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F.-y. Wang - One of the best experts on this subject based on the ideXlab platform.
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Spectral Gap for measure valued diffusion processes
Journal of Mathematical Analysis and Applications, 2020Co-Authors: F.-y. WangAbstract:Abstract The Spectral Gap is estimated for some measure-valued processes, which are induced by the intrinsic/extrinsic derivatives on the space of finite measures over a Riemannian manifold. These processes are symmetric with respect to the Dirichlet and Gamma distributions arising from population genetics. In addition to the evolution of allelic frequencies investigated in the literature, they also describe stochastic movements of individuals.
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Spectral Gap for measure valued diffusion processes
arXiv: Probability, 2019Co-Authors: F.-y. WangAbstract:The Spectral Gap is estimated for measure-valued diffusion processes induced by the intrinsic/extrinsic derivatives on the space of finite measures over a Riemannian manifold. This provides explicit exponential convergence rate for these processes to approximate the Dirichlet and Gamma distributions arising from population genetics.
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Criteria of Spectral Gap for Markov operators
Journal of Functional Analysis, 2014Co-Authors: F.-y. WangAbstract:Let (E,F,μ) be a probability space, and let P be a Markov operator on L2(μ) with 1 a simple eigenvalue such that μP=μ (i.e. μ is an invariant probability measure of P). Then Pˆ:=12(P+P⁎) has a Spectral Gap, i.e. 1 is isolated in the spectrum of Pˆ, if and only if ‖P‖τ:=limR→∞supμ(f2)⩽1μ(f(Pf−R)+)
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Spectral Gap for hyperbounded operators
Proceedings of the American Mathematical Society, 2004Co-Authors: F.-y. WangAbstract:Let (E, F, μ) be a probability space, and P a symmetric linear contraction operator on L 2 (μ) with P1 = 1 and ∥P∥ L 2 (μ)→L 4 (μ) < ∞. We prove that ∥P∥ 4 L 2 (μ)→L 4 (μ) < 2 is the optimal sufficient condition for P to have a Spectral Gap. Moreover, the optimal sufficient conditions are obtained, respectively, for the defective log-Sobolev and for the defective Poincare inequality to imply the existence of a spectra] Gap. Finally, we construct a symmetric, hyperbounded, ergodic contraction C 0 -semigroup without a Spectral Gap.
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On estimation of the Dirichlet Spectral Gap
Archiv der Mathematik, 2000Co-Authors: F.-y. WangAbstract:By using a general lower bound formula of Spectral Gap, the lower bound estimate of the Dirichlet Spectral Gap is studied. The resulting estimates improve and recover known ones in the literature.
Toby S Cubitt - One of the best experts on this subject based on the ideXlab platform.
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Undecidability of the Spectral Gap in One Dimension.
arXiv: Quantum Physics, 2018Co-Authors: Johannes Bausch, Toby S Cubitt, Angelo Lucia, David Pérez-garcíaAbstract:The Spectral Gap problem - determining whether the energy spectrum of a system has an energy Gap above ground state, or if there is a continuous range of low-energy excitations - pervades quantum many-body physics. Recently, this important problem was shown to be undecidable for quantum systems in two (or more) spatial dimensions: it is provably impossible to determine in general whether a system is Gapped or Gapless, a result which has many unexpected consequences for the physics of such systems. However, there are many indications that one dimensional systems are simpler than their higher-dimensional counterparts: for example, they cannot have thermal phase transitions or topological order, and there exist highly-effective numerical algorithms such as DMRG for Gapped 1D systems, exploiting the fact that such systems obey an entropy area-law. Furthermore, the Spectral Gap undecidability construction crucially relied on aperiodic tilings, which are easily seen to be impossible in 1D. So does the Spectral Gap problem become decidable in 1D? In this paper we prove this is not the case, by constructing a family of 1D spin chains with translationally-invariant nearest neighbour interactions with undecidable Spectral Gap. This not only proves that the Spectral Gap of 1D systems is just as intractable, but also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with constant Spectral Gap and unique classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a Gapless behaviour with dense spectrum.
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Comment on "On the uncomputability of the Spectral Gap"
arXiv: Quantum Physics, 2016Co-Authors: Toby S Cubitt, David Pérez-garcía, Michael M WolfAbstract:The aim of this short note is to clarify some of the claims made in the comparison made in [S. Lloyd, On the uncomputability of the Spectral Gap, arXiv:1602.05924] between our recent result [T.S. Cubitt, D. Perez-Garcia, M.M. Wolf, Undecidability of the Spectral Gap, Nature 528, 207-211 (2015), arXiv:1502.04573] and his 1994 paper [S. Lloyd, Necessary and sufficient conditions for quantum computation, J. Mod. Opt. 41(12), 2503-2520 (1994)].
