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Feedback-Based Quantum Algorithms for Optimization

Salahuddin Abdul Rahman

Published Jan 1, 2026• 0 citations
Publisher

Abstract

This thesis develops a control-theoretic framework for the design and analysis of quantum optimization algorithms.The central idea is to model quantum algorithms as controlled dynamical systems in which quantum gates act as control inputs that steer the system's state toward the ground or excited states of a target Hamiltonian.Within this framework, the thesis advances the family of feedback-based quantum algorithms (FQAs), building on Lyapunov-inspired feedback principles introduced in prior work, and develops new algorithmic extensions for constrained optimization and excited state preparation.This work is motivated by the potential of FQAs to tackle complex optimization problems, such as preparing ground states of Hamiltonians, which underpin a wide range of applications in engineering, chemistry, and combinatorial optimization.This work enhances the performance of FQAs and extends their applicability to a broader class of quantum optimization problems.It consists of three main parts.The first part focuses on enhancing the performance and robustness of FQAs.It introduces an adaptive sampling noise mitigation technique that incorporates a new control law into the algorithm design to improve robustness and convergence with limited measurements.The second part extends the feedback-based framework to excited state preparation, yielding two new algorithms: the feedback-based quantum algorithm for excited states preparation (FQAE) and the weighted feedback-based quantum algorithm for excited states preparation (WFQAE), with applications in quantum chemistry and many-body physics.The third part generalizes the feedback-based framework to tackle more complex combinatorial optimization problems, specifically those with inequality and invalid configuration constraints, by proposing the feedback-based algorithm for quantum optimization with constraints (FALQON-C) and the feedback-based algorithm for quantum optimization with invalid configuration constraints (FALQON-IC).The design of these algorithms leverages control theory to reduce circuit depth while maintaining convergence efficiency within the feedback framework.The thesis demonstrates that control-theoretic tools such as Lyapunov stability analysis and feedback design provide a systematic foundation for developing quantum algorithms with improved robustness, generalization, and scalability.The results offer new insights into quantum optimization and highlight the role of control theory as a guiding principle for the development of efficient feedback-based quantum algorithms.iii ResuméDenne afhandling handler om, hvordan man kan bruge kontrolteori til at udvikle og analysere kvantealgoritmer til optimering.Hovedideen er at betragte kvantealgoritmer som dynamiske systemer, der kan styres, hvor kvanteoperatorer fungerer som styringssignaler, der gradvist bringer systemet hen mod en ønsket tilstand, for eksempel grundtilstanden af en given Hamilton-operator.Inden for denne ramme videreudvikler afhandlingen såkaldte feedback-baserede kvantealgoritmer (FQAs).Disse algoritmer bruger løbende målinger og feedback til at justere sig selv, inspireret af klassiske principper fra kontrolteori.Afhandlingen bygger videre på tidligere arbejde og introducerer nye metoder, som gør det muligt både at løse optimeringsproblemer med begrænsninger og at beregne exciterede tilstande.Motivation for arbejdet er, at denne type algoritmer har potentiale til at løse svære optimeringsproblemer, som f.eks.at finde grundtilstande af Hamiltonoperatorer.Sådanne problemer er centrale i mange anvendelser inden for ingeniørvidenskab, kemi og kombinatorisk optimering.Afhandlingen forbedrer både ydeevnen og anvendeligheden af feedback-baserede kvantealgoritmer og er opdelt i tre hoveddele.Den første del fokuserer på at gøre algoritmerne mere robuste og mindre følsomme over for støj i målinger, blandt andet ved at indføre en adaptiv metode, som fungerer bedre, når antallet af målinger er begrænset.Den anden del udvider metoden til også at kunne beregne exciterede tilstande.Dette fører til to nye algoritmer, som kan anvendes inden for blandt andet kvantekemi og mangelegeme-fysik.Den tredje del viser, hvordan den feedback-baserede tilgang kan bruges til mere komplekse optimeringsproblemer, hvor der er begrænsninger eller ugyldige løsninger.Her introduceres nye algoritmer, der er designet til at håndtere sådanne problemer effektivt, samtidig med at kvantekredsløbene holdes så simple som muligt.Samlet set viser afhandlingen, at værktøjer fra kontrolteori, som stabilitetsanalyse og feedback-design, giver et solidt og systematisk grundlag for at udvikle mere robuste, fleksible og skalerbare kvantealgoritmer.Resultaterne giver ny indsigt i kvanteoptimering og understreger kontrolteoriens vigtige rolle i udviklingen af effektive feedback-baserede kvantealgoritmer.

