Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

On renormalization and the local gap method for proving frustration-free local spin chains are gapped

Ari Mizel, Van Molino

Published Dec 1, 2023• 0 citations
Publisher

Abstract

Key properties of a physical system depend on whether it is gapped, i.e., whether its spectral gap has a positive lower bound that is independent of system size. Here, we provide a prescription for renormalizing a spin chain Hamiltonian in such a way that the renormalized Hamiltonian is gapped if and only if the original Hamiltonian is gapped. Then, we articulate a set of conditions that guarantees the renormalized Hamiltonian is gapped. These conditions are built on a certain strong notion of decaying correlations involving an operator norm of non-commuting terms in the renormalized Hamiltonian. We apply the method to show that two interesting models, with forms motivated by quantum circuits, are gapped. We also confirm the generality of the method by successfully applying it to a somewhat different case, the well-known Affleck-Kennedy-Lieb-Tasaki (AKLT) model.

DOI
10.1063/5.0139298

Related papers

Renormalization method for proving frustration-free local spin chains are gapped

Ari Mizel, Van Molino

Key properties of a physical system depend on whether it is gapped, i.e. whether its spectral gap has a positive lower bound that is independent of system size. In quantum information theory, the question of whether a system is gapped has essential computational significance as well. Here, we introduce a rigorous renormalization method to prove that a spin chain is gapped. This approach exploits the fact that ground states of gapped systems exhibit decaying correlations. We apply the method to show that two interesting models are gapped, successfully completing proofs even where the previously established methods are inconclusive.

arXiv:2111.09358•Nov 17, 2021•0 citations

Divide and conquer method for proving gaps of frustration free Hamiltonians

Michael J Kastoryano, Angelo Lucia

Abstract 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.

arXiv:1705.09491•Journal of Statistical Mechanics Theory and Experiment•Mar 1, 2018•18 citations

Gapped and gapless phases of frustration-free spin-1/2 chains

Sergey B. Bravyi, David Gosset

We consider a family of translation-invariant quantum spin chains with nearest-neighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let $ψ$ be an arbitrary two-qubit state. We consider a chain of $n$ qubits with open boundary conditions and Hamiltonian $H_n(ψ)$ which is defined as the sum of rank-1 projectors onto $ψ$ applied to consecutive pairs of qubits. We show that the spectral gap of $H_n(ψ)$ is upper bounded by $1/(n-1)$ if the eigenvalues of a certain two-by-two matrix simply related to $ψ$ have equal non-zero absolute value. Otherwise, the spectral gap is lower bounded by a positive constant independent of $n$ (depending only on $ψ$). A key ingredient in the proof is a new operator inequality for the ground space projector which expresses a monotonicity under the partial trace. This monotonicity property appears to be very general and might be interesting in its own right. As an extension of our main result, we obtain a complete classification of gapped and gapless phases of frustration-free translation-invariant spin-1/2 chains with nearest-neighbor interactions.

arXiv:1503.04035•Mar 13, 2015•0 citations

Gapped and gapless phases of frustration-free spin-12 chains

Sergey B. Bravyi, David Gosset

We consider a family of translation-invariant quantum spin chains with nearest-neighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let ψ be an arbitrary two-qubit state. We consider a chain of n qubits with open boundary conditions and Hamiltonian Hn(ψ) which is defined as the sum of rank-1 projectors onto ψ applied to consecutive pairs of qubits. We show that the spectral gap of Hn(ψ) is upper bounded by 1/(n − 1) if the eigenvalues of a certain 2 × 2 matrix simply related to ψ have equal non-zero absolute value. Otherwise, the spectral gap is lower bounded by a positive constant independent of n (depending only on ψ). A key ingredient in the proof is a new operator inequality for the ground space projector which expresses a monotonicity under the partial trace. This monotonicity property appears to be very general and might be interesting in its own right. As an extension of our main result, we obtain a complete classification of gapped and gapless phases of frustration-free translation-invariant spin-1/2 chains with nearest-neighbor interactions.

Journal of Mathematical Physics•Jun 1, 2015•40 citations

Spectral gaps of Affleck-Kennedy-Lieb-Tasaki Hamiltonians using tensor network methods

Artur García-Sáez, V. Murg, Tzu-Chieh Wei

Using exact diagonalization and tensor network techniques, we compute the gap for the Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian in one and two spatial dimensions. Tensor network methods are used to extract physical properties directly in the thermodynamic limit, and we support these results using finite-size scalings from exact diagonalization. Studying the AKLT Hamiltonian perturbed by an external field, we show how to obtain an accurate value of the gap of the original AKLT Hamiltonian from the field value at which the ground state verifies ${e}_{0}<0$, which is a quantum critical point. With the tensor network renormalization group methods we provide direct evidence of a finite gap in the thermodynamic limit for the AKLT models in the one-dimensional chain and two-dimensional hexagonal and square lattices. This method can be applied generally to Hamiltonians with rotational symmetry, and we also show results beyond the AKLT model.

