We present an upper bound on the energy gap above the ground energy in terms of the variance of the particle position in one-particle tight-binding systems. Our inequality gives a complementary relationship between the energy gap and the localization of the particle; that is, the energy gap narrows as the ground state spreads spatially, while the energy gap can broaden as the ground state becomes localized. We first consider the ground state in one-dimensional tight-binding systems and relate the energy gap to the variance of the particle position in the ground state. We then extend the result to higher-dimensional systems. We compare our upper bound with the exact energy gap as calculated numerically in one and two dimensions; this comparison confirms that our upper bound places a tight restriction on the energy gap.
We present an upper bound on the energy gap above the ground energy in terms of the variance of the particle position in one-particle tight-binding systems. Our inequality gives a complementary relationship between the energy gap and the localization of the particle; that is, the energy gap narrows as the ground state spreads spatially, while the energy gap can broaden as the ground state becomes localized. We first consider the ground state in one-dimensional tight-binding systems and relate the energy gap to the variance of the particle position in the ground state. We then extend the result to higher-dimensional systems. We compare our upper bound with the exact energy gap as calculated numerically in one and two dimensions; this comparison confirms that our upper bound places a tight restriction on the energy gap.
We investigate the relationship between the spectral gap delta E_0 and the localization length xi in general one-particle systems. A relationship for many-body systems between the spectral gap and the exponential clustering has been derived from the Lieb-Robinson bound, which reduces to the inequality xi le const. times delta E_0^{-1} for one-particle systems. This inequality, however, turned out not to be optimal qualitatively. As a refined upper bound, we here prove the inequality xi le const. times delta E_0^{-1/2} in general one-particle systems. Our proof is not based on the Lieb-Robinson bound, but on our complementary inequality related to the uncertainty principle [T. Kuwahara, J. Phys. A: Math. Theor. 46 (2013)]. We give a specific form of the upper bound and test its tightness in the tight-binding Hamiltonian with a diagonal impurity, where the localization length behaves as xi ~ delta E_0^{-1/2}. We ensure that our upper bound is quantitatively tight in the case of nearest-neighbor hopping.
We investigate the relationship between the energy spectrum of a local Hamiltonian and the geometric properties of its ground state. By generalizing a standard framework from the analysis of Markov chains to arbitrary (non-stoquastic) Hamiltonians we are naturally led to see that the spectral gap can always be upper bounded by an isoperimetric ratio that depends only on the ground state probability distribution and the range of the terms in the Hamiltonian, but not on any other details of the interaction couplings. This means that for a given probability distribution the inequality constrains the spectral gap of any local Hamiltonian with this distribution as its ground state probability distribution in some basis (Eldar and Harrow derived a similar result in order to characterize the output of low-depth quantum circuits). Going further, we relate the Hilbert space localization properties of the ground state to higher energy eigenvalues by showing that the presence of k strongly localized ground state modes (i.e. clusters of probability, or subsets with small expansion) in Hilbert space implies the presence of k energy eigenvalues that are close to the ground state energy. Our results suggest that quantum adiabatic optimization using local Hamiltonians will inevitably encounter small spectral gaps when attempting to prepare ground states corresponding to multi-modal probability distributions with strongly localized modes, and this problem cannot necessarily be alleviated with the inclusion of non-stoquastic couplings.
We derive eigenstate comparisons from ordered Schrödinger operators, extending energy bounds to overlaps, spatial distributions and nodes. For three form-ordered operators with discrete comparison ground states, suppose the upper ground-state energy lies in the lower operator's ground-state gap. Three comparison spectral values then bound all three squared ground-state overlaps and guarantee a simple discrete intermediate ground state. Related gap estimates control low-energy subspaces and individual excited-state overlaps under additional gap conditions. In one dimension and radially, a single crossing of the potential difference relative to the energy shift orders ground-state likelihood ratios and monotone observables; a nondecreasing difference suffices. The results cover asymmetric wells and shortened right Dirichlet walls, with coordinatewise comparisons for submodular reference potentials and coordinatewise nondecreasing perturbations. A two-sided Sturm argument orders corresponding internal nodes for nondecreasing potential differences, even between orthogonal one-dimensional excited states. For Yukawa ground states, Hulthén comparisons bound overlaps, radial moments and the density at the nucleus; the latter two-sided bounds hold for screening 0<δ<2/6. Optimizing scaled Hulthén majorants gives a lower bound of approximately 1.13611 for n2δc(n,0), where δc(n,0) is the critical screening for the nth s-wave bound state. This bound is optimal within that two-parameter comparison family.
A numerical bootstrap method is proposed to provide rigorous and nontrivial bounds in general quantum many-body systems with locality. In particular, lower bounds on ground state energies of local lattice systems are obtained by imposing positivity constraints on certain operator expectation values. Complemented with variational upper bounds, ground state observables are constrained to be within a narrow range. The method is demonstrated with the Hubbard model in one and two dimensions, and bounds on ground state double occupancy and magnetization are discussed.
Suppose we can apply the unitary U=eiH for some Hamiltonian H, and are given access to a unitary that prepares a guiding state promised to have overlap at least γ>0 with the ground space of H. Our goal is to estimate the ground-state energy of H within additive error δ>0 and success probability at least 1−ε, ε>0. How many applications of U and its inverse U−1 are necessary and sufficient? This quantity corresponds to the total Hamiltonian-simulation time needed. An upper bound O(log(1/ε)log(1/γ)/γδ) was known, and was improved to O(log(1/ε)/γδ) very recently [JW26]. A matching lower bound was known whenever one of the three parameters δ,γ,ε was held constant [MdW26]. In this paper we prove the joint lower bound Ω(log(1/ε)/γδ) with the tight ε-dependence provided the dimension of H is at least log(1/ε)/γ2. Furthermore, we show that this same lower bound (with slightly larger dimension) holds for both the special case in which the ground state is guaranteed to be unique and H has a gap of δ between its first and second eigenvalue; and for ground-state preparation, where δ denotes the spectral gap and ε now is the approximation error. The lower bounds also apply when the Hamiltonian can be accessed via its block-encoding, and when fractional powers of U are allowed, as in continuous-time Hamiltonian simulation. Lastly, improved upper bounds are known when H is nonnegative and presented as a sum of squares; and our results imply the lower bound Ω(log(1/ε)/γδ) for this case.
It has been proved that in gapped ground states of locally interacting lattice quantum systems with a finite local Hilbert space, the effect of local perturbations decays exponentially with distance.However, in systems with power-law-(1/r α ) decaying interactions, no analogous statement has been shown and there are serious mathematical obstacles to proving it with existing methods.In this paper, we prove that when α exceeds the spatial dimension D, the effect of local perturbations on local properties a distance r away is upper bounded by a power law 1/r α 1 in gapped ground states, provided that the perturbations do not close the spectral gap.The power-law exponent α 1 is tight if α > 2D and interactions are two-body, where we have α 1 = α.The proof is enabled by a method that avoids the use of quasiadiabatic continuation and incorporates techniques of complex analysis.This method also improves bounds on ground-state correlation decay, even in short-range interacting systems.Our work generalizes the fundamental notion that local perturbations have local effects to power-law interacting systems, with broad implications for numerical simulations and experiments.