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arXiv:2512.17713 · Quantum

Certified bounds on optimization problems in quantum theory

Younes Naceur, Jie Wang, Victor Magron, Antonio Acín

quant-phcs.SCmath.OCSubmitted Dec 19, 2025• Announced Dec 22, 2025• 0 citations
v1 · Dec 19, 2025v2 · Oct 26, 2026
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Abstract

Semidefinite relaxations of polynomial optimization have become a central tool for addressing the non-convex optimization problems over non-commutative operators that are ubiquitous in quantum information theory and, more in general, quantum physics. Yet, as these global relaxation methods rely on floating-point methods, the bounds issued by the semidefinite solver can - and often do - exceed the global optimum, undermining their certifiability. To counter this issue, we introduce a rigorous framework for extracting exact rational bounds on non-commutative optimization problems from numerical data, and apply it to several paradigmatic problems in quantum information theory. An extension to sparsity and symmetry-adapted semidefinite relaxations is also provided and compared to the general dense scheme. Our results establish rational post-processing as a practical route to reliable certification, pushing semidefinite optimization toward a certifiable standard for quantum information science.

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29 pages, 11 figures

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Semidefinite relaxations of polynomial optimization have become a central tool for addressing the non-convex optimization problems over non-commutative operators that are ubiquitous in quantum information theory and, more in general, quantum physics. Yet, as these global relaxation methods rely on floating-point methods, the bounds issued by the semidefinite solver can - and often do - exceed the global optimum, undermining their certifiability. To counter this issue, we introduce a rigorous framework for extracting exact rational bounds on non-commutative optimization problems from numerical data, and apply it to several paradigmatic problems in quantum information theory. An extension to sparsity and symmetry-adapted semidefinite relaxations is also provided and compared to the general dense scheme. Our results establish rational post-processing as a practical route to reliable certification, pushing semidefinite optimization toward a certifiable standard for quantum information science.
A fundamental problem in quantum coding theory is to determine the maximum size of quantum codes of given block length and distance. A recent work introduced bounds based on semidefinite programming, strengthening the well-known quantum linear programming bounds. However, floating-point inaccuracies prevent the extraction of rigorous non-existence proofs from the numerical methods. Here, we address this by providing rational infeasibility certificates for a range of quantum codes. Using a clustered low-rank solver with heuristic rounding to algebraic expressions, we can improve upon 181818 upper bounds on the maximum size of nnn-qubit codes with 6≤n≤196 \leq n \leq 196≤n≤19. Our work highlights the practicality and scalability of semidefinite programming for quantum coding bounds.
This thesis focuses on the intersection of mathematical and computational optimization and quantum information. Main contributions are open-source software code: A hybrid approach mixing "traditional" nonconvex and convex methods can make difficult problems more accessible. A demonstration of how to efficiently implement such an algorithm, avoiding interfacial bottlenecks, is provided, finding optimal protocols to establish entanglement through a lossy channel. The central software package developed addresses polynomial optimization problems. Many problems naturally involve only a polynomial objective and constraint polynomials. Such problems can automatically be cast into semidefinite programs that provide a hierarchy of outer approximations. The resulting problems are often so large and scale so unfavorably with respect to the variable number and degree involved that the boundary of the doable is reached quickly. However, technical progress both in hardware and algorithms has pushed this boundary - but software frameworks for polynomial optimization have not followed in the same manner, often now making them the bottleneck that before was the solver. The package PolynomialOptimization.jl developed during this thesis aims to fill the gap and provide a very resource-efficient intermediate layer together with a wide number of algorithms to reduce the problem size, and naturally supporting complex numbers and semidefinite constraints ubiquitous in quantum information problems. Its application on an entanglement distribution problem is demonstrated, showing that even relaxations with semidefinite matrices of three- and four-digit size can be solved conveniently. Finally, a new way to calculate interior-point barriers for the cone of sums-of-squares matrices in a nearly time-optimal way is developed, whose efficient implementation has the potential of further reducing resource consumption.
