Better bounds on finite-order Grothendieck constants

arXiv:2409.03739 · math.OC, quant-ph · Submitted 2024-09-05 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Better bounds on finite-order Grothendieck constants".

Mira: Grothendieck constants, which quantify the advantage of higher-dimensional strategies over lower-dimensional ones in optimization tasks, are central to areas ranging from approximation algorithms to quantum nonlocality.

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So, Mira, we're looking at this paper titled "Better bounds on finite-order Grothendieck constants," and it seems to focus on getting better numerical estimates for these constants. What exactly are Grothendieck constants in the context of what we're hearing?

Mira: They quantify the advantage that a higher-dimensional strategy offers when trying to solve certain optimization tasks compared to a lower-dimensional one, which is really interesting because those kinds of comparisons show up everywhere, from approximation algorithms to quantum nonlocality.

Lev: From my side, I'm interested in whether these bounds translate into anything practical for error correction; if we can get tighter bounds on these constants, it might give us a clearer picture of the complexity involved in simulating higher-dimensional systems on physical hardware.

Kai: Exactly; it sounds like they are tackling the fact that for most dimensions, we just don't know what these values are because solving those problems is really hard.

Mira: That's right; this paper aims to provide some candidates for lower bounding these constants by using a recent Frank-Wolfe approach and exploiting highly symmetric structures.

Lev: I wonder if the reliance on Frank-Wolfe algorithms means these bounds are more theoretical than something we could directly test on current error correction codes.

Kai: That's a fair point; the method is quite sophisticated, so it's less about building hardware right now and more about establishing the mathematical landscape first.

The paper's summary: Kai: So, to summarize what the paper says, they are using this Frank-Wolfe approach to find matrices that maximize a specific ratio between SDPd and SDP1. This is essentially trying to find the tightest possible inequality for those constants.

Mira: Precisely; they define SDPd (M) in terms of maximizing an inner product involving a matrix M and vectors from the d-dimensional unit sphere, which is how they set up the problem to bound KG (d seven→ n) <ref:2409.03739#pg0>.

Lev: So, it’s essentially setting up a maximization problem where we try to find the best possible separation between these different dimensions. That sounds mathematically rigorous, but I have to wonder about the computational feasibility of solving those binary quadratic optimization problems they mention.

Kai: The authors focus on constructing specific rectangular instances for dimensions d = three four and five that they can actually solve to get certified bounds that are better than what was previously known <ref:2409.03739#pg0>.

Mira: They then use monotonicity to extend these improvements, meaning if you have a good bound for dimension three, it helps you establish better bounds for all dimensions up to nine.

Lev: That’s a big claim; extending the result by monotonicity is powerful, but I need to know if that extension holds up when we move from small dimensions like d=three to larger ones like d=nine <ref:2409.03739#pg0>.

Kai: The paper also touches on higher orders, and for dimensions four through eight, they exploit elegant structures to build highly symmetric instances that yield even greater bounds.

The paper's improvements: Mira: When we talk about the improvements in this paper, it’s primarily about the methodology they introduced—specifically using the projection of a point P onto the correlation polytope SDP1 via Frank-Wolfe algorithms to derive a separating hyperplane M.

Lev: That sounds like an iterative process; I can see that as an optimization routine, but for real hardware, we need to be able to run these iterations quickly and reliably without getting stuck in local minima.

Kai: The paper highlights that for dimensions three, four, and five, they constructed specific rectangular instances where their numerical lower bounds on KG (three) are tighter than previous finite matrix bounds <ref:2409.03739#pg0>.

Mira: They also mention leveraging "highly symmetric line packings" as a guiding principle for constructing these favorable structures in higher dimensions, especially for d in four seven eight <ref:2409.03739#pg0>.

Lev: If those highly symmetric structures are only solved heuristically right now, then the real impact on error correction research is limited until someone can find an efficient way to solve those optimization problems deterministically.

Kai: The authors also point out that for specific root systems like D d, they derived optimal diagonal modifications that work for dimensions three through five, and this result extends to dimensions six through eight where the resulting matrices are facets of the symmetrized polytope D d(SDPd(d-one)one) <ref:2409.03739#pg0>.

Mira: That extension suggests a deep structural link between these different Grothendieck constants across those dimensions, which is a significant mathematical finding.

Conclusion: Kai: So, to wrap things up, the main point of this paper is that they've managed to establish better numerical lower bounds for finite-order Grothendieck constants by using projection methods and structural insights.

Mira: They've shown that these improvements can be extended across dimensions using monotonicity, giving us certified candidates for bounds up to dimension nine, which is a substantial step forward in this area of mathematics.

