We introduce a simple and efficient variation of the tangent-space excitation Ansatz used to compute elementary excitation spectra of one-dimensional quantum lattice systems using matrix product states (MPS). A small basis for the excitation tensors is formed based on a single diagonalization analogous to a single site DMRG step but for multiple states. Once overlap and Hamiltonian matrix elements are found, obtaining the excitation for any momentum only requires diagonalization of a tiny matrix, akin to a non-orthogonal band-theory diagonalization. The approach is based on an infinite MPS description of the ground state, and we introduce an extremely simple alternative to variational uniform matrix product states (VUMPS) based on finite system DMRG. For the S=1 S = 1 Heisenberg chain, our method—site basis excitation Ansatz (SBEA)—efficiently produces the one-magnon dispersion with high accuracy. We also examine the role of MPS gauge choices, finding that not imposing a gauge condition—leaving the basis nonorthogonal—is crucial for the approach, whereas imposing a left-orthonormal gauge (as in prior work) severely hampers convergence. We also show how one can construct Wannier excitations, analogous to the Wannier functions of band theory, where one Wannier excitation, translated to all sites, can reconstruct the single magnon modes exactly for all momenta.We introduce a simple and efficient variation of the tangent-space excitation Ansatz used to compute elementary excitation spectra of one-dimensional quantum lattice systems using matrix product states (MPS). A small basis for the excitation tensors is formed based on a single diagonalization analogous to a single site DMRG step but for multiple states. Once overlap and Hamiltonian matrix elements are found, obtaining the excitation for any momentum only requires diagonalization of a tiny matrix, akin to a non-orthogonal band-theory diagonalization. The approach is based on an infinite MPS description of the ground state, and we introduce an extremely simple alternative to variational uniform matrix product states (VUMPS) based on finite system DMRG. For the S=1 S = 1 Heisenberg chain, our method—site basis excitation Ansatz (SBEA)—efficiently produces the one-magnon dispersion with high accuracy. We also examine the role of MPS gauge choices, finding that not imposing a gauge condition—leaving the basis nonorthogonal—is crucial for the approach, whereas imposing a left-orthonormal gauge (as in prior work) severely hampers convergence. We also show how one can construct Wannier excitations, analogous to the Wannier functions of band theory, where one Wannier excitation, translated to all sites, can reconstruct the single magnon modes exactly for all momenta.

arXiv:2509.06241SciPost Physics CoreApr 8, 20260 citations

Matrix Product States

Timo Felser, Simone Montangero

Graduate texts in physicsJan 1, 20260 citations

Space Reduction in Matrix Product State

Lihua Wang, Kwang S. Kim

We reconstruct a matrix product state (MPS) in reduced spaces using density matrix. This scheme applies to a MPS built on a blocked quantum lattice. Each block contains $N$ physical sites that have a local space of rank $R$ . The simulation in the original spaces of rank $R^N$ is used to construct density matrices for every block. They are diagonalized and only the eigenvectors corresponding to significant diagonal elements are used to transform the original spaces to smaller ones and to reconstruct the MPS in those smaller spaces accordingly. Simulations in the reduced spaces are used to reliably extrapolate the result in unreduced spaces. Moreover, to obtain a required accuracy, the ratio of the reduced space rank over the original decreases quickly with $N$ . The reduced space has a saturated rank to obtain a demanded accuracy when $N\rightarrow \infty$ .We reconstruct a matrix product state (MPS) in reduced spaces using density matrix. This scheme applies to a MPS built on a blocked quantum lattice. Each block contains $N$ physical sites that have a local space of rank $R$. The simulation in the original spaces of rank $R^N$ is used to construct density matrices for every block. They are diagonalized and only the eigenvectors corresponding to significant diagonal elements are used to transform the original spaces to smaller ones and to reconstruct the MPS in those smaller spaces accordingly. Simulations in the reduced spaces are used to reliably extrapolate the result in unreduced spaces. Moreover, to obtain a required accuracy, the ratio of the reduced space rank over the original decreases quickly with $N$. The reduced space has a saturated rank to obtain a demanded accuracy when $N\rightarrow \infty$.

