Walter F.

doi:10.1103/physrevlett.75.930

Cited by 72 publications

(135 citation statements)

References 9 publications

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“…This spin chain can be viewed as a q-analogue of the ferromagnetic Heisenberg chain; it has a remarkable SU q (2) quantum group symmetry which is a deformation of the SU(2) symmetry of the Heisenberg ferromagnet. Its spectral gap, groundspace [17] and excitations are known [17,18]. In the Supplementary Material we derive an expression for the zero energy groundstate of H XXZ (λ) in the sector with Hamming weight n. Using this expression and (4) we immediately obtain a spanning basis for the √ 1 − λ 2 energy groundspace of H string + H circuit (λ), given by (up to normalization)…”

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confidence: 99%

Universal Adiabatic Quantum Computation via the Space-Time Circuit-to-Hamiltonian Construction

2015

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We show how to perform universal adiabatic quantum computation using a Hamiltonian which describes a set of particles with local interactions on a two-dimensional grid. A single parameter in the Hamiltonian is adiabatically changed as a function of time to simulate the quantum circuit. We bound the eigenvalue gap above the unique groundstate by mapping our model onto the ferromagnetic XXZ chain with kink boundary conditions; the gap of this spin chain was computed exactly by Koma and Nachtergaele using its q-deformed version of SU(2) symmetry. We also discuss a related time-independent Hamiltonian which was shown by Janzing to be capable of universal computation. We observe that in the limit of large system size, the time evolution is equivalent to the exactly solvable quantum walk on Young's lattice. [3,4]. Although the class of universal Hamiltonians originally considered (nearest neighbor interactions between six dimensional particles in two dimensions) is not practically viable, perturbation gadget techniques [5,6] were later used to massage it into simpler forms [7,8]. However, these techniques have the disadvantage of requiring impractically high variability in the coupling strengths which appear in the Hamiltonian (see, e.g., the analysis in [9]). Given this state of affairs, it is of interest to consider how to construct a universal adiabatic quantum computer with a simple Hamiltonian without using perturbative gadgets.An alternative type of circuit-to-Hamiltonian mapping which is conceptually distinct from the Feynman-Kitaev construction has been used by some authors [10][11][12][13][14][15][16]. In these works a quantum circuit is mapped to a Hamiltonian which acts on a Hilbert space with computational and "local" clock degrees of freedom associated with every qubit in the circuit. This idea was first explored by Margolus in 1989 [10], just four years after Feynman's celebrated paper on Hamiltonian computation [3]. Margolus showed how to simulate a one-dimensional cellular automaton by Schrödinger time evolution with a time-independent Hamiltonian. More recently, Janzing [11] presented a scheme for universal computation with a time-independent Hamiltonian. In reference [14] it was claimed that an approach along these lines can be used to perform universal adiabatic quantum computation; unfortunately, the analysis presented by Mizel et al. does not establish the claimed results. The local clock idea was developed further in the recent "space-time circuitto-Hamiltonian construction" and was used to prove that approximating the ground energy of a certain class of interacting particle systems is QMA-complete [16].Our main result is a new method which achieves efficient universal adiabatic quantum computation using the space-time circuit-to-Hamiltonian construction. The Hamiltonian we use describes a simple system of interacting particles which live on the edges of a two dimensional grid. To prove that the resulting algorithm is efficient we use a mapping from our Hamiltonian to the ferromagnetic XXZ model...

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Universal Adiabatic Quantum Computation via the Space-Time Circuit-to-Hamiltonian Construction

2015

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“…In addition to its two ferromagnetically ordered, translation invariant ground states, this model has ground states corresponding to an interface between two domains of opposite magnetization. The stability of this interface was proved independently by Alcaraz, Salinas and Wreszinski [1] and Gottstein and Werner [15]. This stability is a direct consequence of the conservation of the total z-component of the spin.…”

Section: Introductionmentioning

confidence: 85%

“…In contrast, for the anisotropic, ferromagnetic XXZ chain, we prove that the dispersion relation is "flat" (i.e., k-independent) in the infinite length limit. This provides another approach to studying the stability of the interface in this model at zero temperature to complement the approaches of [1,15,4,3]. The XZ chain is exactly solvable, and Araki and Matsui used this to prove the absence of non-translationally invariant infinite volume ground states [2].…”

Section: Introductionmentioning

confidence: 99%

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Instability of Interfaces in the Antiferromagnetic XXZ Chain at Zero Temperature

Datta¹,

Kennedy²

2003

Communications in Mathematical Physics

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For the antiferromagnetic, highly anisotropic XZ and XXZ quantum spin chains, we impose periodic boundary conditions on chains with an odd number of sites to force an interface (or kink) into the chain. We prove that the energy of the interface depends on the momentum of the state. This shows that at zero temperature the interface in such chains is not stable. This is in contrast to the ferromagnetic XXZ chain for which the existence of localized interface ground states has been proven for any amount of anisotropy in the Ising-like regime.

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“…Different values of M correspond to different positions of the domain walls, which in one dimension are sometimes referred to as kinks. In [3] and [4] the ground states for this type boundary conditions were further analyzed and generalized to higher spin. A careful analysis of the Ising limit, see Section 4, reveals that for J ≥ 1 one or more low-lying excitations, each with a finite degeneracy, closely resemble the domain wall, i.e.…”

Section: Introductionmentioning

confidence: 99%

Isolated eigenvalues of the ferromagnetic spin-J XXZchain with kink boundary conditions

et al. 2008

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Abstract:We investigate the low-lying excited states of the spin J ferromagnetic XXZ chain with Ising anisotropy ∆ and kink boundary conditions. Since the third component of the total magnetization, M , is conserved, it is meaningful to study the spectrum for each fixed value of M . We prove that for J ≥ 3/2 the lowest excited eigenvalues are separated by a gap from the rest of the spectrum, uniformly in the length of the chain. In the thermodynamic limit, this means that there are a positive number of excitations above the ground state and below the essential spectrum.

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Anisotropic Ferromagnetic Quantum Domains

Cited by 72 publications

References 9 publications

Universal Adiabatic Quantum Computation via the Space-Time Circuit-to-Hamiltonian Construction

Universal Adiabatic Quantum Computation via the Space-Time Circuit-to-Hamiltonian Construction

Instability of Interfaces in the Antiferromagnetic XXZ Chain at Zero Temperature

Isolated eigenvalues of the ferromagnetic spin-J XXZchain with kink boundary conditions

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