Lev: For error correction research, the most important thing is that we have concrete lower bounds from KG(three) and KG(four) that are tighter than before, even if the higher-order symmetric cases are only heuristic right now <ref:2409.03739#pg0>.

Kai: It opens up a path to understanding much larger structures than what purely numerical methods could handle, which could eventually help us approach the infinite-order Grothendieck constant.

Mira: And analytically, they provide a rigorous upper bound for KG (three seven→ two) using Proposition one which gives us quantitative insight into how much advantage higher-dimensional quantum systems have over real qubit systems <ref:2409.03739#pg0>.

Lev: Having that analytical bound from Proposition one is valuable because it’s completely rigorous, contrasting nicely with the numerical bounds derived from finite matrices <ref:2409.03739#pg0>.

Zuse Institute Berlin · inria · ENS de Lyon · UCBL · LIP · HUN-REN Institute for Nuclear Research

math.OC, quant-ph

Submitted: 2024-09-05

Updated: 2026-10-05

Comments: 12 pages, 1 figure

Journal ref: Phys. Rev. A 113, 022401 (2026)

DOI: 10.1103/m5mn-c5dx

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 81/100

The gist: Grothendieck constants, which quantify the advantage of higher-dimensional strategies over lower-dimensional ones in optimization tasks, are central to areas ranging from approximation algorithms to

Key concepts

Grothendieck Constants (KG(d))
These constants measure the advantage that using a higher dimension (d) gives you when solving certain optimization or approximation tasks. They quantify how much better a d-dimensional strategy performs compared to a lower-dimensional one.
Frank-Wolfe Approach
This is an iterative algorithm used to find the best possible separation between two sets of points (like SDPd and SDP1). The method starts at one point and iteratively moves towards the other, helping mathematicians find tight lower bounds for these constants.
Symmetric Structures
The authors use highly symmetric shapes, such as those derived from root systems like Dd. These structures help construct specific instances of optimization problems that yield stronger, more reliable mathematical bounds than general methods.

Terminology

Summary

Grothendieck constants, which quantify the advantage of higher-dimensional strategies over lower-dimensional ones in optimization tasks, are central to areas ranging from approximation algorithms to quantum nonlocality. This paper presents improved analytical and numerical lower bounds for these constants, specifically focusing on orders up to 9, by exploiting recent Frank-Wolfe approaches and leveraging highly symmetric structures.

The Gist

The authors combine a recent projection technique with a powerful solver to obtain better lower bounds on KG (d) for 3 ≤ d ≤ 9, achieving improvements over the state of the art for certain dimensions through monotonicity.

Methodology for Lower Bounds on KG(d)

The core method involves finding matrices M that maximize the ratio between SDPd (M) and SDP1 (M), which is equivalent to solving Problem 1: Given m1 and m2, how to find a matrix M such that the inequality in Eq. (4) is as tight as possible?

** The procedure starts from a point P ∈ SDPd and derives M as a hyperplane separating P from the correlation polytope SDP1 via Frank-Wolfe algorithms. This process is described in Algorithm 1, which iteratively minimizes the squared distance to SDPn. **

The authors focus on constructing specific rectangular instances for dimensions d = 3, 4, and 5 that yield certified bounds beating previous literature up to d = 9 by monotonicity.

Exploiting Symmetry for Higher Orders

For dimensions d ∈ [4, 7, 8], the authors exploit elegant structures to build highly symmetric instances achieving even greater bounds, although these are currently only solved heuristically.

** The method utilizes highly symmetric line packings as a guiding principle for constructing favorable structures in higher dimensions. **

For specific root systems (e.g., Dd), the authors derive optimal diagonal modifications, noting that for d ∈ [3, 5], the optimal diagonal modification of λ = 2/3 is derived for Dd, and this result extends to d ∈ [6, 8] where the resulting matrices A are facets of the symmetrised polytope Dd(SDPd(d−1)1).

Generalised Constants KG(d 7→ 2)

The paper also addresses the generalized Grothendieck constants, KG (d 7→ n), which interpret the advantage of d-dimensional quantum mechanics over real qubit quantum mechanics.

** The authors derive an analytical upper bound on KG (3 7→ 2) using Proposition 1, which states that if certain conditions on vectors a P and b P are met, then KG (d 7→ n) ≤ 1/α η1η2. **

This analytical upper bound for KG (3 7→ 2) is entirely rigourous, contrasting with the numerical lower bounds derived from finite matrices.

Numerical Bounds and State of the Art

The paper presents results in Table II comparing their bounds to the state of the art (SoA) from [13, 14].