arXiv:1803.04844Mar 13, 20180 citations

Symmetry Breaking in Matrix-Product States

Chen Liu, Anders W. Sandvik, Yu-Cheng Su +1 more

Bulletin of the American Physical SocietyMar 15, 20100 citations

Exact Diagonalization Techniques

Alexander Weiße, Holger Fehske

Springer eBooksDec 8, 200754 citations

Comparative study of matrix product state/quantized tensor-train algorithms for solving time-independent partial differential equations

Paula García-Molina, Luca Tagliacozzo, Juan José García‐Ripoll

This work presents a comparative study of new and existing optimization and diagonalization methods for solving time-independent partial differential equations (PDEs) using matrix product states (MPS) in the quantized tensor-train formalism (QTT). This study focuses on Hamiltonian equations, for which five algorithms are introduced: explicit imaginary-time evolution methods, steepest gradient descent in conventional and optimized forms, a power method, and an explicitly restarted Arnoldi method. The first five methods are engineered using a framework of limited-precision linear algebra, in which operators -- i.e., the equation itself -- and vectors are represented using matrix product operator (MPO) and matrix product state (MPS) formalisms, and where operator-vector multiplication and vector addition are approximated with limited resources. All methods are benchmarked using an exactly solvable PDE for a quantum harmonic oscillator in one and two dimensions over a regular grid with up to $2^{30}$ points and compared with the density matrix renormalization group (DMRG) method. Our study reveals that all MPS-based techniques exponentially outperform exact diagonalization techniques based on vectors regarding memory usage. Imaginary-time algorithms are shown to underperform any gradient descent in terms of calibration needs and costs. Finally, MPS DMRG and interpolated Arnoldi-like asymptotically outperform all other methods, including state-of-the-art vector-based exact diagonalization, with significant advantages in time and memory use.This work presents a comparative study of new and existing optimization and diagonalization methods for solving time-independent partial differential equations (PDEs) using matrix product states (MPS) in the quantized tensor-train formalism (QTT). This study focuses on Hamiltonian equations, for which five algorithms are introduced: explicit imaginary-time evolution methods, steepest gradient descent in conventional and optimized forms, a power method, and an explicitly restarted Arnoldi method. The first five methods are engineered using a framework of limited-precision linear algebra, in which operators -- i.e., the equation itself -- and vectors are represented using matrix product operator (MPO) and matrix product state (MPS) formalisms, and where operator-vector multiplication and vector addition are approximated with limited resources. All methods are benchmarked using an exactly solvable PDE for a quantum harmonic oscillator in one and two dimensions over a regular grid with up to $2^{30}$ points and compared with the density matrix renormalization group (DMRG) method. Our study reveals that all MPS-based techniques exponentially outperform exact diagonalization techniques based on vectors regarding memory usage. Imaginary-time algorithms are shown to underperform any gradient descent in terms of calibration needs and costs. Finally, MPS DMRG and interpolated Arnoldi-like asymptotically outperform all other methods, including state-of-the-art vector-based exact diagonalization, with significant advantages in time and memory use.

arXiv:2303.09430Mar 16, 20231 citation

We reconstruct a matrix product state (MPS) in reduced spaces using density matrix. This scheme applies to a MPS built on a blocked quantum lattice. Each block contains

N

physical sites that have a local space of rank

R

. The simulation in the original spaces of rank

R^N

is used to construct density matrices for every block. They are diagonalized and only the eigenvectors corresponding to significant diagonal elements are used to transform the original spaces to smaller ones and to reconstruct the MPS in those smaller spaces accordingly. Simulations in the reduced spaces are used to reliably extrapolate the result in unreduced spaces. Moreover, to obtain a required accuracy, the ratio of the reduced space rank over the original decreases quickly with

N

. The reduced space has a saturated rank to obtain a demanded accuracy when

N\rightarrow \infty

2^{30}

points and compared with the density matrix renormalization group (DMRG) method. Our study reveals that all MPS-based techniques exponentially outperform exact diagonalization techniques based on vectors regarding memory usage. Imaginary-time algorithms are shown to underperform any gradient descent in terms of calibration needs and costs. Finally, MPS DMRG and interpolated Arnoldi-like asymptotically outperform all other methods, including state-of-the-art vector-based exact diagonalization, with significant advantages in time and memory use.