Recently, feedback-based quantum algorithms have been introduced to calculate the ground states of Hamiltonians, inspired by quantum Lyapunov control theory. This paper aims to generalize these algorithms to the problem of calculating an eigenstate of a given Hamiltonian, assuming that the lower energy eigenstates are known. To this aim, we propose a new design methodology that combines the layer wise construction of the quantum circuit in feedback-based quantum algorithms with a new feedback law based on a new Lyapunov function to assign the quantum circuit parameters. We present two approaches for evaluating the circuit parameters: one based on the expectation and overlap estimation of the terms in the feedback law and another based on the gradient of the Lyapunov function. We demonstrate the algorithm through an illustrative example and through an application in quantum chemistry. To assess its performance, we conduct numerical simulations and execution on IBM's superconducting quantum computer.
Recently, feedback-based quantum algorithms have been introduced to calculate the ground states of Hamiltonians, inspired by quantum Lyapunov control theory. This paper aims to generalize these algorithms to the problem of calculating an eigenstate of a given Hamiltonian, assuming that the lower energy eigenstates are known. To this aim, we propose a new design methodology that combines the layer wise construction of the quantum circuit in feedback-based quantum algorithms with a new feedback law based on a new Lyapunov function to assign the quantum circuit parameters. We present two approaches for evaluating the circuit parameters: one based on the expectation and overlap estimation of the terms in the feedback law and another based on the gradient of the Lyapunov function. We demonstrate the algorithm through an illustrative example and through an application in quantum chemistry. To assess its performance, we conduct numerical simulations and execution on IBM's superconducting quantum computer.
Drawing inspiration from the Lyapunov control technique for quantum systems, feedback-based quantum al-gorithms have been proposed for calculating the ground states of Hamiltonians. In this work, we consider extending these algorithms to tackle calculating excited states. Inspired by the weighted subspace-search variational quantum eigensolver al-gorithm, we propose a novel weighted feedback-based quantum algorithm for excited state calculation. We show that depending on how we design the weights and the feedback law, we can prepare the pth excited state or lowest energy states up to the pth excited state. Through an application in quantum chemistry, we show the effectiveness of the proposed algorithm, evaluating its efficacy via 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
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.
In recent quantum algorithmic developments, a feedback-based approach has shown promise for preparing quantum many-body system ground states and solving combinatorial optimization problems. This method utilizes quantum Lyapunov control to iteratively construct quantum circuits. Here, we propose a substantial enhancement by implementing a protocol that uses ideas from quantum Lyapunov control and the counterdiabatic driving protocol, a key concept from quantum adiabaticity. Our approach introduces an additional control field inspired by counterdiabatic driving. We apply our algorithm to prepare ground states in one-dimensional quantum Ising spin chains. Comprehensive simulations demonstrate a remarkable acceleration in population transfer to low-energy states within a significantly reduced time frame compared to conventional feedback-based quantum algorithms. This acceleration translates to a reduced quantum circuit depth, a critical metric for potential quantum computer implementation. We validate our algorithm on the IBM cloud computer, highlighting its efficacy in expediting quantum computations for many-body systems and combinatorial optimization problems. Published by the American Physical Society 2024
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 methods have gained significant attention as an alternative training paradigm for the Quantum Approximate Optimization Algorithm (QAOA) in solving combinatorial optimization problems such as MAX-CUT. In particular, Quantum Lyapunov Control (QLC) employs feedback-driven control laws that guarantee monotonic non-decreasing objective values, can substantially reduce the training overhead of QAOA, and mitigate barren plateaus. However, these methods might require long control sequences, leading to sub-optimal convergence rates. In this work, we propose a hybrid method that incorporates per-layer gradient estimation to accelerate the convergence of QLC while preserving its low training overhead and stability guarantees. By leveraging layer-wise gradient information, the proposed approach selects near-optimal control parameters, resulting in significantly faster convergence and improved robustness. We validate the effectiveness of the method through extensive numerical experiments across a range of problem instances and optimization settings.