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A fundamental challenge in multiparameter persistent homology is the absence of a complete and discrete invariant. To address this issue, we propose an enhanced framework that realizes a holistic understanding of a fully commutative…
The envelope theory is a method to compute approximate eigensolutions of quantum $N$-body Hamiltonians with a quite general structure in $D$ dimensions. The advantages of the method are that it is easy to implement and that $N$ is treated…
I first give an overview of the thesis and Matrix Product States (MPS) representation of quantum spin chains with an improvement on the conventional notation. The rest of this thesis is divided into two parts. The first part is devoted to…
In this thesis, we study three physically relevant models of strongly correlated random variables: trapped fermions, random matrices and random walks. In the first part, we show several exact mappings between the ground state of a trapped…
We analyze the problem of quantifying entanglement in pure and mixed states of fermionic systems with fixed number parity yet not necessarily fixed particle number. The "mode entanglement" between one single-particle level and its…
Obtaining high-precision guaranteed lower eigenvalue bounds remains difficult, even though the standard high-order conforming finite element (FEM) easily yields extremely sharp upper bounds. Recently developed rigorous approaches using such…
We prove that the eigenvalues of a 2-body operator $\gamma_{2}^{\Psi}$ associated to a fermionic $N$-particle state $\Psi$ are highly constrained by the structure of the corresponding eigenvectors: If…
The variational quantum-classical algorithms are the most promising approach for achieving quantum advantage on near-term quantum simulators. Among these methods, the variational quantum eigensolver has attracted a lot of attention in…
Self-Consistent RPA is extended in a way so that it is compatable with a variational ansatz for the ground state wave function as a fermionic many-body vacuum. Employing the usual equation of motion technique, we arrive at extended RPA…
We prove quadratic eigenvalue perturbation bounds for generalized Hermitian eigenvalue problems. The bounds are proportional to the square of the norm of the perturbation matrices divided by the gap between the spectrums. Using the results…
We have investigated high-precision measurements, beyond the standard quantum limit, utilizing non-classical states. Although entanglement has been considered a resource for achieving the Heisenberg limit in measurements, we show that any…
We consider the plasmonic eigenvalue problem for a general 2D domain with a curvilinear corner, studying the spectral theory of the Neumann--Poincar\'e operator of the boundary. A limiting absorption principle is proved, valid when the…
We introduce a homology-based technique for the analysis of multiqubit state vectors. In our approach, we associate state vectors to data sets by introducing a metric-like measure in terms of bipartite entanglement, and investigate the…
In this article, we introduce a general theoretical framework to analyze non-consistent approximations of the discrete eigenmodes of a self-adjoint operator. We focus in particular on the discrete eigenvalues laying in spectral gaps. We…
The aim of this paper is to propose an efficient adaptive finite element method for eigenvalue problems based on the multilevel correction scheme and inverse power method. This method involves solving associated boundary value problems on…
Particle number conservation in fermionic systems restricts the allowed local operations on bi-partite systems. We show how this restriction is related to measurement entropy of particle fluctuations and compute it for several regimes of…
For ordinary matrix models, the eigenvalue probability density decays rapidly as one goes to infinity, in other words, has "short tails". This ensures that all the multiple trace correlators (multipoint moments) are convergent and…
The nonperturbative Hamiltonian eigenvalue problem for bound states of a quantum field theory is formulated in terms of Dirac's light-front coordinates and then approximated by the exponential-operator technique of the many-body…
The simultaneous estimation of multiple unknown parameters is the most general scenario in quantum sensing. Quantum multi-parameter estimation theory provides fundamental bounds on the achievable precision of simultaneous estimation.…
Relevant metrological scenarios involve the simultaneous estimation of multiple parameters. The fundamental ingredient to achieve quantum-enhanced performances is based on the use of appropriately tailored quantum probes. However, reaching…