Response functions are a key quantity to describe the near-equilibrium dynamics of strongly interacting many-body systems. Recent techniques that attempt to overcome the challenges of calculating these ab initio have employed expansions in terms of orthogonal polynomials. We employ a neural network prediction algorithm to reconstruct a response function $S(\ensuremath{\omega})$ defined over a range in frequencies $\ensuremath{\omega}$. We represent the calculated response function as a truncated Chebyshev series whose coefficients can be optimized to reduce the representation error. We compare the quality of response functions obtained using coefficients calculated using a neural network (NN) algorithm with those computed using the Gaussian integral transform (GIT) method. In the regime where only a small number of terms in the Chebyshev series are retained, we find that the NN scheme outperforms the GIT method.
Response functions are a key quantity to describe the near-equilibrium dynamics of strongly interacting many-body systems. Recent techniques that attempt to overcome the challenges of calculating these ab initio have employed expansions in terms of orthogonal polynomials. We employ a neural network prediction algorithm to reconstruct a response function S(\ensuremathω) defined over a range in frequencies \ensuremathω. We represent the calculated response function as a truncated Chebyshev series whose coefficients can be optimized to reduce the representation error. We compare the quality of response functions obtained using coefficients calculated using a neural network (NN) algorithm with those computed using the Gaussian integral transform (GIT) method. In the regime where only a small number of terms in the Chebyshev series are retained, we find that the NN scheme outperforms the GIT method.
Calculating the spectral function of two-dimensional systems is arguably one of the most pressing challenges in modern computational condensed matter physics. While efficient techniques are available in lower dimensions, two-dimensional systems present insurmountable hurdles, ranging from the sign problem in quantum Monte Carlo methods to the entanglement area law in tensor-network-based methods. We hereby present a variational approach based on a Chebyshev expansion of the spectral function and a neural network representation for the wave functions. The Chebyshev moments are obtained by recursively applying the Hamiltonian and projecting on the space of variational states using a modified natural gradient descent method. We compare this approach with a modified approximation of the spectral function which uses a Krylov subspace constructed from the ``Chebyshev wave functions.'' We present results for the one-dimensional and two-dimensional Heisenberg model on the square lattice and compare them with those obtained by other methods in the literature.
Understanding strongly correlated quantum many-body systems requires dealing with a largeconfiguration space comprised of a set of all possible measurable configurations. The total number of configurations grows exponentially with the size of the system, which makes exact numerical calculations impossible for even moderately sized systems. Since the advent of high-temperature superconductivity and the subsequent interest in strongly correlated phases of matter, progress in the field has been marked by ingenuity of the field to overcome these computational limitations. The first techniques to emerge were inspired by quantum chemistry and based on exact diagonalization[1], which is limited to small clusters, and different variants of quantum Monte Carlo[2], that suffers from the sign problem in fermionic and frustrated systems. Circa 1992, the density matrix renormalization group (DMRG) [3, 4, 5, 6, 7] became one of the major breakthroughs of the past few decades. Despite the success of DMRG for one-dimensional(1D) and quasi-one-dimensional geometries, extensions to actual two-dimensional systems remain challenging and applications are constrained to long cylinders and strips. One promising alternative is to combine one the oldest methods, variational wave functions (VWFs), with neural networks, which have been used to recently with great success in machine learning. The focus of this dissertation is on developing novel variational Monte Carlo algorithms to make neural network variational wave functions applicable to many problems in quantum many-body physics. These algorithms draw from existing variational wave-function, tensor network, and machine learning methods. We calculate dynamical spectral functions using the vector correction and Chebyshev expansion method. We do finite temperature calculations with minimally entangled thermal states. Finally, we do ground state calculations for the frustrated J1−J2 Heisenberg model. We benchmark these results using restricted Boltzmann machines as the neural network for the the variational wave-functions.--Author's abstract
Computing dynamical distributions in quantum many-body systems represents one of the paradigmatic open problems in theoretical condensed matter physics. Despite the existence of different techniques both in real-time and frequency space, computational limitations often dramatically constrain the physical regimes in which quantum many-body dynamics can be efficiently solved. Here we show that the combination of machine-learning methods and complementary many-body tensor network techniques substantially decreases the computational cost of quantum many-body dynamics. We demonstrate that combining kernel polynomial techniques and real-time evolution, together with deep neural networks, allows to compute dynamical quantities faithfully. Focusing on many-body dynamical distributions, we show that this hybrid neural-network many-body algorithm, trained with single-particle data only, can efficiently extrapolate dynamics for many-body systems without prior knowledge. Importantly, this algorithm is shown to be substantially resilient to numerical noise, a feature of major importance when using this algorithm together with noisy many-body methods. Ultimately, our results provide a starting point towards neural-network powered algorithms to support a variety of quantum many-body dynamical methods, that could potentially solve computationally expensive many-body systems in a more efficient manner.
