Estimate Of Ground-state Energy Research Articles

More quantum chemistry with fewer qubits

Quantum computation is one of the most promising new paradigms for the simulation of physical systems composed of electrons and atomic nuclei, with applications in chemistry, solid-state physics, materials science, and molecular biology. This requires a truncated representation of the electronic structure Hamiltonian using a finite number of orbitals. While it is, in principle, obvious how to improve on the representation by including more orbitals, this is usually unfeasible in practice (e.g., because of the limited number of qubits available) and severely compromises the accuracy of the obtained results. Here, we propose a quantum algorithm that improves on the representation of the physical problem by virtue of second-order perturbation theory. In particular, our quantum algorithm evaluates the second-order energy correction through a series of time-evolution steps under the unperturbed Hamiltonian. An important application is to go beyond the active-space approximation, allowing to include corrections of virtual orbitals, known as multireference perturbation theory. Here, we exploit the fact that the unperturbed Hamiltonian is diagonal for virtual orbitals and show that the number of qubits is independent of the number of virtual orbitals. This gives rise to more accurate energy estimates without increasing the number of qubits. Moreover, we demonstrate numerically for realistic chemical systems that the total runtime has highly favorable scaling in the number of virtual orbitals compared to previous studies. Numerical calculations confirm the necessity of the multireference perturbation theory energy corrections to reach accurate ground-state energy estimates. Our perturbation theory quantum algorithm can also be applied to symmetry-adapted perturbation theory. As such, we demonstrate that perturbation theory can help to reduce the quantum hardware requirements for quantum chemistry. Published by the American Physical Society 2024

Rayleigh-Gauss-Newton optimization with enhanced sampling for variational Monte Carlo

Variational Monte Carlo (VMC) is an approach for computing ground-state wave functions that has recently become more powerful due to the introduction of neural network-based wave-function parametrizations. However, efficiently training neural wave functions to converge to an energy minimum remains a difficult problem. In this work, we analyze optimization and sampling methods used in VMC and introduce alterations to improve their performance. First, based on theoretical convergence analysis in a noiseless setting, we motivate a new optimizer that we call the Rayleigh-Gauss-Newton method, which can improve upon gradient descent and natural gradient descent to achieve superlinear convergence at no more than twice the computational cost. Second, to realize this favorable comparison in the presence of stochastic noise, we analyze the effect of sampling error on VMC parameter updates and experimentally demonstrate that it can be reduced by the parallel tempering method. In particular, we demonstrate that RGN can be made robust to energy spikes that occur when the sampler moves between metastable regions of configuration space. Finally, putting theory into practice, we apply our enhanced optimization and sampling methods to the transverse-field Ising and $XXZ$ models on large lattices, yielding ground-state energy estimates with remarkably high accuracy after just 200 parameter updates.

Two-body bound states through Yukawa forces and perspectives on hydrogen and deuterium

We present and discuss numerical solutions to a two-body quantum bound state problem closely related to that of the hydrogen atom and the deuterium nucleus. The forces binding the particles of our system are Yukawa forces, which fall off with distance faster than the Coulomb force and arise for non-relativistic particles whose interactions are mediated by massive scalar or vector particles; the Coulomb force arises in the limit of massless mediators such as photons. We use the solutions to explain several features of deuterium. We also present heuristic estimates of ground state energies and energy level degeneracy breaking. Studying this Yukawa two-body system provides insights into hydrogen, deuterium, and two-body bound states, generally. We describe our undergraduate-accessible procedure for numerically solving non-relativistic two-body bound state problems with mathematica in the appendix.

Efficient method for grand-canonical twist averaging in quantum Monte Carlo calculations

We introduce a simple but efficient method for grand-canonical twist averaging in quantum Monte Carlo calculations. By evaluating the thermodynamic grand potential instead of the ground state total energy, we greatly reduce the sampling errors caused by twist-dependent fluctuations in the particle number. We apply this method to the electron gas and to metallic lithium, aluminum, and solid atomic hydrogen. We show that, even when using a small number of twists, grand-canonical twist averaging of the grand potential produces better estimates of ground state energies than the widely used canonical twist-averaging approach.

Noise can speed Markov chain Monte Carlo estimation and quantum annealing.