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undecidability of the Spectral Gap
Nature, 2015Co-Authors: Toby S Cubitt, David Perezgarcia, Michael M WolfAbstract:The Spectral Gap—the energy difference between the ground state and first excited state of a system—is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of Gapped topological spin liquid phases, and the Yang–Mills Gap conjecture, concern Spectral Gaps. These and other problems are particular cases of the general Spectral Gap problem: given the Hamiltonian of a quantum many-body system, is it Gapped or Gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the Spectral Gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The Spectral Gap depends on the outcome of the corresponding ‘halting problem’. Our result implies that there exists no algorithm to determine whether an arbitrary model is Gapped or Gapless, and that there exist models for which the presence or absence of a Spectral Gap is independent of the axioms of mathematics. The Spectral Gap problem—whether the Hamiltonian of a quantum many-body problem is Gapped or Gapless—is rigorously proved to be undecidable; there exists no algorithm to determine whether an arbitrary quantum many-body model is Gapped or Gapless, and there exist models for which the presence or absence of a Spectral Gap is independent of the axioms of mathematics. In quantum many-body physics, the Spectral Gap is the energy difference between the ground state of a system and its first excited state. Establishing whether it is possible to make a decision about the system being Gapped or Gapless, given a specific model Hamiltonian, is a long-standing problem in physics known as the Spectral Gap problem. Here, Toby Cubitt et al. prove that the Spectral Gap problem is undecidable. Although it had been known before that deciding about the existence of a Spectral Gap is difficult, this result proves the strongest possible form of algorithmic difficulty for a core problem of many-body physics.
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Supplementary Information. Undecidability of the Spectral Gap
2015Co-Authors: Toby S Cubitt, David Pérez García, Michael M WolfAbstract:This is the supplementary information associated to the article: T.S. Cubitt, D. Perez-Garcia, M.M. Wolf, Undecidability of the Spectral Gap, Nature 528, 207-211 (2015), doi:10.1038/nature16059
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undecidability of the Spectral Gap full version
Nature, 2015Co-Authors: Toby S Cubitt, David Perezgarcia, Michael M WolfAbstract:We show that the Spectral Gap problem is undecidable. Specifically, we construct families of translationally-invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d-level quantum systems (d constant), for which determining whether the system is Gapped or Gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either Gapped or Gapless in the strongest sense: it is promised to either have continuous spectrum above the ground state in the thermodynamic limit, or its Spectral Gap is lower-bounded by a constant in the thermodynamic limit. Moreover, this constant can be taken equal to the local interaction strength of the Hamiltonian.
Michael M Wolf - One of the best experts on this subject based on the ideXlab platform.
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Comment on "On the uncomputability of the Spectral Gap"
arXiv: Quantum Physics, 2016Co-Authors: Toby S Cubitt, David Pérez-garcía, Michael M WolfAbstract:The aim of this short note is to clarify some of the claims made in the comparison made in [S. Lloyd, On the uncomputability of the Spectral Gap, arXiv:1602.05924] between our recent result [T.S. Cubitt, D. Perez-Garcia, M.M. Wolf, Undecidability of the Spectral Gap, Nature 528, 207-211 (2015), arXiv:1502.04573] and his 1994 paper [S. Lloyd, Necessary and sufficient conditions for quantum computation, J. Mod. Opt. 41(12), 2503-2520 (1994)].
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undecidability of the Spectral Gap
Nature, 2015Co-Authors: Toby S Cubitt, David Perezgarcia, Michael M WolfAbstract:The Spectral Gap—the energy difference between the ground state and first excited state of a system—is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of Gapped topological spin liquid phases, and the Yang–Mills Gap conjecture, concern Spectral Gaps. These and other problems are particular cases of the general Spectral Gap problem: given the Hamiltonian of a quantum many-body system, is it Gapped or Gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the Spectral Gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The Spectral Gap depends on the outcome of the corresponding ‘halting problem’. Our result implies that there exists no algorithm to determine whether an arbitrary model is Gapped or Gapless, and that there exist models for which the presence or absence of a Spectral Gap is independent of the axioms of mathematics. The Spectral Gap problem—whether the Hamiltonian of a quantum many-body problem is Gapped or Gapless—is rigorously proved to be undecidable; there exists no algorithm to determine whether an arbitrary quantum many-body model is Gapped or Gapless, and there exist models for which the presence or absence of a Spectral Gap is independent of the axioms of mathematics. In quantum many-body physics, the Spectral Gap is the energy difference between the ground state of a system and its first excited state. Establishing whether it is possible to make a decision about the system being Gapped or Gapless, given a specific model Hamiltonian, is a long-standing problem in physics known as the Spectral Gap problem. Here, Toby Cubitt et al. prove that the Spectral Gap problem is undecidable. Although it had been known before that deciding about the existence of a Spectral Gap is difficult, this result proves the strongest possible form of algorithmic difficulty for a core problem of many-body physics.
