Response functions are key observables for probing the structure and dynamics of many-body systems. We present and demonstrate a quantum-classical computational framework for computing response functions of general many-fermion systems that also provides the full bound-state spectrum. The framework employs the Lorentz integral transform and a recently developed Hamiltonian input scheme that enables practical and scalable circuit constructions for general many-fermion Hamiltonians. Within this framework, we evaluate the Lorentz integral and propose three protocols to extract response functions and bound-state structural information. We apply the method to 19 O with realistic internucleon interactions, computing both the bound-state spectrum and the response function. This demonstration indicates opportunities for exploring the structure and dynamics of a broad class of many-body systems across diverse fields.
Response functions are key observables for probing the structure and dynamics of many-body systems. We present and demonstrate a quantum-classical computational framework for computing response functions of general many-fermion systems that also provides the full bound-state spectrum. The framework employs the Lorentz integral transform and a recently developed Hamiltonian input scheme that enables practical and scalable circuit constructions for general many-fermion Hamiltonians. Within this framework, we evaluate the Lorentz integral and propose three protocols to extract response functions and bound-state structural information. We apply the method to 19 O with realistic internucleon interactions, computing both the bound-state spectrum and the response function. This demonstration indicates opportunities for exploring the structure and dynamics of a broad class of many-body systems across diverse fields.
Response functions are key observables for probing the structure and dynamics of many-body systems. We introduce and demonstrate a quantum-classical framework for computing response functions of general many-fermion systems that also provides the full bound-state spectrum. The framework employs the Lorentz integral transform and a new Hamiltonian input scheme that enables practical and scalable circuit constructions for general many-fermion Hamiltonians. Within this framework, we develop a hybrid strategy to evaluate the Lorentz integral and propose three protocols to extract response functions and bound-state structural information. As a demonstration, we apply the method to 19O with realistic internucleon interactions, computing both the bound-state spectrum and the response function. We envision that our approach will open new avenues for exploring the structure and dynamics of a broad class of many-body systems across diverse fields.
To overcome the limitations of existing algorithms for solving self-bound quantum many-body problems -- such as those encountered in nuclear and particle physics -- that access only a restricted subset of energy levels and provide limited structural information, we introduce and demonstrate a novel quantum-classical approach capable of resolving the complete bound-state spectrum. This method also provides the total angular momentum J associated with each eigenstate. Our approach is based on expressing the Hamiltonian in second-quantized form within a novel input model combined with a scan scheme, enabling broad applicability to configuration-interaction calculations across diverse fields. We apply this hybrid method to compute, for the first time, the bound-state spectrum together with corresponding J values of 20O using a realistic strong-interaction Hamiltonian. Our approach applies to hadron spectra and J values solved in the relativistic Basis Light-Front Quantization approach.
This dissertation concerns the quantum many-body problem, which is the problem of predicting the properties of systems of several quantum particles from the first principles of quantum mechanics. Included under this umbrella are various problems of fundamental importance in quantum chemistry, condensed matter physics, and materials science. Of particular interest is the electronic structure problem, the problem of determining the state of the electrons in a system with fixed atomic nuclei. Since direct numerical solution of the many-body Schrödinger equation is intractable even for systems of moderate size, a diverse array of approximate methods has been developed. The broad goals of this dissertation are to improve the mathematical understanding of certain widely-used approximations, as well as to propose new methods. Roughly speaking, we consider three (overlapping) categories of methods: Green's function methods, embedding methods, and variational methods.One can understand Green's function methods in terms of many-body perturbation theory, which computes series expansions of physical quantities about a non-interacting reference system. These expansions can be expressed graphically in terms of Feynman diagrams, which can in turn be reorganized, in some cases, into an expansion in terms of so-called bold diagrams. Green's function methods can be specified by choosing a subset of bold diagrams to approximate the sum. At the same time, such methods can be understood in terms of an object known as the Luttinger-Ward (LW) functional, which admits a representation in terms of the bold diagrams. Many aspects of these constructions are purely formal, and indeed the existence of the fermionic LW functional as a single-valued functional has recently been called into question. To contribute to the understanding of these issues, we provide rigorous proofs of the combinatorial construction and analytic interpretation of the bold diagrams in the simplified setting of a classical field theory. In this setting we also provide a rigorous non-perturbative construction of the LW functional via convex duality and prove several key properties, including continuity up to the boundary of its domain and asymptotics in the limit of large interaction.Quantum embedding methods, meanwhile, view a large system as being composed of smaller fragments that are treated with high accuracy and embedded in the larger system in a mutually consistent way. Inspired by a connection between the boundary analysis of the LW functional and embedding, we perform similar analysis for the 1-RDM theory for fermionic systems, which is also developed via convex duality, illustrating a relation to fermionic embedding methods such as the density matrix embedding theory (DMET).Another embedding method of note is the dynamical mean-field theory (DMFT), which is at the same time a Green's function method that can be understood in terms of the LW functional. DMFT relies on the solution of impurity problems, which specify the embedding of an interacting system into a non-interacting bath. Underlying DMFT is a result about the sparsity pattern of the self-energy matrix for impurity problems, which to our knowledge has not been proved in the literature. We provide a rigorous proof of this result in various classical and quantum settings. We go on to investigate the fermionic DMFT in depth, identifying the key mathematical structures that appear in the algorithmic loop for solving it and using these to prove the well-posedness of this loop, in a certain sense.Finally, we introduce a suite of new approaches to the quantum many-body problem that provide variational lower bounds to the ground-state energy. These methods, which combine the themes of convexity and embedding, are based on novel convex relaxations of the variational principles for the ground-state energies of many-body systems. To begin, we recover a second-quantized version of the formalism of strictly correlated electrons (SCE), which yields an exact expression for the exchange-correlation functional in Kohn-Sham density functional theory in the limit of infinite Coulomb repulsion in terms of the solution of a multi-marginal optimal transport problem. We introduce a semidefinite relaxation method for approximately solving this problem and obtaining a lower bound for the ground-state energy. The ideas underlying this relaxation are generalized considerably, outside the context of SCE, to yield much tighter lower bounds, which we validate numerically for both quantum spin systems and fermionic systems. We also describe how these relaxation methods can be interpreted as embedding methods via convex duality.
