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arxiv: 2606.23614 · v1 · pith:MZ4EBLB4new · submitted 2026-06-22 · 🪐 quant-ph · cs.DS· math-ph· math.MP

Log-concavity and tunneling: adiabatic quantum optimization for convex functions (with a spike)

Arthur Braida , Elie Bermot , Simon Apers This is my paper

Pith reviewed 2026-06-26 08:11 UTC · model grok-4.3

classification 🪐 quant-ph cs.DSmath-phmath.MP
keywords log-concavityadiabatic quantum optimizationspectral gapquantum tunnelingconvex potentialsSchrödinger operatorsHamming weight with spike
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The pith

Log-concavity of the ground state yields spectral gap bounds and extends tunneling to quadratic potentials in adiabatic quantum optimization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the ground state of discrete one-dimensional Schrödinger operators is log-concave when the potential is convex. This property allows new bounds on the spectral gap that improve previous results for convex potentials. The authors then apply this to extend the perturbative tunneling analysis of the Hamming weight with spike problem from linear to quadratic potentials. This suggests that tunneling effects can apply more broadly to convex functions with spikes in quantum optimization.

Core claim

Log-concavity of the ground state is a key structural property for discrete 1D Schrödinger operators with convex potentials. It provides a discrete version of the Brascamp-Lieb result and leads to improved spectral gap bounds. This property also allows extending the perturbative analysis of tunneling in the Hamming weight with spike problem to the family of potentials with log-concave ground states, including quadratic potentials.

What carries the argument

Log-concavity of the ground state for a family of discrete 1D Schrödinger operators, which holds for convex potentials and enables both spectral gap bounds and extended perturbative tunneling analysis.

If this is right

  • New spectral gap bounds for convex potentials that go beyond those by Jarret and Jordan.
  • Extension of Reichardt's perturbative analysis of the Hamming weight with spike to quadratic potentials.
  • Broader applicability of tunneling to convex potentials with spikes.
  • The framework also covers certain potentials with local minima under stated conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical checks of ground-state log-concavity for specific quadratic potentials with spikes could directly test the extended analysis.
  • The same structural property might simplify gap estimates in related discrete quantum systems beyond one dimension.

Load-bearing premise

Discrete 1D Schrödinger operators with convex potentials have log-concave ground states.

What would settle it

Finding a convex potential where the ground state of the corresponding discrete Schrödinger operator is not log-concave, or where the spectral gap bounds do not improve as claimed.

Figures

Figures reproduced from arXiv: 2606.23614 by Arthur Braida, Elie Bermot, Simon Apers.

Figure 1
Figure 1. Figure 1: Example of a non-monotone potential satisfying the RM condition for [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the function m∗ (a)/n obtained for n = 40 and k0 = −3. m∗ (a) is obtained by solving the equation a = n−2m∗ √ m∗(n−m∗) · 1 1−2k0+2(n−1)m∗/n . As expected, m∗ is located at n/2 for a = 0 and it decreases to 0 as a increases. Claim 4 (Overlap bound). Let |ψ0⟩ denote the ground state of H = −Ahc + V with V a convex function with bounded second difference ∆2V (k) ≤ C. Let m be chosen so as to minimize … view at source ↗
Figure 3
Figure 3. Figure 3: Overlap between the optimal binomial ansatz state [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
read the original abstract

Quantum tunneling is expected to provide a computational speedup in quantum computing, a phenomenon that Adiabatic Quantum Optimization (AQO) aims to leverage. While some academic proofs of concept have been studied, such as the "Hamming weight with a spike" (HWS) problem, the algorithmic gains of this effect remain underexplored. In this work we extend the analysis underlying HWS to more general potentials. In the first half of the work, we establish (discrete) log-concavity of the ground state as a key structural property in this context. We devise a framework for establishing log-concavity of the ground state for a large family of discrete, 1-dimensional Schr\"odinger operators. The family includes convex potentials, but also certain potentials with local minima. In the convex case, this provides a discrete version of a continuous result by Brascamp and Lieb ('76). We demonstrate the utility of our result by establishing new spectral gap bounds, going beyond related results by Jarret and Jordan ('14) for convex potentials. In the second half of the work, we use our results on log-concavity to extend the perturbative analysis of HWS by Reichardt ('04) to the larger family of potentials with log-concave ground state. As a concrete instantiation, we use our result to extend the HWS analysis from a linear potential (which is exactly solvable) to a quadratic potential (which is no longer solvable). Our result strongly suggests the broader applicability of tunneling to convex potentials with spikes

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper establishes a framework proving log-concavity of the ground state for a family of discrete 1D Schrödinger operators (including convex potentials, as a discrete Brascamp-Lieb analog, and certain potentials with local minima). It applies this to derive new spectral gap bounds beyond Jarret-Jordan (2014) for convex potentials, and extends Reichardt's (2004) perturbative tunneling analysis from the linear Hamming-weight-with-spike problem to quadratic potentials with log-concave ground states, suggesting broader applicability of tunneling in AQO for convex functions with spikes.

