The Kernel Polynomial Method (KPM) is a method to approximate any function as a polynomial of finite order. In this research the KPM will be used to approximate the density of states and expectation values of physical observables in a tight binding system. To be able to use the KPM a tight binding Hamiltonian is obtained by using the python package Kwant. A python implementation of KPM combined with a time propagator then results in time-dependent expectation values of any operator. This approach benefits from large system sizes, since KPM uses a statistical approximation that increases in accuracy with larger size. The statistical approximation always converges, but there is limited guarantee as to how fast it converges. In further research the KPM could be compared to other available approaches. Another possibility is to use the KPM to get a general overview of the system and combine it with more exact approaches to find specific results.
The Kernel Polynomial Method (KPM) is a method to approximate any function as a polynomial of finite order. In this research the KPM will be used to approximate the density of states and expectation values of physical observables in a tight binding system. To be able to use the KPM a tight binding Hamiltonian is obtained by using the python package Kwant. A python implementation of KPM combined with a time propagator then results in time-dependent expectation values of any operator. This approach benefits from large system sizes, since KPM uses a statistical approximation that increases in accuracy with larger size. The statistical approximation always converges, but there is limited guarantee as to how fast it converges. In further research the KPM could be compared to other available approaches. Another possibility is to use the KPM to get a general overview of the system and combine it with more exact approaches to find specific results.
Computing finite temperature properties of a quantum many-body system is key to describing a broad range of correlated quantum many-body physics from quantum chemistry and condensed matter to thermal quantum field theories. Quantum computing, which has seen rapid developments in recent years, has a huge potential to impact the computation of quantum thermodynamics. To fulfill these potential impacts, it is crucial to design quantum algorithms that utilize the computation power of quantum computing devices. Here, we present a quantum kernel function expansion (QKFE) algorithm for solving thermodynamic properties of quantum many-body systems. In this quantum algorithm, the many-body density of states is approximated by a kernel Fourier expansion, whose expansion moments are obtained by random state sampling and quantum interferometric measurements. Compared to its classical counterpart, namely, the kernel polynomial method (KPM), QKFE has an exponential advantage in the cost of both time and memory. In computing low temperature properties, QKFE becomes inefficient, similar to classical KPM. To resolve this difficulty, we further construct a thermal ensemble iteration (THEI) protocol which starts from the trivial limit of an infinite temperature ensemble and approaches the low temperature regime step by step. For quantum Hamiltonians, whose ground states are preparable with polynomial quantum circuits, THEI has an overall polynomial complexity. We demonstrate its efficiency with applications to one- and two-dimensional quantum spin models and a fermionic lattice. With our analysis of the realization with digital and analog quantum devices, we believe the quantum algorithm is accessible to current quantum technology.
An oscillation-free Fourier expansion is proposed and applied to the polynomial expansion of resolvent operator, as an alternative treatment for Gibbs phenomenon to the kernel polynomial method (KPM) by Voter et al. [Phys. Rev. B 53 (1996) 12733]. Adopting regulated Legendre polynomial as basis functions, the recurrence algorithm underlying the KPM is practically preserved, while it eliminates unphysical oscillations entirely. The resolution is uniform in the frequency domain, and improved without limitation by extending the truncation range of the expansion. The method can also be used to calculate the eigenvector with remarkably high precision. It is further extended to the time domain, being able to deal with the quantum evolution in the same algorithm. A numerical test was performed against a lattice dynamics of 19 3 atoms, as well as of a computer generated Lennard–Jones glass of 1000 atoms. Not only it serves as a most efficient simulation scheme for a bulk, confirmed was also that the calculation converges to exact results within realistic computation time.
Linear-scaling electronic structure methods based on the calculation of moments of the underlying electronic Hamiltonian offer a computationally efficient and numerically robust scheme to drive large-scale atomistic simulations, in which the quantum-mechanical nature of the electrons is explicitly taken into account. We compare the kernel polynomial method to the Fermi operator expansion method and establish a formal connection between the two approaches. We show that the convolution of the kernel polynomial method may be understood as an effective electron temperature. The results of a number of possible kernels are formally examined, and then applied to a representative tight-binding model.
Two Chebyshev recursion methods are presented for calculations with very large sparse Hamiltonians, the kernel polynomial method (KPM) and the maximum entropy method (MEM). If limited statistical accuracy and energy resolution are acceptable, they provide linear scaling methods for the calculation of physical properties involving large numbers of eigenstates such as densities of states, spectral functions, thermodynamics, total energies for Monte Carlo simulations and forces for molecular dynamics. KPM provides a uniform approximation to a DOS, with resolution inversely proportional to the number of Chebyshev moments, while MEM can achieve significantly higher, but non-uniform, resolution at the risk of possible artifacts. This paper emphasizes efficient algorithms.
The density of states (DoS) encodes the thermodynamic and spectral properties of quantum many-body systems, yet its reconstruction becomes intractable for Hilbert spaces too large to diagonalize. Classically, the kernel polynomial method (KPM) addresses this by combining stochastic trace estimation with a smoothing kernel. Here we show that the Rodeo algorithm---one of the simplest eigenvalue-location protocols for near-term quantum hardware---provides a direct quantum analogue of this approach. Averaging the Rodeo response over Haar-random input states yields the DoS convolved with a spectral kernel fixed entirely by the distribution of evolution times: the random states play the role of stochastic trace estimation, and the temporal sampling distribution that of the damping kernel. The construction requires only the standard single-ancilla circuit, and quantum typicality suppresses the statistical error as the Hilbert-space dimension grows. We derive the estimator and its uncertainties, establish an explicit dictionary between signal-processing window functions and quantum reconstruction kernels, and validate the method on the one-dimensional transverse-field Ising and spin-1 models.
We study the (Anderson) metal-insulator transition (MIT) in tight binding models (TBM) of disordered systems using the scaling behavior of the typical density of states (GDOS) as localization criterion. The GDOS is obtained as the geometrical mean value of the local density of states (LDOS) averaged over many different lattice sites and disorder realizations. The LDOS can efficiently be obtained within the kernel polynomial method (KPM). To check the validity and accuracy of the method, we apply it here to the standard Anderson model of disordered systems, for which the results (for instance for the critical disorder strength of the Anderson transition) are well known from other methods.