DOI
10.54337/aau828432982

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This thesis develops a control-theoretic framework for the design and analysis of quantum optimization algorithms.The central idea is to model quantum algorithms as controlled dynamical systems in which quantum gates act as control inputs that steer the system's state toward the ground or excited states of a target Hamiltonian.Within this framework, the thesis advances the family of feedback-based quantum algorithms (FQAs), building on Lyapunov-inspired feedback principles introduced in prior work, and develops new algorithmic extensions for constrained optimization and excited state preparation.This work is motivated by the potential of FQAs to tackle complex optimization problems, such as preparing ground states of Hamiltonians, which underpin a wide range of applications in engineering, chemistry, and combinatorial optimization.This work enhances the performance of FQAs and extends their applicability to a broader class of quantum optimization problems.It consists of three main parts.The first part focuses on enhancing the performance and robustness of FQAs.It introduces an adaptive sampling noise mitigation technique that incorporates a new control law into the algorithm design to improve robustness and convergence with limited measurements.The second part extends the feedback-based framework to excited state preparation, yielding two new algorithms: the feedback-based quantum algorithm for excited states preparation (FQAE) and the weighted feedback-based quantum algorithm for excited states preparation (WFQAE), with applications in quantum chemistry and many-body physics.The third part generalizes the feedback-based framework to tackle more complex combinatorial optimization problems, specifically those with inequality and invalid configuration constraints, by proposing the feedback-based algorithm for quantum optimization with constraints (FALQON-C) and the feedback-based algorithm for quantum optimization with invalid configuration constraints (FALQON-IC).The design of these algorithms leverages control theory to reduce circuit depth while maintaining convergence efficiency within the feedback framework.The thesis demonstrates that control-theoretic tools such as Lyapunov stability analysis and feedback design provide a systematic foundation for developing quantum algorithms with improved robustness, generalization, and scalability.The results offer new insights into quantum optimization and highlight the role of control theory as a guiding principle for the development of efficient feedback-based quantum algorithms.iii ResuméDenne afhandling handler om, hvordan man kan bruge kontrolteori til at udvikle og analysere kvantealgoritmer til optimering.Hovedideen er at betragte kvantealgoritmer som dynamiske systemer, der kan styres, hvor kvanteoperatorer fungerer som styringssignaler, der gradvist bringer systemet hen mod en ønsket tilstand, for eksempel grundtilstanden af en given Hamilton-operator.Inden for denne ramme videreudvikler afhandlingen såkaldte feedback-baserede kvantealgoritmer (FQAs).Disse algoritmer bruger løbende målinger og feedback til at justere sig selv, inspireret af klassiske principper fra kontrolteori.Afhandlingen bygger videre på tidligere arbejde og introducerer nye metoder, som gør det muligt både at løse optimeringsproblemer med begrænsninger og at beregne exciterede tilstande.Motivation for arbejdet er, at denne type algoritmer har potentiale til at løse svære optimeringsproblemer, som f.eks.at finde grundtilstande af Hamiltonoperatorer.Sådanne problemer er centrale i mange anvendelser inden for ingeniørvidenskab, kemi og kombinatorisk optimering.Afhandlingen forbedrer både ydeevnen og anvendeligheden af feedback-baserede kvantealgoritmer og er opdelt i tre hoveddele.Den første del fokuserer på at gøre algoritmerne mere robuste og mindre følsomme over for støj i målinger, blandt andet ved at indføre en adaptiv metode, som fungerer bedre, når antallet af målinger er begrænset.Den anden del udvider metoden til også at kunne beregne exciterede tilstande.Dette fører til to nye algoritmer, som kan anvendes inden for blandt andet kvantekemi og mangelegeme-fysik.Den tredje del viser, hvordan den feedback-baserede tilgang kan bruges til mere komplekse optimeringsproblemer, hvor der er begrænsninger eller ugyldige løsninger.Her introduceres nye algoritmer, der er designet til at håndtere sådanne problemer effektivt, samtidig med at kvantekredsløbene holdes så simple som muligt.Samlet set viser afhandlingen, at værktøjer fra kontrolteori, som stabilitetsanalyse og feedback-design, giver et solidt og systematisk grundlag for at udvikle mere robuste, fleksible og skalerbare kvantealgoritmer.Resultaterne giver ny indsigt i kvanteoptimering og understreger kontrolteoriens vigtige rolle i udviklingen af effektive feedback-baserede kvantealgoritmer.
Feedback-based quantum algorithms have recently emerged as potential methods for approximating the ground states of Hamiltonians . One such algorithm, the feedback-based algorithm for quantum optimization (FALQON), is specifically designed to solve quadratic unconstrained binary optimization problems. Its extension, the feedback-based algorithm for quantum optimization with constraints (FALQON-C), was introduced to handle constrained optimization problems with equality and inequality constraints . In this work, we extend the feedback-based quantum algorithms framework to address a broader class of constraints known as invalid configuration (IC) constraints, which explicitly prohibit specific configurations of decision variables. We first present a transformation technique that converts the constrained optimization problem with invalid configuration constraints into an equivalent unconstrained problem by incorporating a penalizing term into the cost function. Then, leaning upon control theory, we propose an alternative method tailored for feedback-based quantum algorithms that directly tackles IC constraints without requiring slack variables. Our approach introduces a new operator that encodes the optimal feasible solution of the constrained optimization problem as its ground state. Then, a controlled quantum system based on the Lyapunov control technique is designed to ensure convergence to the ground state of this operator. Two approaches are introduced in the design of this operator to address IC constraints: the folded spectrum approach and the deflation approach. These methods eliminate the need for slack variables, significantly reducing the quantum circuit depth and the number of qubits required. We show the effectiveness of our proposed algorithms through numerical simulations.
