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.
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.
Many fundamental problems in quantum information theory can be formulated as optimization tasks over quantum states and measurements. While the convex structure of quantum theory often enables powerful semidefinite programming formulations, a major challenge persists: the exponential growth of Hilbert space dimension with system size. This dissertation discusses techniques to address these and related obstacles across several core problems in quantum information. We first study optimization over separable quantum states. Such problems are typically NP-hard and standard semidefinite programming hierarchies suffer from the curse of dimensionality. We introduce a hybrid algorithm that leverages a quantum co-processor to reduce the effective dimension of the problem. While several existing approaches provide only a numerical bound on the optimal value, our method is guaranteed to return an explicit feasible solution: a valid separable state, even in instances where it is not optimal. We apply this framework to approximate the minimum separable energy of many-body Hamiltonians, define an entanglement measure for their ground spaces, and demonstrate numerical scalability to systems of up to 28 qubits. We next consider the general Bayes framework, as introduced by Helstrom, for state discrimination when, instead of a classical description of the candidate states, one has access to their source code: the quantum circuit that prepares them. We show that the typical semidefinite program for the discrimination problem can be reformulated in terms of the Gram matrix of these states, reducing the variable dimensions from dL to NL, where d is the Hilbert space dimension, N is the number of candidate states, and L is the number of possible guesses. Importantly, we further introduce a quantum pre-processing procedure which efficiently constructs the reduced problem from the source code, enabling our method to operate directly on quantum data. We consider two applications. First, we characterize the optimal identifications for quantum changepoint problems under several reward structures whose associated discrimination problems were previously computationally inaccessible. Second, we consider a quantum error classification problem and show how our reduction makes it tractable for systems of hundreds of qubits. We also analyze structural measurement strategies. As a counterpart to the well-known pretty good measurement for minimum-error state discrimination, we introduce the pretty bad measurement for the exclusion variant, in which the goal is to identify a state that was not sent, rather than to identify the one that was, and prove that both are guaranteed to perform at least as well as blind guessing in their respective objectives. Finally, we consider evaluating Boolean functions under severe memory constraints and prove that a simple local "greedy" measurement strategy is globally optimal precisely for affine Boolean functions, while more generally achieving at least the square of the optimal global success probability. Together, these results demonstrate that high-dimensional quantum optimization problems often admit hidden structure enabling dimension reduction, hybrid computation, and provable approximation guarantees.
Quantum information processing (QIP) studies the representation, manipulation, and transmission of information using quantum systems. By leveraging counterintuitive quantum phenomena such as superposition and entanglement, QIP enables new paradigms for computation, communication, and learning that can surpass classical limits in certain regimes. Foundational tasks in QIP include quantum data compression, quantum communication, and quantum learning and optimization, which together form the basis for scalable quantum technologies. This thesis develops new frameworks and algorithms at the intersection of quantum information theory, quantum learning, and quantum-inspired modeling. The thesis is organized into three parts. The first part studies quantum compression problems; the second addresses quantum communication; and the third develops quantum learning and optimization frameworks with applications to complex systems. In the first part, a new formulation for lossy quantum-classical source coding, termed rate-channel theory, is introduced, replacing the conventional distortion-based formulation with a posterior-channel-based framework under a block-error criterion. The framework is further generalized to incorporate quantum side information at the decoder. A connection between rate-channel theory and classical rate-distortion theory is also established, showing that optimal rate-distortion performance can be recovered under suitable conditions. These results provide a unified and flexible framework for data compression, which is essential for scalable quantum computing infrastructure, enabling efficient storage of quantum data, such as outputs from quantum sensors (e.g., MRI) and information encoded in photonic systems for satellite and optical communication. In the second part, the problem of computing functions of distributed sources over classical-quantum multiple access channels is studied. A new coding framework is developed that combines algebraic structured codes with unstructured random coding techniques, enabling efficient communication of functions without reconstructing individual sources. The resulting achievable conditions, expressed in terms of single-letter quantum information quantities, strictly enlarge the class of sources for which reliable computation is possible. These results contribute to the theoretical foundations of quantum network communication and are directly relevant to deep-space satellite-based quantum communication, distributed quantum computing, and the design of scalable quantum network infrastructures. In the final part, new approaches to quantum learning and optimization are developed. A geometry-aware quantum optimization framework is proposed based on a novel quantum information metric and an unbiased estimator constructed from single-shot measurements. This leads to an optimization method with constant sample complexity per iteration and provable exponential convergence under mild conditions, while avoiding saddle points and local minima. Such advances are particularly relevant for challenging optimization tasks in quantum sensing and quantum computational modeling of many-body systems. Building on this optimization framework, a Hamiltonian learning approach is introduced to infer interaction structures from time-resolved measurement data, with finite-sample guarantees and polynomial scaling with system size. This framework is further extended to develop a quantum-inspired generative model for gene regulatory network inference, where interactions are encoded as a parameterized Hamiltonian evolving over pseudotime. The proposed methods demonstrate strong performance on synthetic benchmarks and reveal biologically meaningful structures in real-world data, with applications to multi-omics integration, cancer genomics, and systems biology. Overall, this thesis advances the theoretical and algorithmic foundations of quantum information processing, with contributions spanning compression, communication, and learning. These developments support the design of resource-efficient and scalable quantum technologies, enabling applications in quantum sensing, quantum networks, distributed quantum computation, and the modeling of complex physical and biological systems.