Abstract Dynamical correlation functions are essential for revealing emergent phenomena in quantum matter, since they measure the frequency-resolved response of a system at equilibrium to a perturbation and thereby probe its full spectrum. While neural quantum states provide an accurate and flexible description of correlated ground states, accessing real-frequency dynamical correlators from them remains a major challenge. Here we introduce a general and systematically improvable framework to compute real-frequency linear-response functions directly from neural quantum state ground states. Our method performs the Lanczos subspace construction through an operator recursion and controls operator growth with a suited truncation scheme. We benchmark it on demanding quantum impurity models, where high precision is required close to the Fermi energy, and show that rich spectral structure and the self-energy can be resolved over several orders of magnitude. The approach generalizes straightforwardly, for example providing a direct path toward more broadly applicable dynamical mean-field theory solvers.
Response functions are a fundamental aspect of physics; they represent the link between experimental observations and the underlying quantum many-body state. However, this link is often under-appreciated, as the Lehmann formalism for obtaining response functions in linear response has no direct link to experiment. Within the context of quantum computing, and by using a linear response framework, we restore this link by making the experiment an inextricable part of the quantum simulation. This method can be frequency- and momentum-selective, avoids limitations on operators that can be directly measured, and is ancilla-free. As prototypical examples of response functions, we demonstrate that both bosonic and fermionic Green's functions can be obtained, and apply these ideas to the study of a charge-density-wave material on ibm_auckland. The linear response method provides a robust framework for using quantum computers to study systems in physics and chemistry. It also provides new paradigms for computing response functions on classical computers.
The study of quantum dynamics is an increasingly important aspect of atomic, molecular and optical physics, as well as condensed-matter physics and other disciplines. It is essential in order to understand non-equilibrium phenomena which do not fit within the framework of equilibrium thermodynamics, such as many-body localization, dynamical quantum phase transitions, and non-equilibrium steady states in driven-dissipative systems. However, the exact calculation of dynamical observables is extremely difficult in general for several reasons, which makes it challenging for theoretical predictions and analyses to keep pace with rapidly advancing experiments. In this thesis I use and develop several computational methods to investigate the dynamics of quantum many-body systems, with the two goals of better understanding the evolution of local observables and correlation functions in spin and Fermion systems as well as creating and testing more effective numerical methods for solving quantum dynamics problems. I explore quantum dynamics by using Hamiltonian parameter quenches in several commonly-used models in AMO physics and elsewhere: the Fermi-Hubbard model, the XXZ model, and the transverse-Ising model. The first represents a system of short-range interacting Fermions, while the latter two represent spin systems with different spin-spin couplings and magnetic field terms. All of these models can be realized in current experiments. First, I derive an exact solution for an extreme interaction quench in the Fermi-Hubbard model, from a strongly-interacting initial state to a noninteracting final Hamiltonian. From this result I show that there are nontrivial transient connected correlations, despite the fact that the system has very high temperature and a noninteracting Hamiltonian. Those features would suppress such correlations in equilibrium, but the dynamical correlations are still significant. Then, I develop and examine two numerical methods for solving dynamics of the spin expectation values and correlation functions in the transverse-Ising and XXZ models. Both methods rely on approximating a thermodynamically large system with a much smaller and more computationally tractable cluster of lattice sites, or a series of clusters. The first method is a dynamical numerical linked cluster expansion (NLCE). NLCEs have already been used successfully on systems in thermodynamic equilibrium, and I extended the technique to directly compute the dynamics of an observable after a quantum quench. I review the general NLCE method, describe my dynamical variant of it, and compute the quench dynamics of spin and correlation observables in the transverse-Ising and XXZ models. The second method, which I term the adaptive-boundary cluster method, relies on recently refined bounds on information propagation in quantum systems to choose a cluster of lattice sites that minimizes the finite size error bound of a dynamical computation. I briefly review these bounds, and describe a new algorithm which uses them to construct a cluster whose boundary is chosen according to a related error estimate. Such an adaptive-boundary cluster has the property that there exists a computable upper bound on the finite size error of observables measured at the cluster center, and this error bound is nearly minimal when compared to that of other clusters of the same size. I then compute the quench dynamics of spin and correlation observables in the transverse-Ising model using the adaptive boundary clusters. I apply this method to both uniform and strongly disordered systems. To evaluate the effectiveness of both dynamical NLCE and adaptive-boundary clusters, I compare the results obtained using them to results from a standard and general purpose numerical method: exact solvers applied a rectangular cluster with periodic boundary conditions. Both of the new methods usually converge faster and are more accurate than the standard method for comparable cluster sizes. This thesis illustrates that choosing numerical methods which take advantage of the geometric and entanglement structure of the initial state and Hamiltonian can substantially enhance our ability to study quantum dynamics, by reducing the computational cost of numerically exact solutions. I conclude with a discussion of various avenues for future research to build on this work.