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arXiv:2510.08427

A convergent hierarchy of spectral gap certificates for qubit Hamiltonians

Sujit Rao

Published Oct 9, 2025• 0 citations
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Abstract

We give a convergent hierarchy of SDP certificates for lower bounds on the spectral gap of a local qubit Hamiltonian $H$. Our approach is based on the NPA hierarchy applied to a polynomially-sized system of constraints defining the universal enveloping algebra of the Lie algebra $\mathfrak{su}(2^{n})$ along with additional constraints which force the representation to be $\wedge^{2}(\mathbb{C}^{2^{n}})$, one which the induced Hamiltonian has ground energy $λ_{0}(H) + λ_{1}(H)$. Combined with an upper bound on the ground state energy, which can be obtained either using a hierarchy introduced by Fawzi, Fawzi, and Scalet or a noncommutative analog developed by Klep, Magron, Massé, and Volčič of a hierarchy introduced by Lasserre, we obtain lower bounds on the spectral gap of $H$ without assuming frustration-freeness or geometric locality. We prove that the resulting certificates have polynomial size at fixed degree and converge at level $n$. As illustrations of the certificates which can be obtained, we show that for a commuting 1-local Hamiltonian the hierarchy certifies the optimal bound on its spectral gap and prove that similar hierarchies require degree growing with $n$ to certify the same bound. We then extend this to prove a lower bound on the spectral gap of the transverse-field $XY$ model, which is not frustration-free, on arbitrary regular graphs without any assumption of geometric locality.

DOI
10.48550/arxiv.2510.08427

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We give a convergent hierarchy of SDP certificates for lower bounds on the spectral gap of a local qubit Hamiltonian HHH. Our approach is based on the NPA hierarchy applied to a polynomially-sized system of constraints defining the universal enveloping algebra of the Lie algebra su(2n)\mathfrak{su}(2^{n})su(2n) along with additional constraints which force the representation to be ∧2(C2n)\wedge^{2}(\mathbb{C}^{2^{n}})∧2(C2n), one which the induced Hamiltonian has ground energy λ0(H)+λ1(H)λ_{0}(H) + λ_{1}(H)λ0​(H)+λ1​(H). Combined with an upper bound on the ground state energy, which can be obtained either using a hierarchy introduced by Fawzi, Fawzi, and Scalet or a noncommutative analog developed by Klep, Magron, Massé, and Volčič of a hierarchy introduced by Lasserre, we obtain lower bounds on the spectral gap of HHH without assuming frustration-freeness or geometric locality. We prove that the resulting certificates have polynomial size at fixed degree and converge at level nnn. As illustrations of the certificates which can be obtained, we show that for a commuting 1-local Hamiltonian the hierarchy certifies the optimal bound on its spectral gap and prove that similar hierarchies require degree growing with nnn to certify the same bound. We then extend this to prove a lower bound on the spectral gap of the transverse-field XYXYXY model, which is not frustration-free, on arbitrary regular graphs without any assumption of geometric locality.
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A central question in quantum many-body physics is estimating the ground energy of a kkk-local Hamiltonian system. In this work, we present a spectral technique for certifying a lower bound on the ground energy of a random nnn-qubit Hamiltonian system defined as the sum of signed kkk-local Pauli operators. In particular, we prove that for any constant ℓ\ellℓ, there exists an efficiently computable length nO(ℓ)n^{O(\ell)}nO(ℓ) certificate that is always a lower bound on the ground energy with the promise that, with high probability over the random Hamiltonian distribution, the certificate value is an ε\varepsilonε-good approximation of the true ground energy when the number of terms is sufficiently large. Second, we show by construction that this technique, while successful on average over random Hamiltonian systems, can fail to produce good certificates on worst-case instances. Our spectral technique for producing these certificates comes from extending classical results on kkk-XOR refutations to kkk-local Pauli Hamiltonians by crafting a quantum variant of the Kikuchi matrix for CSP refutations. To show the limitations of this technique, we prove non-commutative Sum-of-Squares lower bounds for worst-case signed kkk-local Pauli operators. More generally, we explore how the non-commutative Sum-of-Squares relaxation can be understood as augmenting the standard Sum-of-Squares relaxation with the commutation relations between the Pauli operators. We instantiate the resulting framework with a modification to prior quantum code-based NLTS Hamiltonians that yields stronger complexity guarantees for the low-energy space; our Hamiltonian family satisfies simultaneously (1) Ω(log⁡n)Ω(\log n)Ω(logn)-circuit depth lower bounds for all low-energy states, (2) constant-gap NP-hardness to approximate the ground energy, and (3) a constant-gap non-commutative Sum-of-Squares integrality gap up to Ω(n)Ω(n)Ω(n)-levels.
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We develop a bootstrap approach to deriving lower bounds on the spectral gap of the first Haldane pseudopotential Hamiltonian on the sphere with NΦN_ΦNΦ​ flux quanta for arbitrary NΦN_ΦNΦ​, including in the thermodynamic limit NΦ→∞N_Φ\to \inftyNΦ​→∞. This method can in principle be generalized to pseudopotentials describing other model wavefunctions such as the Moore--Read state or Read--Rezayi sequence. Our approach is based on Knabe's observation that if HHH is a Hamiltonian with ground state energy zero then the spectral gap is at least ε>0ε> 0ε>0 if and only if H2⪰εHH^2 \succeq εHH2⪰εH. To show this inequality, we split H2H^2H2 into two parts, one that can be analytically bounded and another that can be bounded using a finite semidefinite program (SDP). Crucially, the size of this program is determined only by cutoff parameters, and the program depends on NΦN_ΦNΦ​ only parametrically, allowing us to obtain numerically exact (up to solver precision) lower bounds on the gap. In order to turn these numerical bounds into mathematically rigorous results, we introduce an algorithm that can produce nearly-optimal exact interior feasible points for large rational SDPs even if the affine constraints are highly redundant and ill-conditioned. We additionally provide an algorithm that establishes the rank stability of the SDP in the NΦ→∞N_Φ\to \inftyNΦ​→∞ limit, which is needed to certify the bound in the thermodynamic limit. Beyond establishing a Laughlin gap greater than 1/1001/1001/100 in the thermodynamic limit and revealing a potential method for deriving spectral gap lower bounds for other pseudopotentials, our work provides a general method for turning numerical SDP bounds into rigorous mathematical proofs, which may have wide applicability. In particular, this work demonstrates that bootstrap methods can be used to rigorously resolve major open problems in quantum many-body physics.