We introduce QICS (Quantum Information Conic Solver), an open-source primal-dual interior point solver fully implemented in Python, which is focused on solving optimization problems arising in quantum information theory. QICS has the ability to solve optimization problems involving the quantum relative entropy, noncommutative perspectives of operator convex functions, and related functions. It also includes an efficient semidefinite programming solver which exploits sparsity, as well as support for Hermitian matrices. QICS is also currently supported by the Python optimization modelling software PICOS. This paper aims to document the implementation details of the algorithm and cone oracles used in QICS, and serve as a reference guide for the software. Additionally, we showcase extensive numerical experiments which demonstrate that QICS outperforms state-of-the-art quantum relative entropy programming solvers, and has comparable performance to state-of-the-art semidefinite programming solvers.
Quantum computing is a rapidly developing field that has seen a huge amount of interest in the last couple of decades due to its promise of revolutionizing several domains of business and science. It presents a new way of doing computations by making use of fundamental properties of quantum mechanics such as superposition and entanglement. Optimization, on the other hand, is a field that is omnipresent in the industry and where small improvements can have a significant impact. This thesis aims to tackle optimization problems using quantum algorithms.NP-hard optimization problems are not believed to be exactly solvable through general polynomial time algorithms. Variational quantum algorithms (VQAs) to address such combinatorial problems have been of great interest recently. Such algorithms are heuristic and aim to obtain an approximate solution. The hardware, however, is still in its infancy and the current Noisy Intermediate Scale Quantum (NISQ) computers are not able to optimize industrially relevant problems. Moreover, the storage of qubits and introduction of entanglement require extreme physical conditions.An issue with contemporary quantum optimization algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) is that they scale linearly with problem size. To tackle this issue, we present the LogQ encoding, using which we can design quantum variational algorithms that scale logarithmically with problem size - opening an avenue for treating optimization problems of unprecedented scale on gate-based quantum computers. We show how this algorithm can be applied to several combinatorial optimization problems such as Maximum Cut, Minimum Partition, Maximum Clique and Maximum Weighted Independent Set (MWIS). Subsequently, these algorithms are tested on a quantum simulator with graph sizes of over a hundred nodes and on a real quantum computer up to graph sizes of 256. To our knowledge, these constitute the largest realistic combinatorial optimization problems ever run on a NISQ device, overcoming previous problem sizes by almost tenfold.Next, we apply the LogQ encoding to two use-cases for large companies such as TotalEnergies. Fleet conversion is the process of transitioning a fleet of vehicles to more sustainable and environmentally friendly alternatives. It is modeled as a column generation scheme with the MWIS problem as the sub-problem or worker problem. We use the LogQ method to solve the MWIS Workers and demonstrate how quantum and classical solvers can be used together in a hybrid manner to approach an industrial-sized use-case. Mesh segmentation refers to the process of dividing a complex mesh (composed of vertices, edges, and faces) into meaningful and semantically coherent parts or regions. Mesh segmentation plays an important part in computer modeling, which is extensively used in the core domains of TotalEnergies' activities such as Earth imaging, physical modeling for reservoirs, and others. We define the problem as a graph optimization problem and use the LogQ encoding to solve it.
This chapter covers applications of quantum computing in the area of continuous optimization, including both convex and nonconvex optimization. We discuss quantum algorithms for computing Nash equilibria for zero-sum games and for solving linear, second-order, and semidefinite programs. These algorithms are based on quantum implementations of the multiplicative weights update method or interior point methods. We also discuss general quantum algorithms for convex optimization which can provide a speedup in cases where the objective function is much easier to evaluate than the gradient of the objective function. Finally, we cover quantum algorithms for escaping saddle points and finding local minima in nonconvex optimization problems.
