Simple proof of the detectability lemma and spectral gap amplification

Anshu, Anurag; Arad, Itai; Vidick, Thomas

doi:10.1103/physrevb.93.205142

Cited by 32 publications

(48 citation statements)

References 30 publications

(57 reference statements)

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“…Another tool that we will use is the notion of coarse-grained Hamiltonian [ALV12,AAV16]. Let Q S be the projector orthogonal to the ground space of h S .…”

Section: B2 Coarse-grained Hamiltonianmentioning

confidence: 99%

Improved local spectral gap thresholds for lattices of finite size

Anshu

2020

Phys. Rev. B

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Knabe's theorem lower bounds the spectral gap of a one dimensional frustration-free local hamiltonian in terms of the local spectral gaps of finite regions. It also provides a local spectral gap threshold for hamiltonians that are gapless in the thermodynamic limit, showing that the local spectral gap much scale inverse linearly with the length of the region for such systems. Recent works have further improved upon this threshold, tightening it in the one dimensional case and extending it to higher dimensions. Here, we show a local spectral gap threshold for frustration-free hamiltonians on a finite dimensional lattice, that is optimal up to a constant factor that depends on the dimension of the lattice. Our proof is based on the detectability lemma framework and uses the notion of coarse-grained hamiltonian (introduced in [Phys. Rev. B 93, 205142]) as a link connecting the (global) spectral gap and the local spectral gap.

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“…Another tool that we will use is the notion of coarse-grained Hamiltonian [ALV12,AAV16]. Let Q S be the projector orthogonal to the ground space of h S .…”

Section: B2 Coarse-grained Hamiltonianmentioning

confidence: 99%

Improved local spectral gap thresholds for lattices of finite size

Anshu

2020

Phys. Rev. B

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“…It has since then become a useful tool in many-body problems. A converse result is known as the Converse Detectability Lemma [11,4], and will also be used later. At the same time as we recall them, we will reformulate them in terms of inequalities between some quadratic functionals.…”

Section: The Detectability Lemma and Its Conversementioning

confidence: 99%

Divide and conquer method for proving gaps of frustration free Hamiltonians

Kastoryano¹,

Lucia²

2018

J. Stat. Mech.

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Providing system-size independent lower bounds on the spectral gap of local Hamiltonian is in general a hard problem. For the case of finite-range, frustration free Hamiltonians on a spin lattice of arbitrary dimension, we show that a property of the ground state space is sufficient to obtain such a bound. We furthermore show that such a condition is necessary and equivalent to a constant spectral gap. Thanks to this equivalence, we can prove that for gapless models in any dimension, the spectral gap on regions of diameter n is at most o log(n) 2+ε n for any positive ε.

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“…It also has a much simpler proof, which is given in [2]. In addition to the detectability lemma, we will use a converse statement which gives a lower bound on the norm of DL(H )|ψ .…”

Section: Definition 9 (Thementioning

confidence: 99%

“…In addition to the detectability lemma, we will use a converse statement which gives a lower bound on the norm of DL(H )|ψ . The converse, and its proof, appear in [2].…”

Section: Definition 9 (Thementioning

confidence: 99%

Rigorous RG Algorithms and Area Laws for Low Energy Eigenstates in 1D

Arad

Landau

Vazirani

et al. 2017

Commun. Math. Phys.

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144

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Abstract:One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unsolved, including whether there exist efficient algorithms when the ground space is degenerate (and of polynomial dimension in the system size), or for the polynomially many lowest energy states, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of n particles. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an n O(log n) algorithm for the poly(n) lowest energy states (under a mild density condition). For these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time isÕ(nM(n)), where M(n) is the time required to multiply two n × n matrices.

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Simple proof of the detectability lemma and spectral gap amplification

Cited by 32 publications

References 30 publications

Improved local spectral gap thresholds for lattices of finite size

Improved local spectral gap thresholds for lattices of finite size

Divide and conquer method for proving gaps of frustration free Hamiltonians

Rigorous RG Algorithms and Area Laws for Low Energy Eigenstates in 1D

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