The ground state properties of quantum many-body systems are a subject of interest across chemistry, materials science, and physics. Thus, algorithms for finding ground states can have broad impacts. Variational quantum algorithms are one class of ground state algorithms that has received significant attention in recent years. These algorithms utilize a hybrid quantum-classical computing framework to prepare ground states on quantum computers. However, this requires solving a classical optimization problem that can become prohibitively expensive in high dimensions. Here, we develop formulations of feedback-based quantum algorithms for ground state preparation that can be used to address this challenge for two broad classes of Hamiltonians: Fermi-Hubbard Hamiltonians, and molecular Hamiltonians represented in second quantization. Feedback-based quantum algorithms are optimization-free; in place of classical optimization, quantum circuit parameters are set according to a deterministic feedback law derived from quantum Lyapunov control principles. This feedback law guarantees a monotonic improvement in solution quality with respect to the depth of the quantum circuit. A variety of numerical illustrations are provided that analyze the convergence and robustness of feedback-based quantum algorithms for these problem classes. Published by the American Physical Society 2024
Quantum optimization, for both classical and quantum functions, is one of the most well-studied applications of quantum computing, but recent trends have relied on hybrid methods that push much of the fine-tuning off onto costly classical algorithms. Feedback-based quantum algorithms, such as FALQON, avoid these fine-tuning problems but at the cost of additional circuit depth and a lack of convergence guarantees. In this work we take the local greedy information collected by Lyapunov feedback control and develop an analytic framework to use it to perturbatively update previous control layers, similar to the global optimal control achievable using Pontryagin optimal control. This perturbative methodology, which we call feedback-based optimally controlled quantum states (FOCQS), can be used to improve the results of feedback-based algorithms, like FALQON, providing better convergence but still no formal guarantees of convergence. Furthermore, this perturbative method can be used to push smooth annealing-like control protocol closer to the control optimum, even providing an iterative approach, albeit with diminishing returns. In numerical testing, we show improvements in convergence and required depth due to these methods over existing quantum feedback control methods.
Quantum optimization, both for classical and quantum functions, is one of the most well-studied applications of quantum computing, but recent trends have relied on hybrid methods that push much of the fine-tuning off onto costly classical algorithms. Feedback-based quantum algorithms, such as FALQON, avoid these fine-tuning problems but at the cost of additional circuit depth and a lack of convergence guarantees. In this work, we take the local greedy information collected by Lyapunov feedback control and develop an analytic framework to use it to perturbatively update previous control layers, similar to the global optimal control achievable using Pontryagin optimal control. This perturbative methodology, which we call Feedback Optimally Controlled Quantum States (FOCQS), can be used to improve the results of feedback-based algorithms, like FALQON. Furthermore, this perturbative method can be used to push smooth annealing-like control protocol closer to the control optimum, even providing and iterative approach, albeit with diminishing returns. In numerical testing, we show improvements in convergence and required depth due to these methods over existing quantum feedback control methods.
We introduce a new hybrid quantum-classical algorithm inspired by an advanced control strategy known as model predictive control (MPC). This algorithm unifies the optimization-based design of variational quantum algorithms (VQAs) with the feedback-based design of feedback-based quantum algorithms (FQAs). Variational circuit parameters are optimized using a layer-wise receding horizon strategy, where observable measurements after every layer initialize a classically simulated dynamic model used to predict quantum state evolution and optimize over future parameterized gates. This hybrid algorithm can be used for applications such as ground state preparation and approximate combinatorial optimization, and presents an ideal use case for the integration of quantum computers with high-performance computing, where the latter resource can be used to increase the scale and efficiency of the predictions critical to MPC. We show through mathematical proof and numerical evidence that the MPC-based algorithm can be guaranteed to at least match the performance of FQAs. Through simulations on Max-Cut problems and a two-dimensional transverse-field Ising model, we demonstrate that relaxed implementations of the MPC-based algorithm can also provide improved performance in practice compared to an FQA.
The Feedback-based Algorithm for Quantum Optimization (FALQON) is a Lyapunov inspired quantum algorithm proposed to tackle combinatorial optimization problems. In this paper, we examine the robustness of FALQON against coherent control errors, a class of multiplicative errors that affect the control input. We show that the algorithm is asymptotically robust with respect to systematic errors, and we derive robustness bounds for independent errors. Finally, we propose a robust version of FALQON which minimizes a regularized Lyapunov function. Our theoretical results are supported through simulations.