(English) The thesis considers a number of optimisation techniques applied in the context of quantum information theory. After a pedagogical introduction of both quantum information theory and optimisation, it considers three main avenues of research. The first is the well-known foundational open problem of mutually unbiased bases, which consists of finding sets of orthonormal bases that are each unbiased with one another. More specifically, it remains unknown whether one can find a set of 4 mutually unbiased bases in dimension 6. A variety of optimisation techniques are applied, including non-linear semidefinite programming, see-saw optimisation, semidefinite programming relaxations, branch-and-cut, gradient descent methods and the method of Lagrange multipliers, each providing further insights into the problem. The second avenue is that of Bell nonlocality, more specifically attempting to simplify the hierarchy of semidefinite programs known as the NPA (Navascués-Pironio-Acín) hierarchy used to find bounds on the maximum quantum violation of Bell inequalities. For the case in which one has a large number of inputs per party, advantage in both memory and time versus state-of-the-art solvers is demonstrated using a combination several optimisation techniques. The third avenue is that of many-body quantum physics, which encompasses a wide range of topics. The thesis considers the problems of bounding expectation values of observables over the steady-states of open quantum systems, finding improved Fermion-to-qubit mappings and solving the graph colouring problem with a novel qudit-inspired optimisation algorithm. In each case, advantage versus comparable methods is demonstrated. (Català) La tesi discorre sobre un nombre de tècniques d’optimització que poden aplicar-se en el context de la teoria de la informació quàntica. Després d’una introducció pedagògica tant de la teoria de la informació quàntica com de l’optimització, té en compte tres línies principals de recerca. En primer lloc, una de prou coneguda, el problema obert i fonamental de les bases mútuament imparcials, que consisteix en trobar conjunts de bases ortonormals que siguin imparcials entre si. En concret, encara es desconeix si hom pot trobar un conjunt de 4 bases mútuament imparcials amb la dimensió 6. S’hi apliquen diverses tècniques d’optimització, entre les quals hi ha la programació semidefinida no lineal, l’optimització cíclica, les relaxacions en programació semidefinida, el mètode Branch and cut, mètodes de gradient descendent i el mètode dels multiplicadors de Lagrange, i cadascuna aprofundeix amb noves perspectives al problema. La segona línia és la de la no localitat de Bell, en particular amb la intenció de simplificar la jerarquia de la programació semidefinida que rep el nom de jerarquia d'NPA (Navascués-Pironio-Acín), que s’utilitzen per trobar fites als valors que maximitzen quànticament la violació de les desigualtats de Bell. Pel cas en què hi ha un gran nombre d'entrades per cada part, es demostra l’avantatge tant en memòria com en temps envers els solucionadors més avançats utilitzant una combinació de diverses tècniques d’optimització. La tercera línia és la de la física quàntica de molts cossos, que abraça un ampli ventall de temes. La tesi considera els problemes de fitar els valors esperats dels observables en els estats estacionaris de sistemes quàntics oberts, per trobar millors aplicacions de fermions a qubits i resoldre el problema de coloració de grafs mitjançant un nou algorisme d’optimització inspirat en qudits. En cada cas, es demostra un avantatge respecte a mètodes comparables. (Español) La tesis considera varias técnicas de optimización aplicadas en el contexto de la teoría de la información cuántica. Luego de una introducción pedagógica a la teoría de la información cuántica y a la optimización, se abordan tres líneas principales de investigación. La primera es el afamado problema abierto fundamental de las bases mutuamente insesgadas, que consiste en encontrar conjuntos de bases ortonormales que sean iinsesgadas entre sí. Más específicamente, aún se desconoce si es posible encontrar un conjunto de 4 bases mutuamente insesgadas en dimensión 6. Se aplican diversas técnicas de optimización, incluyendo programación semidefinida no lineal, optimización see-saw, relajaciones de programación semidefinida, branch-and-cut, métodos de descenso de gradiente y el método de los multiplicadores de Lagrange, que ofrecen nuevas perspectivas sobre el problema. La segunda línea de investigación se centra en la no localidad de Bell, y más concretamente, en la simplificación de la jerarquía de programas semidefinidos conocida como jerarquía NPA (Navascués-Pironio-Acín), utilizada para encontrar cotas sobre la máxima violación cuántica de las desigualdades de Bell. En el caso de un gran número de entradas por parte, se demuestra una ventaja tanto en memoria como en tiempo frente a los solucionadores más avanzados, combinando diversas técnicas de optimización. La tercera línea de investigación trata sobre la física cuántica de muchos cuerpos, un campo que abarca una amplia variedad de temas. La tesis considera los problemas de acotar los valores esperados de observables en los estados estacionarios de sistemas cuánticos abiertos, encontrar mejores mapeos de fermiones a cúbits y resolver el problema de coloreo de grafos mediante un novedoso algoritmo de optimización inspirado en qudits. En cada caso, se muestra una ventaja en comparación con métodos similares.