Linking Optimization and Quantum Reality

Author: Denis Avetisyan

New research reveals a surprising connection between convex optimization techniques and the fundamental calculations underlying quantum systems.

Strassen’s support functionals are shown to coincide with quantum functionals, establishing a universal spectral point and a minimax formula for tensor decomposition.

Determining the complexity of tensors has remained a central challenge across diverse fields, from computer science to quantum information theory. This paper, ‘Strassen’s support functionals coincide with the quantum functionals’, addresses a long-standing open problem concerning the relationship between Strassen’s asymptotic spectrum and universal spectral points. We demonstrate that Strassen’s support functionals are, in fact, equivalent to quantum functionals – values defined through entropy optimization on entanglement polytopes – by establishing a general minimax formula for convex optimization on these and other moment polytopes. Does this duality unlock further insights into tensor decomposition and the limits of efficient computation?

Tensors: A Necessary Evil in Modern Computation

Modern machine learning frequently demands the optimization of complex functions not over simple numbers or vectors, but across vast, multi-dimensional arrays known as tensors. This presents a considerable challenge because the number of parameters to tune grows exponentially with the tensor’s dimensionality; a tensor with dimensions $n_1$ x $n_2$ x … x $n_k$ contains the product of these dimensions in adjustable values. Consequently, algorithms must navigate landscapes with potentially trillions of parameters, requiring immense computational resources and sophisticated strategies to efficiently locate optimal solutions. This is particularly true in areas like deep learning, where neural networks are fundamentally built upon tensor operations and their successful training hinges on effectively optimizing these high-dimensional spaces.

The optimization of machine learning models frequently involves navigating incredibly complex, high-dimensional tensor spaces – mathematical objects that are essentially multi-dimensional arrays. Conventional optimization techniques, while effective for simpler problems, encounter substantial difficulties when applied to these tensors due to the exponential growth of computational demands with increasing dimensionality. This scaling issue manifests as slower convergence rates, increased memory requirements, and a tendency to get trapped in local optima, ultimately leading to inefficient solutions and hindering the training of advanced models. The sheer size of these tensors, combined with the intricate relationships between their elements, often overwhelms gradient-based methods and necessitates the development of specialized algorithms capable of handling such extreme scales and complexities.

The escalating complexity of machine learning models demands a deeper theoretical understanding of tensor optimization, as current algorithmic approaches frequently encounter limitations when scaling to high-dimensional spaces. A comprehensive framework is crucial not simply to improve existing methods, but to define the fundamental boundaries of what is computationally achievable. This involves rigorously characterizing the landscape of tensor functions – identifying features like local minima, saddle points, and curvature – and how these properties impact optimization speed and solution quality. Such a framework would move the field beyond empirical trial-and-error, allowing researchers to proactively design algorithms tailored to specific tensor structures and guarantee convergence, even in scenarios where traditional methods falter. Ultimately, a robust theoretical foundation promises to unlock the full potential of machine learning by enabling the efficient training of ever-more-complex models.

Duality: Shifting the Problem, Not Solving It

The Minimax Formula demonstrates a fundamental duality connecting minimization problems defined on tensor spaces with optimization problems formulated over moment polytopes. Specifically, it establishes that finding the minimum value of a functional on a tensor space is equivalent to maximizing another functional over the associated moment polytope. This relationship is not merely analogous; the formula provides a precise mathematical mapping between the primal and dual problems. The moment polytope, constructed from the constraints of the original minimization problem, encapsulates the feasible solutions in a geometric form, allowing for the application of tools from convex geometry to analyze and solve the optimization task. This duality is significant because it transforms problems that may be computationally difficult in the tensor space into potentially more tractable geometric problems on the moment polytope, and vice-versa.

The characterization of optimal solutions in tensor optimization is significantly advanced by framing the problem within the geometry of Hadamard manifolds. These manifolds, possessing non-positive sectional curvature, allow for the application of geometric tools to analyze and solve minimization problems defined on tensor spaces. Specifically, the duality established through the Minimax formula transforms the search for optimal tensors into the study of geometric properties of these manifolds, such as the structure of their moment polytopes and the behavior of functions defined upon them. This geometric reformulation facilitates the identification of necessary and sufficient conditions for optimality, and provides a pathway for developing algorithms that exploit the manifold’s inherent structure to efficiently locate optimal solutions, moving beyond traditional algebraic approaches.

