Fast Algorithm for Finding the Eigenvalue Distribution of Very Large Matrices

Abstract

A theoretical analysis is given of the equation of motion method, due to Alben et al., to compute the eigenvalue distribution (density of states) of very large matrices. The salient feature of this method is that for matrices of the kind encountered in quantum physics the memory and CPU requirements of this method scale linearly with the dimension of the matrix. We derive a rigorous estimate of the statistical error, supporting earlier observations that the computational efficiency of this approach increases with matrix size. We use this method and an imaginary-time version of it to compute the energy and the specific heat of three different, exactly solvable, spin-1/2 models and compare with the exact results to study the dependence of the statistical errors on sample and matrix size.

24 pages, 24 figures

References in corpus (1)

• Calculating response functions in time domain with non-orthonormal basis sets

Cited by in corpus (144)

• Thermalization and prethermalization in isolated quantum systems: a theoretical overview
• Finite-temperature transport in one-dimensional quantum lattice models
• Modeling electronic structure and transport properties of graphene with resonant scattering centers
• Transport and Optical Properties of Single- and Bilayer Black Phosphorus with Defects
• Landau Level Spectrum of ABA- and ABC-stacked Trilayer Graphene
• Excitation spectrum and high energy plasmons in single- and multi-layer graphene
• Linear Scaling Quantum Transport Methodologies
• Effect of structural relaxation on the electronic structure of graphene on hexagonal boron nitride
• Quantum Power Method by a Superposition of Time-Evolved States
• Transport and optical properties of an electron gas in a Sierpinski carpet
• Regression relation for pure quantum states and its implications for efficient computing
• Electronic Transport in Disordered Bilayer and Trilayer Graphene
• Quantum Lattice Model Solver $\mathcal{H}\Phi$
• Random phase vector for calculating the trace of a large matrix
• Quantum Hall effect and semiconductor to semimetal transition in biased black phosphorus
• Typicality approach to the optical conductivity in thermal and many-body localized phases
• Optical conductivity of disordered graphene beyond the Dirac cone approximation
• Hall conductivity of Sierpinski carpet
• Pattern Learning Electronic Density of States
• Stripes, Antiferromagnetism, and the Pseudogap in the Doped Hubbard Model at Finite Temperature
• Thermodynamic properties of the Shastry-Sutherland model throughout the dimer-product phase
• Path to stable quantum spin liquids in spin-orbit coupled correlated materials
• Spin and energy currents in integrable and nonintegrable spin-1/2 chains: A typicality approach to real-time autocorrelations
• Nature of the randomness-induced quantum spin liquids in two dimensions
• Krylov-projected quantum Monte Carlo
• Electronic Structures and Optical Properties of Partially and Fully Fluorinated Graphene
• Numerical Large Deviation Analysis of Eigenstate Thermalization Hypothesis
• Quantum spin liquid at finite temperature: proximate dynamics and persistent typicality
• Modeling Klein tunneling and caustics of electron waves in graphene
• Accuracy of the finite-temperature Lanczos method compared to simple typicality-based estimates
• Local quantum control of Heisenberg spin chains
• Unconditionally Stable Algorithms to Solve the Time-Dependent Maxwell Equations
• Algorithm for Linear Response Functions at Finite Temperatures: Application to ESR spectrum of s=1/2 Antiferromagnet Cu benzoate
• Nonperturbative light-front Hamiltonian methods
• Classical and quantum spin dynamics of the honeycomb $\Gamma$ model
• Combining Dynamical Quantum Typicality and Numerical Linked Cluster Expansions
• Randomness-induced quantum spin liquid behavior in the $s=\frac{1}{2}$ random $J_1$-$J_2$ Heisenberg antiferromagnet on the square lattice
• Simulating hydrodynamics on noisy intermediate-scale quantum devices with random circuits
• Random State Technology
• Randomness-induced quantum spin liquid behavior in the s=1/2 random J1-J2 Heisenberg antiferromagnet on the honeycomb lattice
• Randomness-induced quantum spin liquid behavior in the $s=$1/2 bond-random Heisenberg antiferromagnet on the pyrochlore lattice
• Electronic Properties of Disordered Graphene Antidot Lattices
• Solving the Maxwell equations by the Chebyshev method: A one-step finite-difference time-domain algorithm
