A Spectral Condition for Spectral Gap: Fast Mixing in High-Temperature Ising Models

Eldan, Ronen; Koehler, Frederic; Zeitouni, Ofer

doi:10.48550/arxiv.2007.08200

Cited by 7 publications

(18 citation statements)

References 15 publications

(22 reference statements)

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“…This result can be applied to a number of other models of interest; see [EKZ20] for the example of a d-regular diluted SK model. Informally, the difference between the results we obtain from Theorem 6 and the results implied by Dobrushin's condition is analogous to the difference between the broadcasting threshold and uniqueness threshold on trees, see e.g.…”

Section: (B) the Glauber Dynamics Mixes In Time Omentioning

confidence: 90%

“…Building on a breakthrough work of Bauerschmidt and Bodineau [BB19], this was established with c = 1/4 in recent work of Eldan, Koehler, and Zeitouni [EKZ20] as a consequence of the following general result: if the interaction matrix J is positive semidefinite, then the spectral gap of the Glauber dynamics for the Ising model is at least…”

Section: Application: High-temperature Ising Modelsmentioning

confidence: 98%

“…Note that the interaction matrix J can always be assumed to be positive semidefinite without loss of generality, because adding a multiple of the identity matrix to J does not change the measure µ J,h . By the well-known relationship between the spectral gap and the mixing time, the result of [EKZ20] implies that the mixing time of the Glauber dynamics is…”

Section: Application: High-temperature Ising Modelsmentioning

confidence: 99%

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Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models

Anari¹,

Jain²,

Koehler³

et al. 2021

Preprint

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We introduce a notion called entropic independence for distributions µ defined on pure simplicial complexes, i.e., subsets of size k of a ground set of elements. Informally, we call a background measure µ entropically independent if for any (possibly randomly chosen) set S, the relative entropy of an element of S drawn uniformly at random carries at most O(1/k) fraction of the relative entropy of S, a constant multiple of its "share of entropy." Entropic independence is the natural analog of spectral independence, another recently established notion, if one replaces variance by entropy.In our main result, we show that µ is entropically independent exactly when a transformed version of the generating polynomial of µ can be upper bounded by its linear tangent, a property implied by concavity of the said transformation. We further show that this concavity is equivalent to spectral independence under arbitrary external fields, an assumption that also goes by the name of fractional log-concavity. Our result can be seen as a new tool to establish entropy contraction from the much simpler variance contraction inequalities. A key differentiating feature of our result is that we make no assumptions on marginals of µ or the degrees of the underlying graphical model when µ is based on one. We leverage our results to derive tight modified log-Sobolev inequalities for multi-step down-up walks on fractionally log-concave distributions. As our main application, we establish the tight mixing time of O(n log n) for Glauber dynamics on Ising models with interaction matrix of operator norm smaller than 1, improving upon the prior quadratic dependence on n.

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Section: (B) the Glauber Dynamics Mixes In Time Omentioning

confidence: 90%

Section: Application: High-temperature Ising Modelsmentioning

confidence: 98%

Section: Application: High-temperature Ising Modelsmentioning

confidence: 99%

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Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models

Anari¹,

Jain²,

Koehler³

et al. 2021

Preprint

Self Cite

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“…can fall outside the rapid mixing regime β > ∼ 1/ J , in which state-of-the-art MCMC could approximate the pseudo-Boltzmann state [15]. As illustration of this fact, Fig.…”

mentioning

confidence: 91%

Quantum Approximate Optimization Algorithm pseudo-Boltzmann states

Díez-Valle,

Porras,

García-Ripoll

2022

Preprint

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In this letter, we provide analytical and numerical evidence that the single-layer Quantum Approximate Optimization Algorithm (QAOA) on universal Ising spin models produces thermal states with Gaussian perturbations. We find that these pseudo-Boltzman states can not be efficiently simulated on classical computers according to state-of-art techniques, and we relate this distribution to the optimization potential of QAOA. Moreover, we observe that the temperature depends on a hidden universal correlation between the energy of a state and the covariance of other energy levels and the Hamming distances of the state to those energies.

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“…However, sampling from a general Gibbs distribution is a computationally hard task for classical computers, due to the complexity of calculating the partition function [12]. Most techniques rely on Monte-Carlo Markov Chain (MCMC) algorithms, which are often provably efficient only above a certain threshold temperature [13].…”

Section: Introductionmentioning

confidence: 99%

Noise-Assisted Variational Quantum Thermalization

Foldager¹,

Pesah²,

Hansen³

2021

Preprint

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Preparing thermal states on a quantum computer can have a variety of applications, from simulating many-body quantum systems to training machine learning models. Variational circuits have been proposed for this task on near-term quantum computers, but several challenges remain, such as finding a scalable cost-function, avoiding the need of purification, and mitigating noise effects. We propose a new algorithm for thermal state preparation that tackles those three challenges by exploiting the noise of quantum circuits. We consider a variational architecture containing a depolarizing channel after each unitary layer, with the ability to directly control the level of noise. We derive a closed-form approximation for the free-energy of such circuit and use it as a cost function for our variational algorithm. By evaluating our method on a variety of Hamiltonians and system sizes, we find several systems for which the thermal state can be approximated with a high fidelity. However, we also show that the ability for our algorithm to learn the thermal state strongly depends on the temperature: while a high fidelity can be obtained for high and low temperatures, we identify a specific range for which the problem becomes more challenging. We hope that this first study on noise-assisted thermal state preparation will inspire future research on exploiting noise in variational algorithms.

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A Spectral Condition for Spectral Gap: Fast Mixing in High-Temperature Ising Models

Cited by 7 publications

References 15 publications

Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models

Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models

Quantum Approximate Optimization Algorithm pseudo-Boltzmann states

Noise-Assisted Variational Quantum Thermalization

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