Carefully injected noise can speed the average convergence of Markov chain Monte Carlo (MCMC) estimates and simulated annealing optimization. This includes quantum annealing and the MCMC special case of the Metropolis-Hastings algorithm. MCMC seeks the solution to a computational problem as the equilibrium probability density of a reversible Markov chain. The algorithm must cycle through a long burn-in phase until it reaches equilibrium because the Markov samples are statistically correlated. The special injected noise reduces this burn-in period in MCMC. A related theorem shows that it reduces the cooling time in simulated annealing. Simulations showed that optimal noise gave a 76% speed-up in finding the global minimum in the Schwefel optimization benchmark. The noise-boosted simulations found the global minimum in 99.8% of trials compared with only 95.4% of trials in noiseless simulated annealing. Simulations also showed that the noise boost is robust to accelerated cooling schedules and that noise decreased convergence times by more than 32% under aggressive geometric cooling. Molecular dynamics simulations showed that optimal noise gave a 42% speed-up in finding the minimum potential energy configuration of an eight-argon-atom gas system with a Lennard-Jones 12-6 potential. The annealing speed-up also extends to quantum Monte Carlo implementations of quantum annealing. Noise improved ground-state energy estimates in a 1024-spin simulated quantum annealing simulation by 25.6%. The quantum noise flips spins along a Trotter ring. The noisy MCMC algorithm brings each Markov step closer on average to equilibrium if an inequality holds between two expectations. Gaussian or Cauchy jump probabilities reduce the noise-benefit inequality to a simple quadratic inequality. Simulations show that noise-boosted simulated annealing is more likely than noiseless annealing to sample high probability regions of the search space and to accept solutions that increase the search breadth.

Using a variational method to obtain the ground state of the quantum Hamiltonian: symbolic computation approach

The importance of learning symbolic computation in research in theoretical sciences cannot be overemphasised. While Fortran and/or C programming laboratories have become an essential part of MSc curricula now, symbolic computing is almost never taught. We demonstrate how a freeware, SAGE, can be employed for the variational solution of simple (or complex) Hamiltonians encountered in quantum mechanics in one dimension. One can easily change the trial wavefunction and the Hamiltonian and obtain estimates of ground state energy. This should lead to a qualitative understanding of the physics of the problem. A brief extension to the first excited state for potentials with parity is discussed. Finally, we give a brief overview of the range of problems which can be introduced in a physics laboratory by using SAGE.

The variational method in quantum mechanics: an elementary introduction

Variational methods in quantum mechanics are customarily presented as invaluable techniques to find approximate estimates of ground state energies. In the present paper a short catalogue of different celebrated potential distributions (both 1D and 3D), for which an exact and complete (energy and wavefunction) ground state determination can be achieved in an elementary way, is illustrated. No previous knowledge of calculus of variations is required. Rather, in all presented cases the exact energy functional minimization is achieved by using only a couple of simple mathematical tricks: ‘completion of square’ and integration by parts. This makes our approach particularly suitable for undergraduates. Moreover, the key role played by particle localization is emphasized through the entire analysis. This gentle introduction to the variational method could also be potentially attractive for more expert students as a possible elementary route toward a rather advanced topic on quantum mechanics: the factorization method. Such an unexpected connection is outlined in the final part of the paper.

Using uncertainty principle to find the ground-state energy of the helium and a helium-like Hookean atom

In this paper, we accurately estimate the ground-state energy and the atomic radius of the helium atom and a helium-like Hookean atom by employing the uncertainty principle in conjunction with the variational approach. We show that with the use of the uncertainty principle, electrons are found to be spread over a radial region, giving an electron cloud. Our calculation also shows how the Coulomb interaction between electrons affects their distribution. This leads to a physical picture of how electrons are located with respect to each other in these atoms. Finally, we also obtain through our calculations a general formula for the estimate of ground-state energy and radius of two electron atoms and ions with atomic number Z.

Frozen Gaussian Wavepacket Study of the Ground State of the He Atom.

The Rayleigh-Ritz functional is used in conjunction with an approximate time evolution to improve ab initio estimates of ground-state energies. The improvement is due in part to the introduction of a novel variational "normalization function" for the approximate propagator. An additional variational parameter was introduced in the form of a constant shift energy of the Hamiltonian. The approximate propagator used was the frozen Gaussian propagator; however, the trajectories evolved on the coherent-state averaged Hamiltonian (Q representation). For Coulombic forces, this removes the singularity, easing the computation. An additional variational parameter was the width parameter used for the coherent states appearing in the frozen Gaussian propagator. Using an initial combination of nine Gaussian functions for He, with an initial energy of -2.5115 au, the variational method, with a very short time interval of integration, led to an improved energy of -2.81 ± 0.04 au.