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Supplementary Information. Undecidability of the Spectral Gap
2015Co-Authors: Toby S Cubitt, David Pérez García, Michael M WolfAbstract:This is the supplementary information associated to the article: T.S. Cubitt, D. Perez-Garcia, M.M. Wolf, Undecidability of the Spectral Gap, Nature 528, 207-211 (2015), doi:10.1038/nature16059
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undecidability of the Spectral Gap full version
Nature, 2015Co-Authors: Toby S Cubitt, David Perezgarcia, Michael M WolfAbstract:We show that the Spectral Gap problem is undecidable. Specifically, we construct families of translationally-invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d-level quantum systems (d constant), for which determining whether the system is Gapped or Gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either Gapped or Gapless in the strongest sense: it is promised to either have continuous spectrum above the ground state in the thermodynamic limit, or its Spectral Gap is lower-bounded by a constant in the thermodynamic limit. Moreover, this constant can be taken equal to the local interaction strength of the Hamiltonian.
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undecidability of the Spectral Gap short version
Nature, 2015Co-Authors: Toby S Cubitt, David Perezgarcia, Michael M WolfAbstract:The Spectral Gap - the energy difference between the ground state and first excited state - is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, existence of Gapped topological spin liquid phases, or the Yang-Mills Gap conjecture, concern Spectral Gaps. These and other problems are particular cases of the general Spectral Gap problem: given a quantum many-body Hamiltonian, is it Gapped or Gapless? Here we prove that this is an undecidable problem. We construct families of quantum spin systems on a 2D lattice with translationally-invariant, nearest-neighbour interactions for which the Spectral Gap problem is undecidable. This result extends to undecidability of other low energy properties, such as existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing Machine. The Spectral Gap depends on the outcome of the corresponding Halting Problem. Our result implies that there exists no algorithm to determine whether an arbitrary model is Gapped or Gapless. It also implies that there exist models for which the presence or absence of a Spectral Gap is independent of the axioms of mathematics.
Nicolas Saxcé - One of the best experts on this subject based on the ideXlab platform.
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A Spectral Gap theorem in simple Lie groups
Inventiones mathematicae, 2016Co-Authors: Yves Benoist, Nicolas SaxcéAbstract:We establish the Spectral Gap property for dense subgroups generated by algebraic elements in any compact simple Lie group, generalizing earlier results of Bourgain and Gamburd for unitary groups.
Tadeusz Kulczycki - One of the best experts on this subject based on the ideXlab platform.
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Spectral Gap for Stable Process on Convex Planar Double Symmetric Domains
Potential Analysis, 2007Co-Authors: Bartłomiej Dyda, Tadeusz KulczyckiAbstract:We study the semigroup of the symmetric α -stable process in bounded domains in R ^2. We obtain a variational formula for the Spectral Gap, i.e. the difference between two first eigenvalues of the generator of this semigroup. This variational formula allows us to obtain lower bound estimates of the Spectral Gap for convex planar domains which are symmetric with respect to both coordinate axes. For rectangles, using “midconcavity” of the first eigenfunction (Bañuelos et al., Potential Anal. 24(3): 205–221, 2006 ), we obtain sharp upper and lower bound estimates of the Spectral Gap.
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Spectral Gap for stable process on convex planar double symmetric domains
arXiv: Spectral Theory, 2006Co-Authors: Bartłomiej Dyda, Tadeusz KulczyckiAbstract:We study the semigroup of the symmetric $\alpha$-stable process in bounded domains in $\R^2$. We obtain a variational formula for the Spectral Gap, i.e. the difference between two first eigenvalues of the generator of this semigroup. This variational formula allows us to obtain lower bound estimates of the Spectral Gap for convex planar domains which are symmetric with respect to both coordinate axes. For rectangles, using "midconcavity" of the first eigenfunction, we obtain sharp upper and lower bound estimates of the Spectral Gap.
