We show that limit-periodic Jacobi matrices in the weak exponential class $EC(η)$ have purely absolutely continuous spectral type when $η$ is sufficiently large, depending only on the $R$ -boundedness constant. The proof first establishes homogeneity of the spectrum from the weak exponential approximation rate. A closure theorem for reflectionless Jacobi matrices then transfers reflectionlessness from the periodic approximants to the limit, and a theorem of Poltoratski and Remling rules out singular spectrum on a homogeneous set.We show that limit-periodic Jacobi matrices in the weak exponential class $EC(η)$ have purely absolutely continuous spectral type when $η$ is sufficiently large, depending only on the $R$-boundedness constant. The proof first establishes homogeneity of the spectrum from the weak exponential approximation rate. A closure theorem for reflectionless Jacobi matrices then transfers reflectionlessness from the periodic approximants to the limit, and a theorem of Poltoratski and Remling rules out singular spectrum on a homogeneous set.

math.SPCross-listedarXiv:2610.19416Oct 15, 2026

Quantum Many-Body Dynamics in the Gaussian Approximation and Beyond

A. F. R. de Toledo Piza, Chi-Yong Lin

Jan 1, 19940 citations

Block encoding of sparse matrices with a periodic diagonal structure

Alessandro Andrea Zecchi, Claudio Sanavio, Luca Cappelli +3 more

Block encoding is a successful technique used in several powerful quantum algorithms. In this work we provide an explicit quantum circuit for block encoding a sparse matrix with a periodic diagonal structure. The proposed methodology is based on the linear combination of unitaries (LCU) framework and on an efficient unitary operator used to project the complex exponential at a frequency $ω$ multiplied by the computational basis into its real and imaginary components. We demonstrate a distinct computational advantage with a $\mathcal{O}(\text{poly}(n))$ gate complexity, where $n$ is the number of qubits, in the worst-case scenario used for banded matrices, and $\mathcal{O}(n)$ when dealing with a simple diagonal matrix, compared to the exponential scaling of general-purpose methods for dense matrices. Various applications for the presented methodology are discussed in the context of solving differential problems such as the advection-diffusion-reaction (ADR) dynamics, using quantum algorithms with optimal scaling, e.g., quantum singular value transformation (QSVT). Numerical results are used to validate the analytical formulation.Block encoding is a successful technique used in several powerful quantum algorithms. In this work we provide an explicit quantum circuit for block encoding a sparse matrix with a periodic diagonal structure. The proposed methodology is based on the linear combination of unitaries (LCU) framework and on an efficient unitary operator used to project the complex exponential at a frequency $ω$ multiplied by the computational basis into its real and imaginary components. We demonstrate a distinct computational advantage with a $\mathcal{O}(\text{poly}(n))$ gate complexity, where $n$ is the number of qubits, in the worst-case scenario used for banded matrices, and $\mathcal{O}(n)$ when dealing with a simple diagonal matrix, compared to the exponential scaling of general-purpose methods for dense matrices. Various applications for the presented methodology are discussed in the context of solving differential problems such as the advection-diffusion-reaction (ADR) dynamics, using quantum algorithms with optimal scaling, e.g., quantum singular value transformation (QSVT). Numerical results are used to validate the analytical formulation.

quant-pharXiv:2602.10589Feb 12, 20260 citations

Quantum Iterative Methods for Solving Differential Equations with Application to Computational Fluid Dynamics

Chelsea A. Williams, Antonio A. Gentile, Vincent E. Elfving +2 more

Abstract Quantum methods are proposed for solving differential equations that are based on a gradual improvement of the solution via an iterative process, and are targeted at applications in fluid dynamics. First, the Jacobi iteration is implemented on a quantum register that utilizes a linear combination of unitaries (LCU) approach to store the trajectory information. Second, quantum methods are extended to Gauss‐Seidel iterative methods. Additionally, a quantum‐suitable resolvent decomposition is proposed based on the Woodbury identity. From a technical perspective, tools are developed and utilized for the block encoding of specific matrices as well as their multiplication. The approach is benchmarked on paradigmatic fluid dynamics problems. These results stress that instead of inverting large matrices, one can program quantum computers to perform multigrid‐type computations and leverage corresponding advances in scientific computing.Abstract Quantum methods are proposed for solving differential equations that are based on a gradual improvement of the solution via an iterative process, and are targeted at applications in fluid dynamics. First, the Jacobi iteration is implemented on a quantum register that utilizes a linear combination of unitaries (LCU) approach to store the trajectory information. Second, quantum methods are extended to Gauss‐Seidel iterative methods. Additionally, a quantum‐suitable resolvent decomposition is proposed based on the Woodbury identity. From a technical perspective, tools are developed and utilized for the block encoding of specific matrices as well as their multiplication. The approach is benchmarked on paradigmatic fluid dynamics problems. These results stress that instead of inverting large matrices, one can program quantum computers to perform multigrid‐type computations and leverage corresponding advances in scientific computing.

arXiv:2404.08605Advanced Quantum TechnologiesOct 24, 20253 citations

The Spectral Foundations of Quantum Many-Body Dynamics

Stergios Pellis

Oct 25, 20260 citations

We show that limit-periodic Jacobi matrices in the weak exponential class

EC(η)

have purely absolutely continuous spectral type when

η

is sufficiently large, depending only on the

R

-boundedness constant. The proof first establishes homogeneity of the spectrum from the weak exponential approximation rate. A closure theorem for reflectionless Jacobi matrices then transfers reflectionlessness from the periodic approximants to the limit, and a theorem of Poltoratski and Remling rules out singular spectrum on a homogeneous set.

ω

multiplied by the computational basis into its real and imaginary components. We demonstrate a distinct computational advantage with a

\mathcal{O}(\text{poly}(n))

gate complexity, where

n

is the number of qubits, in the worst-case scenario used for banded matrices, and

\mathcal{O}(n)

when dealing with a simple diagonal matrix, compared to the exponential scaling of general-purpose methods for dense matrices. Various applications for the presented methodology are discussed in the context of solving differential problems such as the advection-diffusion-reaction (ADR) dynamics, using quantum algorithms with optimal scaling, e.g., quantum singular value transformation (QSVT). Numerical results are used to validate the analytical formulation.