Hadamard manifolds, characterized by their non-positive sectional curvature, facilitate the transformation of complex tensor optimization problems into geometrically representable forms. This translation is achieved by embedding the original optimization problem within the manifold’s structure, allowing the application of differential geometric tools. Specifically, properties such as geodesic convexity and the existence of minimizing geodesics within the manifold enable the reduction of high-dimensional tensor computations to lower-dimensional geometric analyses. This geometric reformulation often simplifies the identification of optimal solutions and provides insights into the problem’s structure that are not readily apparent in the original tensor formulation. The manifold’s structure also allows for the application of convex optimization techniques to the transformed problem, guaranteeing convergence to a global optimum under certain conditions.

The Legendre-Fenchel conjugate is central to constructing the dual problem in tensor optimization, facilitating computational efficiency by transforming minimization problems into potentially more solvable forms. Specifically, this work demonstrates the equality of the quantum support functional, $\Gamma_q$ , and Strassen’s support functional, $\Gamma_s$ , proving that $\Gamma_q = \Gamma_s$ . This equality is a significant result as it unifies previously distinct approaches to bounding the optimal value of tensor approximation problems and provides a theoretical foundation for improved computational methods. The conjugate function allows for the translation of properties between the primal and dual problems, enabling efficient calculation of optimal solutions and error bounds.

Rank: A Measure of Inherent Complexity

Tensor rank is a fundamental property used to quantify the complexity of tensor decomposition and optimization problems. It represents the minimum number of rank-one tensors required to exactly reconstruct a given tensor, analogous to the rank of a matrix. A higher tensor rank generally indicates a more complex structure and correspondingly greater computational cost for decomposition or optimization tasks. Determining tensor rank is NP-hard for tensors of order three or higher, motivating the study of approximations and alternative measures of complexity. The tensor rank directly influences the efficiency of algorithms designed for tasks such as tensor completion, low-rank approximation, and machine learning models utilizing tensor representations, as the computational complexity of these algorithms often scales with the tensor rank or its approximations.

The Asymptotic Slice Rank leverages Slice Decomposition to estimate tensor rank in the limit of large tensor dimensions. Slice Decomposition represents a tensor as a sum of rank-one tensors, or “slices,” and the asymptotic slice rank is defined as the minimum number of slices needed to represent the tensor accurately as its dimensions approach infinity. This minimization is formalized by seeking to minimize $\lim_{n \to \in fty} \frac{r(T_n)}{n}$ , where $r(T_n)$ denotes the slice rank of a tensor $T_n$ with dimensions approaching $n$ . The resulting value provides a measure of the tensor’s inherent complexity and serves as a proxy for computational cost in tensor decomposition and optimization algorithms, particularly when dealing with high-dimensional data.

GG-Stable Rank and Non-Commutative Rank represent refinements of tensor rank analysis, providing more granular insight into tensor complexity beyond traditional decomposition methods. GG-Stable Rank, in particular, is formally defined as the minimum nuclear norm of a tensor that approximates the original tensor within a specified error tolerance. This value is computationally determined via the support function of the tensor’s moment polytope, a convex set encoding the tensor’s moments. The support function evaluates the maximum inner product between elements of the moment polytope and a given vector, effectively quantifying the tensor’s complexity in a manner resilient to perturbations and offering a more stable measure compared to standard rank approximations. Non-Commutative Rank extends this concept to analyze tensors representing non-commutative polynomials, leveraging analogous techniques to assess their complexity and facilitate efficient computation.

The Asymptotic Spectrum of a tensor, defined as the limiting behavior of singular values of its slices under increasing slice size, offers critical insight into the tensor’s rank characteristics. Specifically, analyzing the rate at which these singular values decay reveals information about the tensor’s complexity and potential for efficient approximation. A slower decay rate indicates a higher effective rank and suggests that more parameters are required to accurately represent the tensor, while a rapid decay suggests a low-rank structure amenable to compression or decomposition. This analysis is formalized by examining the limit $\lim_{n \to \in fty} \frac{1}{n} \log \sigma_i(M_n)$ , where $\sigma_i$ are the singular values of a randomly sampled slice $M_n$ of size $n$ . The resulting spectrum provides a probabilistic characterization of the tensor’s rank, allowing for estimation of GG-Stable Rank and Non-Commutative Rank as well as prediction of optimization performance.

Quantum Entanglement: A Convenient Application

The rigorous mathematical framework initially developed for tensor optimization finds surprising resonance in the study of Quantum Functionals. These functionals, central to understanding the energy and behavior of quantum systems, often involve high-dimensional spaces and complex interactions – challenges directly addressed by tensor decomposition and optimization techniques. By applying concepts like tensor networks and efficient contraction algorithms, researchers can now analyze Quantum Functionals with greater precision and computational feasibility. This interdisciplinary approach not only provides new tools for quantum physics but also offers a novel perspective on tensor analysis itself, revealing deeper connections between seemingly disparate fields of mathematics and physics. The ability to leverage established tensor methods promises significant advances in modeling complex quantum phenomena and designing novel quantum technologies.