Significance. If the discrete log-concavity result and its applications hold, the work supplies a structural property that strengthens gap estimates and extends tunneling analysis to a larger class of potentials. It builds directly on Brascamp-Lieb (1976), Jarret-Jordan (2014), and Reichardt (2004) without circularity or fitted parameters, and the concrete extension from linear to quadratic potentials is a clear incremental advance. The framework for log-concavity under stated conditions is positioned as the enabling step for both contributions.

major comments (2)
  1. [first half of the work] The log-concavity framework (first half) is load-bearing for both the gap bounds and the tunneling extension; the manuscript must explicitly verify that the discrete operator family satisfies the convexity or local-minima conditions without hidden restrictions on boundary terms or lattice size that could invalidate the ground-state property.
  2. [second half of the work] The extension of the perturbative analysis (second half) to quadratic potentials relies on log-concavity to carry over the gap lower bound; a direct comparison (e.g., explicit leading-order tunneling amplitude or gap scaling) between the exactly solvable linear case and the quadratic case is needed to substantiate that the result is not merely formal but quantitatively useful.
minor comments (2)
  1. Notation for the discrete Schrödinger operator (potential term, hopping) should be stated once at the outset with a clear reference to the continuous analog to avoid ambiguity in later sections.
  2. [Abstract] The abstract states the result 'strongly suggests' broader applicability; replace with a more precise statement of what is proved versus what is conjectured for general convex spiked potentials.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [first half of the work] The log-concavity framework (first half) is load-bearing for both the gap bounds and the tunneling extension; the manuscript must explicitly verify that the discrete operator family satisfies the convexity or local-minima conditions without hidden restrictions on boundary terms or lattice size that could invalidate the ground-state property.

    Authors: The framework is stated for finite 1D lattices equipped with standard open (Dirichlet) boundary conditions, and the convexity (or local-minima) assumption is imposed directly on the potential V. The proof of log-concavity is uniform in the lattice size N and does not invoke any special boundary cancellations. We have added an explicit verification paragraph at the beginning of Section 2 that lists the precise assumptions on V and the boundary operator and confirms that they are satisfied for all convex potentials on any finite chain, with no additional hidden restrictions. revision: yes

  2. Referee: [second half of the work] The extension of the perturbative analysis (second half) to quadratic potentials relies on log-concavity to carry over the gap lower bound; a direct comparison (e.g., explicit leading-order tunneling amplitude or gap scaling) between the exactly solvable linear case and the quadratic case is needed to substantiate that the result is not merely formal but quantitatively useful.

    Authors: Log-concavity supplies an identical lower bound on the spectral gap for both potentials, so the leading exponential dependence of the tunneling amplitude on spike height is the same. To make the comparison quantitative we have inserted a new subsection (4.3) that computes the leading-order prefactor explicitly for the quadratic spike and contrasts it with the exactly solvable linear case; the quadratic prefactor differs only by a curvature-dependent constant that remains O(1) independent of system size. This shows the extension is not merely formal. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper devises an independent framework to prove discrete log-concavity of the ground state for 1D Schrödinger operators under convex potentials (or local-minima conditions), explicitly positioned as a discrete analog of the external Brascamp-Lieb 1976 result. This property is then applied to derive new spectral-gap bounds (extending Jarret-Jordan 2014) and to extend Reichardt 2004 perturbative tunneling analysis to quadratic spiked potentials. No step reduces a claimed prediction or central result to a fitted parameter, self-citation chain, or definitional tautology; all load-bearing steps rely on the newly supplied proof framework and external citations whose authors do not overlap with the present work. The structure is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard results from functional analysis and quantum mechanics; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Discrete 1D Schrödinger operators with convex potentials admit log-concave ground states (discrete Brascamp-Lieb analogue)
    Invoked as the key structural property enabling both spectral-gap bounds and the extended perturbative analysis.

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