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.
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.
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.
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.
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
computations of the ground and excited electronic states. While substantial progress has been made in designing physically informed quantum algorithms to address the ground state under the constraints of near-term quantum hardware, relatively fewer developments exist for reliably computing excited state energetics, presumably due to the growing complexity of the quantum circuits for excited eigenroots. In this work, we exploit the inherent point-group and spin symmetries of molecular systems to construct symmetry-adapted sets of orthogonal reference states which are evolved with a common symmetry-scalar unitary, enabling the simultaneous determination of ground and excited eigenroots. By minimizing the energy functional over a restricted symmetry-adapted correlated manifold, the weighted subspace-search variational quantum eigensolver enables us to directly target multiple eigenroots within each spatial and spin symmetry sector. This physically motivated selection of references and the unitary confer substantial flexibility to the weighted energy landscape, largely allowing the optimization to avoid numerical traps and converge directly to the targeted eigenstates. More crucially, symmetry resolution also enables access to a larger number of eigenstates without substantially increasing the complexity of the cost function. Through benchmarking on moderate to strongly correlated molecular systems, we demonstrate the accuracy and advantages of our approach relative to several established excited state algorithms on quantum computers.
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
In ab-initio quantum chemistry, solving the electronic structure problem of a molecular system of interest to obtain an approximation of the ground and, in some cases, the excited state energies is a core and central task in various quantum chemistry processes. These include, but certainly not limited to, the calculation of the activation energy–that is the minimal energy required to launch a given chemical reaction–, the dipole and multipole moments which describe the polarity of the molecule, performing the optimization of the geometry of the molecule and searching for transition states. Furthermore, many quantum chemistry applications are predicated, not merely on the approximation of the ground state energy, but also on the corresponding first order derivatives. The body of literature of classical algorithms approximating the ground state energy and its first order derivative is extensively rich with a plethora of methods such as self-consistent mean-field methods, coupled-cluster methods, density matrix renormalization group and many more, each of which has intrinsic set of properties and characteristics that makes it suitable for specific tasks and cases. However, classical methods face a practically insurmountable challenge: the efficient computation of high accuracy approximation of ground state energies of large system sizes. This is fundamentally because the electronic structure problem provably belongs to the complexity class of QMA-hard problems. Which also means that it is out of reach for quantum algorithms as well in general. However, with certain assumptions that are expected to hold molecules of interest–that so far proved to be classically challenging because the required accuracy set by the standard of chemical accuracy threshold (generally held in quantum chemistry)– quantum algorithms can efficiently treat such cases. While originally theorized by Richard Feynmann as a potentially efficient way of simulating some classically intractable many-body systems, quantum computing has since proven to have a stronger and richer potential paving the way for the inception of new algorithms such as Shor’s algorithm for integer factorization, HHL for solving linear system of equation and Grover’s algorithm for constrained binary search, just to name a few. These algorithms provably exhibit an improvement rate that goes respectively exponentially and quadratically in the time complexity. Quantum chemistry is considered a prominent candidate to benefit from quantum computing given the variety of quantum algorithms put forth to solve the electronic structure problem of particular systems of interest. These quantum algorithm proposals can be broadly categorized in three main categories that span the spectrum: from noisy intermediate scale quantum (NISQ) computers to early and full scale fault tolerant quantum ((E)FTQC) computers. Firstly, the variational quantum eigensolver (VQE) family of quantum algorithms which variationally search for the ground state energy through the optimization of the energy of a parametrized state prepared in a quantum computer using an ansatz. Secondly, the quantum Krylov subspace diagonalization (QKSD) methods, which generate a subspace of the original Hilbert space and efficiently produce the lowest energy in said subspace as its output. Lastly, we have the quantum phase estimation (QPE) algorithms, which provably produce the ground state energy in Heisenberg scaling presenting hence a provable efficient solution. In this thesis, we firstly set to work on the problem of efficient Hamiltonian linear combination of unitaries (LCU) representation. This is mainly motivated by the wide-range impact an efficient compression of the Hamiltonian can potentially have on several quantum algorithms such as VQE, QPE...etc. We designed a method that efficiently compress the Hamiltonian representation in a way that is beneficially impactful across the set of the aforementioned quantum algorithms. Namely, we motivate and present in Sections 1.3 and 1.4 respectively, the regularized compressed double-factorization (RC-DF) and show how it reduces the energy measurement complexity both in terms of number of distinct observables and number of samples. Moreover, it also reduces the total depth of the quantum phase estimation circuit thanks to its low normalization factor. Secondly, given the importance of the energy derivatives, in Sections 2.1 and 2.2, we introduce and present an efficient quantum algorithm for the calculation of the analytic gradient of VQE energies of double-factorized Hamiltonians. Furthermore, in Section 2.1, we expand on the utility of RC-DF by presenting a quantum algorithm that determines the analytic gradient of energies measured with respect to RC-DF Hamiltonains. Third, in Section 3.1, we summarize the theoretical results of purification error-mitigation techniques, namely echo verification and virtual distillation. In Section 3.2, we report on the results of the experimental bench mark, of said techniques, performed on on Google’s Sycamore device to validate the theoretical scaling of error suppression and extrapolate the resources needed for classically intractable system sizes. Fourth, in Section 4 we establish the challenges of computing energy gradients in a quantum Krylov frame work. To overcome the problem, we provide, in Section 4.3, an efficient quantum algorithm for calculating energy derivative obtained through QKSD procedures. Our approach shows how the QKSD ground state can be prepared using quantum signal processing (QSP) and reduces the number of observables to O(1) per reduced density matrix (RDM). Lastly, in light of the importance of quantum RAMs due to their utility in a variety of quantum algorithms that require efficient initial state preparation, as explained in Section 5.5, we address the problematic exponential depth of the bucket-brigade QRAM (when decomposed). We present in Section 5.6 a very efficient Clifford+T decomposition of the bucket-brigade QRAM, which preserves the linear scaling of the depth of the circuit without introducing any additional ancilla.