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arXiv:2105.01201

Spectral independence, coupling with the stationary distribution, and the spectral gap of the Glauber dynamics

Vishesh Jain, Huy Tuan Pham, Thuy Duong Vuong

Published May 3, 2021• 10 citations
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Abstract

We present a new lower bound on the spectral gap of the Glauber dynamics for the Gibbs distribution of a spectrally independent $q$-spin system on a graph $G = (V,E)$ with maximum degree $Δ$. Notably, for several interesting examples, our bound covers the entire regime of $Δ$ excluded by arguments based on coupling with the stationary distribution. As concrete applications, by combining our new lower bound with known spectral independence computations and known coupling arguments: (1) We show that for a triangle-free graph $G = (V,E)$ with maximum degree $Δ\geq 3$, the Glauber dynamics for the uniform distribution on proper $k$-colorings with $k \geq (1.763\dots + δ)Δ$ colors has spectral gap $\tildeΩ_δ(|V|^{-1})$. Previously, such a result was known either if the girth of $G$ is at least $5$ [Dyer et.~al, FOCS 2004], or under restrictions on $Δ$ [Chen et.~al, STOC 2021; Hayes-Vigoda, FOCS 2003]. (2) We show that for a regular graph $G = (V,E)$ with degree $Δ\geq 3$ and girth at least $6$, and for any $\varepsilon, δ> 0$, the partition function of the hardcore model with fugacity $λ\leq (1-δ)λ_{c}(Δ)$ may be approximated within a $(1+\varepsilon)$-multiplicative factor in time $\tilde{O}_δ(n^{2}\varepsilon^{-2})$. Previously, such a result was known if the girth is at least $7$ [Efthymiou et.~al, SICOMP 2019]. (3) We show for the binomial random graph $G(n,d/n)$ with $d = O(1)$, with high probability, an approximately uniformly random matching may be sampled in time $O_{d}(n^{2+o(1)})$. This improves the corresponding running time of $\tilde{O}_{d}(n^{3})$ due to [Jerrum-Sinclair, SICOMP 1989; Jerrum, 2003].

DOI
10.48550/arxiv.2105.01201

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We present a new lower bound on the spectral gap of the Glauber dynamics for the Gibbs distribution of a spectrally independent qqq-spin system on a graph G=(V,E)G = (V,E)G=(V,E) with maximum degree ΔΔΔ. Notably, for several interesting examples, our bound covers the entire regime of ΔΔΔ excluded by arguments based on coupling with the stationary distribution. As concrete applications, by combining our new lower bound with known spectral independence computations and known coupling arguments: (1) We show that for a triangle-free graph G=(V,E)G = (V,E)G=(V,E) with maximum degree Δ≥3Δ\geq 3Δ≥3, the Glauber dynamics for the uniform distribution on proper kkk-colorings with k≥(1.763⋯+δ)Δk \geq (1.763\dots + δ)Δk≥(1.763⋯+δ)Δ colors has spectral gap Ω~δ(∣V∣−1)\tildeΩ_δ(|V|^{-1})Ω~δ​(∣V∣−1). Previously, such a result was known either if the girth of GGG is at least 555 [Dyer et.~al, FOCS 2004], or under restrictions on ΔΔΔ [Chen et.~al, STOC 2021; Hayes-Vigoda, FOCS 2003]. (2) We show that for a regular graph G=(V,E)G = (V,E)G=(V,E) with degree Δ≥3Δ\geq 3Δ≥3 and girth at least 666, and for any ε,δ>0\varepsilon, δ> 0ε,δ>0, the partition function of the hardcore model with fugacity λ≤(1−δ)λc(Δ)λ\leq (1-δ)λ_{c}(Δ)λ≤(1−δ)λc​(Δ) may be approximated within a (1+ε)(1+\varepsilon)(1+ε)-multiplicative factor in time O~δ(n2ε−2)\tilde{O}_δ(n^{2}\varepsilon^{-2})O~δ​(n2ε−2). Previously, such a result was known if the girth is at least 777 [Efthymiou et.~al, SICOMP 2019]. (3) We show for the binomial random graph G(n,d/n)G(n,d/n)G(n,d/n) with d=O(1)d = O(1)d=O(1), with high probability, an approximately uniformly random matching may be sampled in time Od(n2+o(1))O_{d}(n^{2+o(1)})Od​(n2+o(1)). This improves the corresponding running time of O~d(n3)\tilde{O}_{d}(n^{3})O~d​(n3) due to [Jerrum-Sinclair, SICOMP 1989; Jerrum, 2003].
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Recently, there have been several advancements in quantum algorithms for Gibbs sampling. These algorithms simulate the dynamics generated by an artificial Lindbladian, which is meticulously constructed to obey a detailed-balance condition with the Gibbs state of interest, ensuring it is a stationary point of the evolution, while simultaneously having efficiently implementable time steps. The overall complexity then depends primarily on the mixing time of the Lindbladian, which can vary drastically, but which has been previously bounded in the regime of high enough temperatures [Rouzé et al. arXiv:2403.12691 and arXiv:2411.04885]. In this work, we calculate the spectral gap of the Lindbladian for free fermions using third quantisation, and also prove a logarithmic bound on its mixing time by analysing corresponding covariance matrices. Then we prove a constant gap of the perturbed Lindbladian corresponding to interacting fermions up to some maximal coupling strength. This is achieved by using theorems about stability of the gap for lattice fermions. Our methods apply at any constant temperature and independently of the system size. The gap then provides an upper bound on the mixing time, and hence on the overall complexity of the quantum algorithm, proving that the purified Gibbs state of weakly interacting (quasi-)local fermionic systems of any dimension can be prepared in O~(n3polylog⁡(1/ε))\widetilde{\mathcal{O}} (n^3 \operatorname{polylog}(1/ε))O(n3polylog(1/ε)) time on O(n)\mathcal{O}(n)O(n) qubits, where nnn denotes the size of the system and εεε the desired accuracy. As an application, we explain how to calculate partition functions for the considered systems. We provide exact numerical simulations for small system sizes supporting the theory and also identify different suitable jump operators and filter functions for the sought-after regime of intermediate coupling in the Fermi-Hubbard model.
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