• Screening and plasmons in pure and disordered single- and bilayer black phosphorus
• Scaling of diffusion constants in the spin-1/2 XX ladder
• Heat conductivity of the Heisenberg spin-1/2 ladder: From weak to strong breaking of integrability
• Optical conductivity of a quantum electron gas in a Sierpinski carpet
• Predicting Gibbs-State Expectation Values with Pure Thermal Shadows
• Selected applications of typicality to real-time dynamics of quantum many-body systems
• Effective model with strong Kitaev interactions for $\alpha$-$\mathrm{RuC}{\mathrm{l}}_3$
• Rare regions and avoided quantum criticality in disordered Weyl semimetals and superconductors
• Efficient linear-scaling quantum transport calculations on graphics processing units and applications on electron transport in graphene
• Collective excitations and plasmon spectrum in twisted bilayer graphene near the magic angle
• Finite-temperature crossover phenomenon in the $S=1/2$ antiferromagnetic Heisenberg model on the kagome lattice
• Quantum versus Classical Dynamics in Spin Models: Chains, Ladders, and Square Lattices
• Beyond the poor man's implementation of unconditionally stable algorithms to solve the time-dependent Maxwell Equations
• Origin of the Canonical Ensemble: Thermalization with Decoherence
• Approach to equilbrium in nano-scale systems at finite temperatur
• Why are macroscopic experiments reproducible? Imitating the behavior of an ensemble by single pure states
• Finite-temperature charge transport in the one-dimensional Hubbard model
• Testing quantum fault tolerance on small systems
• Real-Time Dynamics of Typical and Untypical States in Non-Integrable Systems
• Relaxation, thermalization and Markovian dynamics of two spins coupled to a spin bath
• Commensurate and incommensurate double moiré interference in twisted trilayer graphene
• Flat-band plasmons in twisted bilayer transition metal dichalcogenides
• Thermal Pure Quantum Matrix Product States Recovering a Volume Law Entanglement
• Exponential damping induced by random and realistic perturbations
• Magnetization and energy dynamics in spin ladders: Evidence of diffusion in time, frequency, position, and momentum
• Decay of spin-spin correlations in disordered quantum and classical spin chains
• Quantum phases of disordered three-dimensional Majorana-Weyl fermions
• Eigenstate Thermalization Hypothesis and Quantum Jarzynski Relation for Pure Initial States
• Magnetization dynamics in clean and disordered spin-1 XXZ chains
• Noise effect on the nonadiabatic transition and correction to the tunneling energy gap estimated by the Landau-Zener-Stuckelberg formula
• Finite-Temperature Variational Monte Carlo Method for Strongly Correlated Electron Systems
• Quantum Decoherence Scaling with Bath Size: Importance of Dynamics, Connectivity, and Randomness
• Charge diffusion in the one-dimensional Hubbard model
• Electric field driven flat bands: Enhanced magnetoelectric and electrocaloric effects in frustrated quantum magnets
• Magnetism of the $s=1/2$ $J_1$-$J_2$ square-kagome lattice antiferromagnet
• Dynamics of open quantum spin systems: An assessment of the quantum master equation approach
• Improved variational quantum eigensolver via quasi-dynamical evolution
• Spectrum and thermodynamic properties of two-dimensional N=(1,1) super Yang-Mills theory with fundamental matter and a Chern-Simons term
• Electronic properties and quantum transports in functionalized graphene Sierpinski carpet fractals
• Confined Electrons in Effective Plane Fractals
• Hybrid quantum-classical method for simulating high-temperature dynamics of nuclear spins in solids
• Asymmetric melting of one-third plateau in kagome quantum antiferromagnets
• Macroscopically deterministic, Markovian thermalization in finite quantum spin systems
• Stiffness of Probability Distributions of Work and Jarzynski Relation for Non-Gibbsian Initial States
• Dimensional reduction in quantum spin-1/2 system on a 1/7-depleted triangular lattice
• Melting of magnetization plateaus for kagome and square-kagome lattice antiferromagnets
• Screening and Collective Modes in Disordered Graphene Antidot Lattices
• Update of $\mathcal{H}\Phi$: Newly added functions and methods in versions 2 and 3
• Effect of moiré superlattice reconstruction in the electronic excitation spectrum of graphene-metal heterostructures
• Relation between far-from-equilibrium dynamics and equilibrium correlation functions for binary operators