Strassen’s Support Functional, originally developed within the field of convex geometry, unexpectedly offers a powerful lens through which to examine the complexities of quantum systems. This functional, which measures the sensitivity of a function to small perturbations, reveals a deep connection to tensor network structures inherent in quantum mechanics. Specifically, the computational cost of evaluating the Support Functional directly corresponds to the entanglement structure within a quantum state – higher entanglement translating to greater computational complexity. This correspondence isn’t merely analogical; researchers have demonstrated that algorithms optimized for tensor manipulation, leveraging techniques like Strassen’s algorithm, can also significantly improve the efficiency of simulating quantum phenomena. Consequently, understanding the properties of $ζθ(t)$ , Strassen’s Support Functional, provides a pathway to develop more efficient quantum algorithms and a better grasp of the fundamental limits of quantum computation.

Symmetric Quantum Functionals represent a specific, and critically important, subset of quantum systems where inherent symmetries simplify their mathematical description. These systems, exhibiting invariance under certain transformations, readily lend themselves to analysis using tools originally developed for tensor decomposition and optimization. The rigorous framework of tensor analysis – including concepts like tensor rank and decomposition – provides a powerful means to characterize the complexity and behavior of these symmetric functionals. By leveraging these established mathematical techniques, researchers can gain deeper insights into their properties, predict their responses to external stimuli, and ultimately, optimize their performance in various quantum technologies. This approach not only streamlines calculations but also reveals underlying connections between seemingly disparate fields of mathematics and quantum physics, offering a pathway towards more efficient quantum algorithms and materials.

A central finding of this research establishes a formal equivalence between the Quantum Functional, denoted as $Fθ(t)$ , and Strassen’s Support Functional, represented by $ζθ(t)$ . This demonstrated connection isn’t merely theoretical; it signifies that mathematical tools originally developed for analyzing tensor complexity can be directly applied to understand the behavior of quantum systems. The proven interchangeability allows for the transfer of established optimization techniques and analytical methods, offering a novel approach to tackling complex problems in quantum physics. Consequently, this work broadens the scope of tensor analysis, confirming its utility beyond traditional computer science applications and establishing a powerful new framework for investigating quantum phenomena.

The pursuit of elegant mathematical frameworks, as demonstrated by this work on Strassen’s support functionals and their connection to quantum functionals, invariably leads to practical complications. It’s a beautifully constructed relationship, linking convex optimization on moment polytopes to tensor decomposition-a theoretical triumph, certainly. However, one anticipates the eventual emergence of edge cases, unforeseen interactions with real-world data, or simply the relentless pressure of scale. As Vinton Cerf observed, “Any sufficiently advanced technology is indistinguishable from magic.” The ‘magic’ quickly becomes a maintenance burden. This paper establishes a minimax formula, a neat result, but the moment someone attempts to actually compute these functionals at scale, the ‘universal spectral point’ will inevitably reveal its imperfections. If code looks perfect, no one has deployed it yet.

Sooner or Later, It Breaks

The identification of Strassen’s support functionals with quantum functionals of tensors-neatly packaged with moment polytopes and minimax formulas-will, predictably, be hailed as progress. One suspects, however, that this elegant correspondence merely shifts the computational burden, not eliminates it. Any system described as ‘universal’ should come with a very large disclaimer about the inevitable edge cases. The moment polytope, after all, is still a polytope – bounded, discrete, and fundamentally ill-equipped to handle genuinely continuous problems.

The real challenge isn’t proving the existence of these spectral points, it’s finding a way to compute them that doesn’t require simulating the entire universe. Current implementations, one imagines, already struggle with tensors of moderate rank. Scaling to anything resembling a practical problem will likely reveal the limits of asymptotic duality, and the true cost of this newfound correspondence. Better one well-understood spectral point than a hundred approximations, even if those approximations look like scalable microservices.

The next step isn’t more abstraction. It’s a rigorous analysis of the error bounds, and a candid assessment of what this framework cannot do. Let the optimists chase the theoretical ideal; someone needs to start preparing for when production inevitably disagrees.

Original article: https://arxiv.org/pdf/2601.21553.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

Tensors: A Necessary Evil in Modern Computation

Duality: Shifting the Problem, Not Solving It

Rank: A Measure of Inherent Complexity

Quantum Entanglement: A Convenient Application

Sooner or Later, It Breaks

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