Today's most popular techniques for accurately calculating the dynamics of the reduced density operator in an open quantum system, either require, or gain great computational benefits, from representing the bath response function a(t) in the form a(t)=Σ_k^K p_k e^{O_k t} . For some of these techniques, the number of terms in the series K plays the lead role in the computational cost of the calculation, and is therefore often a limiting factor in simulating open quantum system dynamics. We present an open source MATLAB program called BATHFIT 1, whose input is any spectral distribution functions J(w) or bath response function, and whose output attempts to be the set of parameters {p_k,w_k}_k=1^K such that for a given value of K, the series Σ_k^k p_k e^{O_k t} is as close as possible to a(t). This should allow the user to represent a(t) as accurately as possible with as few parameters as possible. The program executes non-linear least squares fitting, and for a very wide variety of spectral distribution functions, competent starting parameters are used for these fits. For most forms of J(w), these starting parameters, and the exact a(t) corresponding to the given J(w), are calculated using the recent Pade decomposition technique - therefore this program can also be used to merely implement the Pade decomposition for these spectral distribution functions; and it can also be used just to efficiently and accurately calculate a(t) for any given J(w) . The program also gives the J(w) corresponding to a given a(t), which may allow one to assess the quality (in the w-domain) of a representation of a(t) being used. Finally, the program can calculate the discretized influence functional coefficients for any J(w), and this is computed very efficiently for most forms of J(w) by implementing the recent technique published in [Quantum Physics Letters (2012) 1 (1) pg. 35].
Accurate calculations of the spectral density in a strongly correlated quantum many-body system are of fundamental importance to study its dynamics in the linear response regime. Typical examples are the calculation of inclusive and semiexclusive scattering cross sections in atomic nuclei and transport properties of nuclear and neutron star matter. Integral transform techniques play an important role in accessing the spectral density in a variety of nuclear systems. However, their accuracy is in practice limited by the need to perform a numerical inversion which is often ill-conditioned. In the present work we extend a recently proposed quantum algorithm which circumvents this problem. We show how to perform controllable reconstructions of the spectral density over a finite energy resolution with rigorous error estimates. An appropriate expansion in Chebyshev polynomials allows for efficient simulations also on classical computers. We apply our idea to obtain the local density of states for graphene in a magnetic field as a proof of principle. This paves the way for future applications in nuclear and condensed matter physics.
Accurate electronic spectra require both a flexible description of electron correlation and a tractable treatment of the many states contributing to the response. We combine neural network wave functions with the Lorentz integral transform to calculate electronic spectra directly in continuous coordinates, without truncation error from a fixed one-electron basis and with polynomial computational cost per optimization step. Instead of constructing a prescribed set of excited states, the method solves an inhomogeneous Schrödinger equation at a chosen complex energy. This formulation gives access, in principle, to the entire spectrum coupled to a perturbation, including bound excitations and the ionization continuum, without explicitly determining all lower-lying eigenstates. A finite imaginary energy controls the resolution and keeps the response square integrable. Near an isolated bound excitation, the normalized response also recovers the corresponding eigenstate as the width tends to zero. A helium application illustrates the extraction of an excitation energy and oscillator strength. The formulation provides a route from neural descriptions of electronic correlation to spectra beyond a small manifold of low-lying states.