arXiv:1308.3631•Physical Review B•Dec 16, 2013•38 citations

Efficient Verification of Ground States of Frustration-Free Hamiltonians

Huangjun Zhu, Yunting Li, Tianyi Chen

Ground states of local Hamiltonians are of key interest in many-body physics and also in quantum information processing. Efficient verification of these states are crucial to many applications, but very challenging. Here we propose a simple, but powerful recipe for verifying the ground states of general frustration-free Hamiltonians based on local measurements. Moreover, we derive rigorous bounds on the sample complexity by virtue of the quantum detectability lemma (with improvement) and quantum union bound. Notably, the number of samples required does not increase with the system size when the underlying Hamiltonian is local and gapped, which is the case of most interest. As an application, we propose a general approach for verifying Affleck-Kennedy-Lieb-Tasaki (AKLT) states on arbitrary graphs based on local spin measurements, which requires only a constant number of samples for AKLT states defined on various lattices. Our work is of interest not only to many tasks in quantum information processing, but also to the study of many-body physics.

arXiv:2206.15292•Quantum•Jan 10, 2024•11 citations
Key properties of a physical system depend on whether it is gapped, i.e., whether its spectral gap has a positive lower bound that is independent of system size. Here, we provide a prescription for renormalizing a spin chain Hamiltonian in such a way that the renormalized Hamiltonian is gapped if and only if the original Hamiltonian is gapped. Then, we articulate a set of conditions that guarantees the renormalized Hamiltonian is gapped. These conditions are built on a certain strong notion of decaying correlations involving an operator norm of non-commuting terms in the renormalized Hamiltonian. We apply the method to show that two interesting models, with forms motivated by quantum circuits, are gapped. We also confirm the generality of the method by successfully applying it to a somewhat different case, the well-known Affleck-Kennedy-Lieb-Tasaki (AKLT) model.
Key properties of a physical system depend on whether it is gapped, i.e. whether its spectral gap has a positive lower bound that is independent of system size. In quantum information theory, the question of whether a system is gapped has essential computational significance as well. Here, we introduce a rigorous renormalization method to prove that a spin chain is gapped. This approach exploits the fact that ground states of gapped systems exhibit decaying correlations. We apply the method to show that two interesting models are gapped, successfully completing proofs even where the previously established methods are inconclusive.
Abstract 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.
We consider a family of translation-invariant quantum spin chains with nearest-neighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let ψψψ be an arbitrary two-qubit state. We consider a chain of nnn qubits with open boundary conditions and Hamiltonian Hn(ψ)H_n(ψ)Hn​(ψ) which is defined as the sum of rank-1 projectors onto ψψψ applied to consecutive pairs of qubits. We show that the spectral gap of Hn(ψ)H_n(ψ)Hn​(ψ) is upper bounded by 1/(n−1)1/(n-1)1/(n−1) if the eigenvalues of a certain two-by-two matrix simply related to ψψψ have equal non-zero absolute value. Otherwise, the spectral gap is lower bounded by a positive constant independent of nnn (depending only on ψψψ). A key ingredient in the proof is a new operator inequality for the ground space projector which expresses a monotonicity under the partial trace. This monotonicity property appears to be very general and might be interesting in its own right. As an extension of our main result, we obtain a complete classification of gapped and gapless phases of frustration-free translation-invariant spin-1/2 chains with nearest-neighbor interactions.
We consider a family of translation-invariant quantum spin chains with nearest-neighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let ψ be an arbitrary two-qubit state. We consider a chain of n qubits with open boundary conditions and Hamiltonian Hn(ψ) which is defined as the sum of rank-1 projectors onto ψ applied to consecutive pairs of qubits. We show that the spectral gap of Hn(ψ) is upper bounded by 1/(n − 1) if the eigenvalues of a certain 2 × 2 matrix simply related to ψ have equal non-zero absolute value. Otherwise, the spectral gap is lower bounded by a positive constant independent of n (depending only on ψ). A key ingredient in the proof is a new operator inequality for the ground space projector which expresses a monotonicity under the partial trace. This monotonicity property appears to be very general and might be interesting in its own right. As an extension of our main result, we obtain a complete classification of gapped and gapless phases of frustration-free translation-invariant spin-1/2 chains with nearest-neighbor interactions.
Using exact diagonalization and tensor network techniques, we compute the gap for the Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian in one and two spatial dimensions. Tensor network methods are used to extract physical properties directly in the thermodynamic limit, and we support these results using finite-size scalings from exact diagonalization. Studying the AKLT Hamiltonian perturbed by an external field, we show how to obtain an accurate value of the gap of the original AKLT Hamiltonian from the field value at which the ground state verifies e0<0{e}_{0}<0e0​<0, which is a quantum critical point. With the tensor network renormalization group methods we provide direct evidence of a finite gap in the thermodynamic limit for the AKLT models in the one-dimensional chain and two-dimensional hexagonal and square lattices. This method can be applied generally to Hamiltonians with rotational symmetry, and we also show results beyond the AKLT model.
Ground states of local Hamiltonians are of key interest in many-body physics and also in quantum information processing. Efficient verification of these states are crucial to many applications, but very challenging. Here we propose a simple, but powerful recipe for verifying the ground states of general frustration-free Hamiltonians based on local measurements. Moreover, we derive rigorous bounds on the sample complexity by virtue of the quantum detectability lemma (with improvement) and quantum union bound. Notably, the number of samples required does not increase with the system size when the underlying Hamiltonian is local and gapped, which is the case of most interest. As an application, we propose a general approach for verifying Affleck-Kennedy-Lieb-Tasaki (AKLT) states on arbitrary graphs based on local spin measurements, which requires only a constant number of samples for AKLT states defined on various lattices. Our work is of interest not only to many tasks in quantum information processing, but also to the study of many-body physics.