We recall some properties of convex functions and, in particular, of the sum of the largest eigenvalues of a Hermitian matrix. From these properties a new estimate of an arbitrary eigenvalue of a sum of Hermitian matrices is derived, which in turn is used to compute an approximate associated spectral projector. These estimates are applied for the first time to explain the generic spectral features of quantum systems. As an application of the formalism, we explain the preponderance of certain ground-state angular momenta as observed in the vibron model with random interactions. We show that the evolution of eigenstates can be predicted from the knowledge of a limited number of spectra and investigate the effect of a three-body interaction in the vibron model on eigenenergies and eigenvectors.
We recall some properties of convex functions and, in particular, of the sum of the largest eigenvalues of a Hermitian matrix. From these properties a new estimate of an arbitrary eigenvalue of a sum of Hermitian matrices is derived, which in turn is used to compute an approximate associated spectral projector. These estimates are applied for the first time to explain the generic spectral features of quantum systems. As an application of the formalism, we explain the preponderance of certain ground-state angular momenta as observed in the vibron model with random interactions. We show that the evolution of eigenstates can be predicted from the knowledge of a limited number of spectra and investigate the effect of a three-body interaction in the vibron model on eigenenergies and eigenvectors.
The observed preponderance of ground states with angular momentum L=0 in many-body quantum systems with random two-body interactions is analyzed in terms of correlation coefficients (covariances) among different eigenstates. It is shown that the geometric analysis of Chau et al. can be interpreted in terms of correlations (covariances) between energy eigenvalues, thus providing an entirely statistical explanation of the distribution of ground state angular momenta of randomly interacting quantum systems that, in principle, is valid for both fermionic and bosonic systems. The method is illustrated for the interacting boson model.
A bstract In this paper, we investigate the spectral projection of density matrices in quantum field theory. With appropriate regularization, the spectral projectors of density matrices are expected to be well-defined. These projectors can be obtained using the Riesz projection formula, which allows us to compute both the density of eigenvalues and the expectation values of local operators in the projected states. We find that there are universal divergent terms in the expectation value of the stress energy tensor, where the coefficients depend universally on the density of eigenvalues and a function that describes the dependence of eigenvalues on boundary location. Using projection states, we can construct a series of new states in quantum field theories and discuss their general properties, focusing on the holographic aspects. We observe that quantum fluctuations are suppressed in the semiclassical limit. We also demonstrate that the fixed area state, previously constructed using gravitational path integrals, can be constructed by suitably superposition of appromiate amount of projection states. Additionally, we apply spectral projection to non-Hermitian operators, such as transition matrices, to obtain their eigenvalues and densities. Finally, we highlight potential applications of spectral projections, including the construction of new density and transition matrices and the understanding of superpositions of geometric states.
We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. For a matrix A and a function f, the spectral sum is the trace of f(A), equivalently the sum over eigenvalues of A of f applied to each eigenvalue. Typical examples of spectral sums are the von Neumann entropy, the trace of the inverse of A, the log-determinant, and the Schatten p-norm, where the latter does not require the matrix to be PSD. The current best classical randomized algorithms estimating these quantities have a runtime that is at least linearly in the number of nonzero entries of the matrix and quadratic in the estimation error. Assuming access to a block-encoding of a matrix, our algorithms are sub-linear in the matrix size, and depend at most quadratically on other parameters, like the condition number and the approximation error, and thus can compete with most of the randomized and distributed classical algorithms proposed in the literature, and polynomially improve the runtime of other quantum algorithms proposed for the same problems. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory: approximating the number of triangles, the effective resistance, and the number of spanning trees in a graph.
We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. The spectral sum of an PSD matrix A, for a function f, is defined as $ \text{Tr}[f(A)] = \sum_j f(λ_j)$, where λj are the eigenvalues of A. Typical examples of spectral sums are the von Neumann entropy, the trace of A−1, the log-determinant, and the Schatten p-norm, where the latter does not require the matrix to be PSD. The current best classical randomized algorithms estimating these quantities have a runtime that is at least linearly in the number of nonzero entries of the matrix and quadratic in the estimation error. Assuming access to a block-encoding of a matrix, our algorithms are sub-linear in the matrix size, and depend at most quadratically on other parameters, like the condition number and the approximation error, and thus can compete with most of the randomized and distributed classical algorithms proposed in the literature, and polynomially improve the runtime of other quantum algorithms proposed for the same problems. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory: approximating the number of triangles, the effective resistance, and the number of spanning trees within a graph.
Motivated by the surge of interest in statistics of non-Hermitian random matrices as a framework for description of universal characteristics of dissipative chaotic quantum many-body systems, we address the problem of characterizing the parametric correlations of spectral densities. Considering parameter-dependent ensemble of complex Ginibre matrices we derive an explicit, closed-form expression for the parametric number covariance in the systems of symmetry class A for eigenvalues in a circular domain containing on average a finite number of eigenvalues in the spectral bulk. This behavior is expected to be universal, as further supported by numerical evidence for the real Ginibre ensemble, non-Hermitian Bernoulli Wigner matrices and bi-unitarily invariant ensembles. We also discuss a relation between parametric correlations of spectral densities and the distribution of the so-called eigenvector non-orthogonality factor, which attracted considerable interest in recent years.
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.