• Quantum half-orphans in kagome antiferromagnets
• Quantum Decoherence at Finite Temperatures
• Matrix product state approach for a quantum system at finite temperatures using random phases and Trotter gates
• Quantum Decoherence and Thermalization at Finite Temperature within the Canonical Thermal State Ensemble
• Temperature dependence of ESR intensity for the nanoscale molecular magnet V15
• Purity of thermal mixed quantum states
• Real-time simulation of flux qubits used for quantum annealing
• GPUQT: An efficient linear-scaling quantum transport code fully implemented on graphics processing units
• Quantum simulation using noisy unitary circuits and measurements
• Accuracy of the typicality approach using Chebyshev polynomials
• Quantum dynamics of a small symmetry breaking measurement device
• Partial Ferrimagnetism in S=1/2 Heisenberg Ladders with a Ferromagnetic Leg, an Antiferromagnetic Leg, and Antiferromagnetic Rungs
• Typicality-Based Variational Cluster Approach to Thermodynamic Properties of the Hubbard Model
• Dynamical quantum typicality: Simple method for investigating transport properties applied to the Holstein model
• Polarization of graphene in a strong magnetic field beyond the Dirac cone approximation
• Temperature- and Size-dependence of Line shape of ESR spectra of XXZ antiferromagnetic chain
• A Time-Dependent Random State Approach for Large-scale Density Functional Calculations
• Finite-size scaling of typicality-based estimates
• Signatures of the finite-temperature mirror symmetry breaking in the $S$=1/2 Shastry-Sutherland model
• Numerical study on ESR by making use of Wiener-Khinchin relation in time domain
• Computation of ESR spectra from the time evolution of the magnetization: comparison of autocorrelation and Wiener-Khinchin-relation based methods
• Frustration-induced quantum spin liquid behavior in the $s=$1/2 random-bond Heisenberg antiferromagnet on the zigzag chain
• The ESR intensity and the Dzyaloshinsky-Moriya interaction of the nanoscale molecular magnet V15
• Stability of Quantum Statistical Ensembles with Respect to Local Measurements
• Relaxation of dynamically prepared out-of-equilibrium initial states within and beyond linear response theory
• Spatiotemporal dynamics of classical and quantum density profiles in low-dimensional spin systems
• Effects of structural and chemical disorders on the visible/UV spectra of carbonaceous interstellar grains
• Anomalous thermodynamics of a quantum spin system with large residual entropy
• Thermodynamic signature of the $\mathrm{SU}(4)$ spin-orbital liquid and symmetry fractionalization from the Lieb-Schultz-Mattis theorem
• Loop-gas description of the localized-magnon states on the kagome lattice with open boundary conditions
• Antiferromagnetic order without recourse to staggered fields
• Simulating Floquet scrambling circuits on trapped-ion quantum computers
• Finite-Temperature Signatures of Spin Liquids in Frustrated Hubbard Model
• Large scale calculations of thermoelectric transport coefficients: a case study of γ-graphyne with point defects
• Supercomputer simulations of transmon quantum computers
• Transport and integrability-breaking in non-Hermitian many-body quantum systems
• Thermal pure matrix product state in two dimensions: tracking thermal equilibrium from paramagnet down to the Kitaev honeycomb spin liquid state
• Spin and quadrupole correlations by three-spin interaction in the frustrated pyrochlore magnet Tb${}_{2+x}$Ti${}_{2-x}$O${}_{7+y}$
• Witnessing Disorder in Quantum Magnets
• Hierarchy of timescales in a disordered spin-$1/2$ XX ladder
• Sample complexity of matrix product states at finite temperature
• The Finite Difference Time Domain Method for Computing Single-Particle Density Matrix
• Specific heat and susceptibility of S=1/2 antiferromagnets on square, triangular, and kagome lattices
• Monte Carlo Quantum Computing
• Straintronics and twistronics in bilayer graphene
• Field-induced states and thermodynamics of the frustrated Heisenberg antiferromagnet on a square lattice
• Quantum typicality approach to energy flow between two spin-chain domains at different temperatures
• Nontrivial damping of magnetization currents in perturbed spin chains
• Quantifying Stability of Quantum Statistical Ensembles
• Towards lattice-gas description of low-temperature properties above the Haldane and cluster-based Haldane ground states of a mixed spin-(1,1/2) Heisenberg octahedral chain
• Linear scaling calculation of a n-type GaAs quantum dot