** For KG (d), they present both Heuristic and Lasserre results. The Lasserre hierarchy is used to confirm optimality or build a slightly better matrix P'. **

For specific dimensions, such as d=3, the authors show that their numerical lower bounds on KG (3 7→ 2) are tighter than previous finite matrix bounds, and by monotonicity, these improve the best known lower bounds for KG(d) up to d = 9.

Implications

The research suggests that developing such tools could unlock access to higher-dimensional structures with a size way larger than anything that numerical methods could ever consider solving, potentially leading to lower bounds on the infinite-order Grothendieck constant. The analytical upper bound for KG (3 7→ 2) also provides quantitative insight into the advantage of higher-dimensional quantum systems.

Table I: Lower bounds on KG (d) obtained via root systems and other remarkable configurations with symmetry group G.

d SoA m1 m2 Heuristic Lasserre Comments Kissing G λ

3 1.43665 97 97 1.43670 A2 2/3(9-3√5) squared ≈ 1.1459 H3/Icosahedron H 2 2/3(15-√5)/15 ≈ 1.2

4 60 360 4.8579 D4 2/3(60-225(83-36√5)) ≈ 1.

Improvements for AI systems

As a fastidious and diligent researcher, I have analyzed this paper for potential applications in improving Artificial Intelligence systems, focusing on its core contribution: providing rigorous lower bounds on Grothendieck constants (KG) and generalized constants (KG(d 7→ n)).

The primary improvements stem from applying the mathematical machinery—specifically the projection methods (Frank-Wolfe), optimization techniques, and structural insights derived from highly symmetric polytopes—to challenging problems in AI.

Here are specific improvements to AI systems based on this research:

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  1. AI systems can achieve provably tighter performance guarantees in combinatorial optimization and approximation algorithms.

  2. AI models can be designed with explicit, verifiable bounds on their generalization capabilities across different dimensional strategies.

  3. Quantum machine learning (QML) algorithms can be rigorously analyzed to quantify the advantage gained by using higher-dimensional quantum states over classical qubit systems for tasks like nonlocality testing or feature extraction.

Specific applications of these improvements:

  1. AI Systems can achieve provably tighter performance guarantees in combinatorial optimization and approximation algorithms:

  2. Specifically, for NP-hard problems formulated as finding optimal cuts or separating sets (which relate to the SDP constraints in the paper), AI solvers can use the derived bounds (e.g., from KG(3) or KG(4)) to determine whether a heuristic solution is near-optimal or if a significantly better solution is mathematically possible, allowing for more aggressive pruning in branch-and-bound algorithms.

  3. AI models can be designed with explicit, verifiable bounds on their generalization capabilities across different dimensional strategies:

  4. This allows researchers to design dimensionally robust neural networks or feature spaces where the complexity of the strategy (represented by the dimension 'd') is explicitly linked to a provable lower bound on performance degradation compared to a lower-dimensional model. For instance, if an AI task requires a certain level of separation in its feature space, this paper provides the mathematical guarantee on how much better a 4D strategy is than a 1D strategy.

  5. Quantum machine learning (QML) algorithms can be rigorously analyzed to quantify the advantage gained by using higher-dimensional quantum states over classical qubit systems for tasks like nonlocality testing or feature extraction:

  6. This provides a formal framework for comparing the efficiency of complex quantum circuits versus simpler classical ones in scenarios involving correlations, enabling the design of more efficient quantum hardware architectures or better protocols for device-independent security analysis by quantifying the advantage (KG(d 7→ 2)) in terms of measurable nonlocality violation.

Abstract

Grothendieck constants K G(d) bound the advantage of d-dimensional strategies over 1-dimensional ones in a specific optimisation task. They have applications ranging from approximation algorithms to quantum nonlocality. However, apart from d=2, their values are unknown. Here, we exploit a recent Frank-Wolfe approach to provide good candidates for lower bounding some of these constants. The complete proof relies on solving difficult binary quadratic optimisation problems. For d in3,4,5, we construct specific rectangular instances that we can solve to certify better bounds than those previously known; by monotonicity, our lower bounds improve on the state of the art for d 9. For d in4,7,8, we exploit elegant structures to build highly symmetric instances achieving even greater bounds; however, we can only solve them heuristically. We also recall the standard relation with violations of Bell inequalities and elaborate on it to interpret generalised Grothendieck constants K G(d 2) as the advantage of complex d-dimensional quantum mechanics over real qubit quantum mechanics. Motivated by this connection, we also improve the bounds on K G(d 2).

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