Simple Glass Models and Their Quantum Annealing

Jörg, Thomas; Krząkała, Florent; Kurchan, Jorge; Maggs, A. C.

doi:10.1103/physrevlett.101.147204

Cited by 119 publications

(165 citation statements)

References 28 publications

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“…The energy gap is therefore very small compared to the ground-state energy and is not an extensive quantity. One usually resorts to numerical methods when computing the energy gap in quantum spin-glasses [22][23][24][25][26] .…”

Section: Introductionmentioning

confidence: 99%

Effects of low-lying excitations on ground-state energy and energy gap of the Sherrington-Kirkpatrick model in a transverse field

Koh

2016

Phys. Rev. B

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We present an extensive numerical study of the Sherrington-Kirkpatrick model in transverse field. Recent numerical studies of quantum spin-glasses have focused on exact diagonalization of the full Hamiltonian for small systems (≈ 20 spins). However, such exact numerical treatments are difficult to apply on larger systems. We propose making an approximation by using only a subspace of the full Hilbert space spanned by low-lying excitations consisting of one-spin flipped and two-spin flipped states. The approximation procedure is carried out within the theoretical framework of Hartree-Fock approximation and Configuration Interaction. Although not exact, our approach allows us to study larger system sizes comparable to that achievable by state of the art Quantum Monte Carlo simulations. We calculate two quantities of interest due to recent advances in quantum annealing, the ground-state energy and the energy gap between the ground and first excited state. For the energy gap, we derive a novel formula that enables it to be calculated using just the ground-state wavefunction, thereby circumventing the need to diagonalize the Hamiltonian.We calculate the scalings of the energy gap and the leading correction to the extensive part of the ground-state energy with system size, which are difficult to obtain with current methods.

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Section: Introductionmentioning

confidence: 99%

Effects of low-lying excitations on ground-state energy and energy gap of the Sherrington-Kirkpatrick model in a transverse field

Koh

2016

Phys. Rev. B

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“…However, short-range models have O(1) local fields, and in fact, so do power-law and infinite-range systems with general p-body interactions. This suggests that the eigenstate-localized phase of the QREM is an exceptional case among long-range models: strict configuration-space localization cannot exist in any model with O(1) local fields, since the introduction of quantum dynamics causes resonant fluctuations.In this paper, we study the eigenstate properties of the quantum p-spin models [25][26][27][28]. Over the past four decades, these models have become paradigms for the mean-field theory of spin glasses [29][30][31][32] Sherrington-Kirkpatrick model (p = 2) [33,34].…”

mentioning

confidence: 99%

“…In this paper, we study the eigenstate properties of the quantum p-spin models [25][26][27][28]. Over the past four decades, these models have become paradigms for the mean-field theory of spin glasses [29][30][31][32] Sherrington-Kirkpatrick model (p = 2) [33,34].…”

mentioning

confidence: 99%

Clustering of Nonergodic Eigenstates in Quantum Spin Glasses

et al. 2017

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The two primary categories for eigenstate phases of matter at finite temperature are manybody localization (MBL) and the eigenstate thermalization hypothesis (ETH). We show that in the paradigmatic quantum p-spin models of spin-glass theory, eigenstates violate ETH yet are not MBL either. A mobility edge, which we locate using the forward-scattering approximation and replica techniques, separates the non-ergodic phase at small transverse field from an ergodic phase at large transverse field. The non-ergodic phase is also bounded from above in temperature, by a transition in configuration-space statistics reminiscent of the clustering transition in spin-glass theory. We show that the non-ergodic eigenstates are organized in clusters which exhibit distinct magnetization patterns, as characterized by an eigenstate variant of the Edwards-Anderson order parameter.Many systems under experimental investigation as platforms for many-body localization (MBL) [1][2][3][4][5][6][7][8][9] have long-range interactions that mediate the direct transport of excitations. This includes disordered electronic materials [10,11], ion traps [12], interacting NV centers in diamond [13,14], and superconducting qubit devices developed for adiabatic quantum computing [15][16][17]. In sufficiently long-ranged systems, the proliferation of longdistance resonances precludes quantum mechanical localization [18][19][20][21][22], an intuitive result strongly supported by analytic work over the last half century. Nevertheless, the quantum Random Energy Model (QREM), an infiniterange spin glass, was recently shown to exhibit a phase with localized eigenstates at finite energy density [23,24]. The QREM provides an analytically tractable framework for studying mobility edges and configuration-space localization. This raises the obvious question of how localization survives despite the infinite-range interactions and what role it plays in more realistic long-range systems.Some insight comes from considering the distribution of local fields -i.e., the energy required to flip one of the system's N spins relative to a given configuration. In the QREM, flipping a spin typically changes the energy by O(N ). Thus the quantum fluctuations which lead to the proliferation of resonances are strongly suppressed. However, short-range models have O(1) local fields, and in fact, so do power-law and infinite-range systems with general p-body interactions. This suggests that the eigenstate-localized phase of the QREM is an exceptional case among long-range models: strict configuration-space localization cannot exist in any model with O(1) local fields, since the introduction of quantum dynamics causes resonant fluctuations.In this paper, we study the eigenstate properties of the quantum p-spin models [25][26][27][28]. Over the past four decades, these models have become paradigms for the mean-field theory of spin glasses [29][30][31][32] Sherrington-Kirkpatrick model (p = 2) [33,34]. The QREM corresponds to the p → ∞ limit in many senses, but the local-field distrib...

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“…Previous work [16][17][18] had argued that a first order quantum phase transition occurs for a broad class of random optimization models. To gain further insight into this matter we study here three optimization problems which had previously been suggested [17,19,20] as good potential candidates for detailed investigation.…”

Section: H(s)mentioning

confidence: 99%

Exponential complexity of the quantum adiabatic algorithm for certain satisfiability problems

Hen

Young

2011

Phys. Rev. E

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We determine the complexity of several constraint satisfaction problems using the quantum adiabatic algorithm in its simplest implementation. We do so by studying the size dependence of the gap to the first excited state of "typical" instances. We find that at large sizes N , the complexity increases exponentially for all models that we study. We also compare our results against the complexity of the analogous classical algorithm WalkSAT and show that the harder the problem is for the classical algorithm the harder it is also for the quantum adiabatic algorithm.

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Simple Glass Models and Their Quantum Annealing

Cited by 119 publications

References 28 publications

Effects of low-lying excitations on ground-state energy and energy gap of the Sherrington-Kirkpatrick model in a transverse field

Effects of low-lying excitations on ground-state energy and energy gap of the Sherrington-Kirkpatrick model in a transverse field

Clustering of Nonergodic Eigenstates in Quantum Spin Glasses

Exponential complexity of the quantum adiabatic algorithm for certain satisfiability problems

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