Bounding the correlations predicted by quantum theory is an important challenge in quantum information science. Today's leading approach is semidefinite programming relaxations, but existing methods still cannot account for many relevant types of constraints. Here, we propose a semidefinite relaxation methodology that can incorporate a breadth of constraints needed in various quantum correlation problems, thereby generalising the seminal Navascués-Pironio-Acín hierarchy. It yields useful results at reasonable computational cost. We showcase the methodology and its features by using it to address five different quantum information problems. These are (i) entanglement witnessing from imperfect measurement devices, (ii) certifying measurements from fidelity-constrained sources, (iii) computing dimensionality in genuine multi-particle entangled states, (iv) benchmarking dimensionality for state preparation devices, and (v) finding uncertainty relations for nearly anti-commuting observables. These applications reflect both the usefulness and versatility of the methodology, as well as its potential for broader relevance in the field.
Semidefinite programming hierarchies for commutative and non-commutative polynomial optimization represent a powerful computational tool with many applications in quantum information. In such applications, a given variable is typically not either commuting or non-commuting with all other variables, but instead commutes with some variables and does not commute with others, i.e., the variables satisfy some partial commutation relations. While such partial commutation relations can always be incorporated in a fully non-commutative setting through suitable linear constraints in the semidefinite programming relaxations, exploiting their algebraic properties from the onset can result in more compact relaxations. This leads us to introduce partially-commutative polynomial optimization, a framework that encompasses commutative and non-commutative polynomial optimization, allowing for arbitrary commutation relations among the variables. We point out that the underlying algebraic structure is that of a partially-commutative monoid. We present and review several key aspects of such monoids and show how they can be used to build SDP relaxations for partially-commutative polynomial optimization problems in which the partial commutations are natively implemented in the monomial structure, without the need of additional linear constraints.
We study quantum algorithms for approximating Lasserre's hierarchy values for polynomial optimization. Let f,g1,…,gmf,g_1,\ldots,g_mf,g1​,…,gm​ be real polynomials in nnn variables and f⋆f^\starf⋆ the infimum of fff over the semialgebraic set S(g)={x:gi(x)≥0}S(g)=\{x: g_i(x)\ge 0\}S(g)={x:gi​(x)≥0}. Let λkλ_kλk​ be the value of the order-kkk Lasserre relaxation. Assume either (i) f⋆=λkf^\star=λ_kf⋆=λk​ and the optimum is attained in the ℓ1\ell_1ℓ1​-ball of radius 1/21/21/2, or (ii) S(g)S(g)S(g) lies in the simplex {x≥0:∑jxj≤1/2}\{x\ge 0: \sum_j x_j\le 1/2\}{x≥0:∑j​xj​≤1/2}, and the constraints define this simplex. After an appropriate coefficient rescaling, we give a quantum algorithm based on matrix multiplicative weights that approximates λkλ_kλk​ to accuracy ε>0\varepsilon>0ε>0 with runtime, for fixed kkk, O(nkε−4+nk/2ε−5),O ⁣(sg ⁣[nkε−4+ ⁣(nk+ ⁣∑i=1mnk−di)1/2 ⁣ε−5]), O(n^k\varepsilon^{-4}+n^{k/2}\varepsilon^{-5}),\qquad O\!\left(s_g\!\left[n^k\varepsilon^{-4}+\!\left(n^{k}+\!\sum_{i=1}^m n^{k-d_i}\right)^{1/2}\!\varepsilon^{-5}\right]\right), O(nkε−4+nk/2ε−5),O(sg​[nkε−4+(nk+∑i=1m​nk−di​)1/2ε−5]), where sgs_gsg​ bounds the sparsity of the coefficient-matching matrices associated with the constraints. Classical matrix multiplicative-weights methods scale as O(n3kpoly(1/ε))O(n^{3k}\mathrm{poly}(1/\varepsilon))O(n3kpoly(1/ε)) even in the unconstrained case. As an example, we obtain an O(nε−4+nε−5)O(n\varepsilon^{-4}+\sqrt{n}\varepsilon^{-5})O(nε−4+n​ε−5) quantum algorithm for portfolio optimization, improving over the classical O(nω+1log⁡(1/ε))O(n^{ω+1}\log(1/\varepsilon))O(nω+1log(1/ε)) bound with ω≈2.373ω\approx2.373ω≈2.373. Our approach builds on and sharpens the analysis of Apeldoorn and Gilyén for the SDPs arising in polynomial optimization. We also show how to implement the required block encodings without QRAM. Under the stated assumptions, our method achieves a super-quadratic speedup in the problem dimension for computing Lasserre relaxations.
The generalized quantum Arimoto--Blahut (QAB) algorithm is a powerful derivative-free iterative method in quantum information theory. A key obstacle to its broader use is that existing convergence guarantees typically rely on analytical conditions that are either overly restrictive or difficult to verify for concrete problems. We address this issue by introducing an a posteriori certification viewpoint: instead of requiring fully a priori verifiable assumptions, we provide convergence and error guarantees that can be validated directly from the iterates produced by the algorithm. Specifically, we prove a generalized global convergence theorem showing that, under convexity and a substantially weaker numerically verifiable condition, the QAB iteration converges to the global minimizer. This theorem yields a practical certification procedure: by checking explicit inequalities along the computed trajectory, one can certify global optimality and bound the suboptimality of the obtained value. As an application, we develop a certified iterative scheme for computing the quantum relative entropy of channels, a fundamental measure of distinguishability in quantum dynamics. This quantity is notoriously challenging to evaluate numerically: gradient-based methods are impeded by the complexity of matrix functions such as square roots and logarithms, while recent semidefinite programming approaches can become computationally and memory intensive at high precision. Our method avoids these bottlenecks by combining the QAB iteration with a posteriori certification, yielding an efficient and scalable algorithm. Numerical experiments demonstrate rapid convergence and improved scalability and adaptivity compared with SDP-based approaches.