Low-temperature excitations within the Bethe approximation

Biazzo, Indaco; Ramezanpour, Abolfazl

doi:10.1088/1742-5468/2013/04/p04011

Cited by 6 publications

(7 citation statements)

References 22 publications

(26 reference statements)

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“…. , s N , with occupation numbers s i ∈ {0, 1} that show the absence (s i = 0) or presence (s i = 1) of a local "excitation" at site i [6,24]. Let us represent the above states by |s = σ ψ(s; σ)|σ .…”

Section: Product Wave Functionsmentioning

confidence: 99%

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Characterizing the parent Hamiltonians for a complete set of orthogonal wave functions: An inverse quantum problem

Ramezanpour

2016

Phys. Rev. A

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We study the inverse problem of constructing an appropriate Hamiltonian from a physically reasonable set of orthogonal wave functions for a quantum spin system. Usually, we are given a local Hamiltonian and our goal is to characterize the relevant wave functions and energies (the spectrum) of the system. Here, we take the opposite approach; starting from a reasonable collection of orthogonal wave functions, we try to characterize the associated parent Hamiltonians, to see how the wave functions and the energy values affect the structure of the parent Hamiltonian.Specifically, we obtain (quasi) local Hamiltonians by a complete set of (multilayer) product states and a local mapping of the energy values to the wave functions. On the other hand, a complete set of tree wave functions (having a tree structure) results to nonlocal Hamiltonians and operators which flip simultaneously all the spins in a single branch of the tree graph. We observe that even for a given set of basis states, the energy spectrum can significantly change the nature of interactions in the Hamiltonian. These effects can be exploited in a quantum engineering problem optimizing an objective functional of the Hamiltonian.

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Section: Product Wave Functionsmentioning

confidence: 99%

“…The Λ i and the complex couplings K ij = K R ij + îK I ij define the parameters P(0). From the above state, we obtain a set of orthonormal tree states |s , with s ij ∈ {0, 1} to show the absence or presence of a local "excitation" at edge (ij) [6,24]. The dependence on s enters only in the parameters…”

Section: Symmetric Tree Wave Functionsmentioning

confidence: 99%

Characterizing the parent Hamiltonians for a complete set of orthogonal wave functions: An inverse quantum problem

Ramezanpour

2016

Phys. Rev. A

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“…In this work, we take a variational approach extending the variational quantum cavity method of Refs. [15,18,19] to finite temperature systems; see also [20] and the extension of matrix product states to finite temperatures [21][22][23]. To this end, we first propose an approximate expression for the matrix elements of the density matrix in terms of the matrix elements of a locally consistent set of reduced density matrices.…”

Section: Introductionmentioning

confidence: 99%

Bethe free-energy approximations for disordered quantum systems

Biazzo

Ramezanpour

2014

Phys. Rev. E

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Given a locally consistent set of reduced density matrices, we construct approximate density matrices which are globally consistent with the local density matrices we started from when the trial density matrix has a tree structure. We employ the cavity method of statistical physics to find the optimal density matrix representation by slowly decreasing the temperature in an annealing algorithm, or by minimizing an approximate Bethe free energy depending on the reduced density matrices and some cavity messages originated from the Bethe approximation of the entropy. We obtain the classical Bethe expression for the entropy within a naive (mean-field) approximation of the cavity messages, which is expected to work well at high temperatures. In the next order of the approximation, we obtain another expression for the Bethe entropy depending only on the diagonal elements of the reduced density matrices. In principle, we can improve the entropy approximation by considering more accurate cavity messages in the Bethe approximation of the entropy. We compare the annealing algorithm and the naive approximation of the Bethe entropy with exact and approximate numerical simulations for small and large samples of the random transverse Ising model on random regular graphs.

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“…Moreover, in both the cases we can easily construct an orthonormal set of locally excited states that could be useful in the framework of the coupled cluster method [24]. We remark that the weighted graph states studied in quantum physics and information theory can be represented by application of some two-body unitary operators on initially mean-field states [25].…”

Section: Introductionmentioning

confidence: 99%

“…Symmetric wave functions with a tree structure provide us with another category of computationally tractable states which somehow complement the mean-field states; while the latter wave functions are good candidates for the state of the system in the ordered (ferromagnetic, or localized) phase, the symmetric states are more appropriate in the disordered (paramagnetic, or extended) phase. Moreover, in both the cases we can easily construct an orthonormal set of locally excited states that could be useful in the framework of the coupled cluster method [24]. We remark that the weighted graph states studied in quantum physics and information theory can be represented by application of some two-body unitary operators on initially mean-field states [25].…”

Section: Introductionmentioning

confidence: 99%

Multilayer wave functions: A recursive coupling of local excitations

Ramezanpour¹

2013

EPL

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Finding a succinct representation to describe the ground state of a disordered interacting system could be very helpful in understanding the interplay between the interactions that is manifested in a quantum phase transition. In this work we use some elementary states to construct recursively an ansatz of multilayer wave functions, where in each step the higher-level wave function is represented by a superposition of the locally "excited states" obtained from the lower-level wave function. This allows us to write the Hamiltonian expectation in terms of some local functions of the variational parameters, and employ an efficient message-passing algorithm to find the optimal parameters.We obtain good estimations of the ground-state energy and the phase transition point for the transverse Ising model with a few layers of mean-field and symmetric tree states. The work is the first step towards the application of local and distributed message-passing algorithms in the study of structured variational problems in finite dimensions. *

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Low-temperature excitations within the Bethe approximation

Cited by 6 publications

References 22 publications

Characterizing the parent Hamiltonians for a complete set of orthogonal wave functions: An inverse quantum problem

Characterizing the parent Hamiltonians for a complete set of orthogonal wave functions: An inverse quantum problem

Bethe free-energy approximations for disordered quantum systems

Multilayer wave functions: A recursive coupling of local excitations

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