Phase separation and stripes in a boson version of a doped quantum antiferromagnet

A theoretical investigation of boson versions of the $t\ensuremath{-}J$ and $t\ensuremath{-}{J}_{z}$ models on the square lattice is carried out, by means of Green's function Monte Carlo simulations. Accurate ground-state energy estimates as a function of hole doping are obtained, allowing one to investigate the stability of the uniform phase against separation of the system into hole-rich and hole-free phases. In the boson $t\ensuremath{-}{J}_{z}$ model, such a separation is found to occur for arbitrarily small values of ${J}_{z},$ at sufficiently low hole doping. Phase separation is suppressed in the boson $t\ensuremath{-}J$ model, which features a uniform ground state at any doping, for $J/t\ensuremath{\lesssim}1.5.$ Relevance of this study to the corresponding fermion models is discussed. Fermi statistics enhances the tendency toward phase separation; in particular, phase separation at low doping is predicted in the fermion $t\ensuremath{-}{J}_{z}$ model at any ${J}_{z}>0.$ The possible formation of stripes of holes is investigated for systems featuring both periodic and cylindrical boundary conditions. No evidence of a striped ground state is found in either the $t\ensuremath{-}J$ or $t\ensuremath{-}{J}_{z}$ boson models.

Variational theory for the ground state of linear E-e Jahn-Teller system: Weak- and intermediate-coupling

We present a simple variational treatment for the ground state of the linear E-e Jahn-Teller system. We have argued that the proposed ground state wavefunction is valid for the weak- and intermediate-coupling case, where the dynamical effect, the nonadiabaticity is the most important one. Both the variational estimates of ground state energy and the calculation of Ham's reduction factor have shown that our results are not only in good agreement with the results of exact diagonalization, but also they are better than those of the previous analytical studies. We have discussed the physical meaning of the proposed wavefunctions.

Ground-state real-space renormalization for linear-chain Heisenberg models with alternation

Abstract Ground-state energy estimates are made via real-space renormalization for linear-chain Heisenberg models of general site spins. The tendency for bond alternation (i.e., Peierls distortion or dimerization) is studied and is found to be obligatory only for the s = 1 2 model, which is thus placed in a different universality class than the other spins s ⩾ 1. Variational bounds which become exact either for large s or “strong” alternation are obtained for the ground-state energy.

Improved estimates of few-body ground-state energies

A method for the estimation of ground-state energies of few-body systems is proposed, which is a combination of an inequality due to Post and the 1/d-expansion. The effectiveness of the method is illustrated for power-law potentials.

Inverse energy weighted sum-rules

A new derivation of the inverse energy-weighted sum-rules is given by applying the spectral distribution methods to the Rayleigh-Schrodinger perturbation theory. The scalar space result is then extended to the configurations. This is applied to obtain corrections to the ground-state energy estimates when the effective interaction is approximated by a model Hamiltonian obtained by taking linear combinations of various parts of the pairing and the Q.Q operators.

Extremum principles for the determination of relativistic bound-state energies.

The fact that the Dirac Hamiltonian is not bounded from below has caused a long-standing difficulty in extending to relativistic ground-state energy estimates the powerful Rayleigh-Ritz variational technique extensively used for ground-state energy estimates of nonrelativistic systems. Here we show how the chief source of the difficulty, namely, the likely appearance of numerical instabilities (or ``variational collapse''), in the calculational procedure can be removed, at least for two relatively simple systems: a single Dirac particle in an external potential\char22{}which could be one or more fixed nuclei\char22{}and two Dirac particles bound by their mutual attraction. (The ultimate objective, discussed briefly but not achieved in the present exploratory paper, would be to remove variational collapse in bound-state energy calculations of arbitrary relativistic systems.) The method is based on a formal separation, done with the aid of the Casimir projection operators ${\ensuremath{\Lambda}}_{\mathrm{\ensuremath{-}}}$ and ${\ensuremath{\Lambda}}_{+}$ of the negative-energy and positive-energy continuum branches of the spectrum. A maximum principle is introduced to evaluate that component of an effective potential which accounts for virtual transitions into the negative-energy continuum. This effective potential appears in an eigenvalue equation confined to the subspace spanned by ${\ensuremath{\Lambda}}_{+}$, and here a Rayleigh-Ritz minimum principle can be used to obtain a variational approximation to (but unfortunately not a variational bound on) the ground-state energy; the extension of the method to excited states is straightforward.

One-dimensional gravitational gas: Estimate of ground state energy

The ground state energy E( N) increases with N at most as N 5 3 .