We derive recursively the perturbation series for the ground-state energy of the D-dimensional anharmonic oscillator and resum it using variational perturbation theory (VPT). From the exponentially fast converging approximants, we extract the coefficients of the large-D expansion to higher orders. The calculation effort is much smaller than in the standard field-theoretic approach based on the Hubbard-Stratonovich transformation.
We analyze a binary classification problem by using a support vector machine based on variational quantum-circuit model. We propose to solve a linear equation of the support vector machine by using a $Γ$ matrix expansion. In addition, it is shown that an arbitrary quantum state is prepared by optimizing a universal quantum circuit representing an arbitrary $U(2^N)$ based on the steepest descent method. It may be a quantum generalization of Field-Programmable-Gate Array (FPGA).
Variational Quantum Algorithms are among the most promising systems to implement quantum computing under the Noisy-Intermediate Scale Quantum (NISQ) technology. In variational quantum algorithm, wavefunction represented by a parametrized ansatz and variational parameters are updated iteratively with a classical computer based on the measurement outputs from the quantum computer. In this paper, we investigate perturbation theory with these algorithms and prospect the possibility of using the variational quantum algorithm to simulate quantum dynamics in perturbation theory. We illustrate the use…
Variational weak-coupling perturbation theory yields converging approximations, uniformly in the coupling strength. This allows us to calculate directly the coefficients of `strong-coupling' expansions. For the anharmonic oscillator we explain the physical origin of the empirically observed convergence behavior which is exponentially fast with superimposed oscillations.
We develop convergent variational perturbation theory for quantum statistical density matrices. The theory is applicable to polynomial as well as nonpolynomial interactions. Illustrating the power of the theory, we calculate the temperature-dependent density of a particle in a double-well and of the electron in a hydrogen atom.
In this paper, a variational perturbation scheme for nonrelativistic many-Fermion systems is generalized to a Bosonic system. By calculating the free energy of an anharmonic oscillator model, we investigated this variational expansion scheme for its efficiency. Using the modified Feynman rules for the diagrams, we obtained the analytical expression of the free energy up to the fourth order. Our numerical results at various orders are compared with the exact and other relevant results.
Variational perturbation theory is used to determine the decay rates of metastable states across a cubic barrier of arbitrary height. For high barriers, a variational resummation procedure is applied to the complex energy eigenvalues obtained from a WKB expansion; for low barriers, the variational resummation procedure converts the non-Borel-summable Rayleigh-Schrö\-din\-ger expansion into an exponentially fast convergent one. The results in the two regimes match and yield very accurate imaginary parts of the energy eigenvalues. This is demonstrated by comparison with the complex eigenvalues f…
We evaluate perturbatively the density matrix in the low-temperature limit and thus the ground-state wave function of the anharmonic oscillator up to second order in the coupling constant. We then employ Kleinert's variational perturbation theory to determine the ground-state wave function for all coupling strengths.
Variational perturbation expansions have recently been used to calculate directly the strong-coupling expansion coefficients of the anharmonic oscillator. The convergence is exponentially fast with superimposed oscillations, as recently observed empirically by the authors. In this note, the observed behavior is explained and used to determine accurately the magnitude and phase of the leading Bender-Wu singularity which is responsible for the finite convergence radius in the complex coupling constant plane.
For many physical quantities, theory supplies weak- and strong-coupling expansions of the types $\sum a_n α^n$ and $ α^p\sum b_n (α^{-2/q) ^n$, respectively. Either or both of these may have a zero radius of convergence. We present a simple interpolation algorithm which rapidly converges for an increasing number of known expansion coefficients. The accuracy is illustrated by calculating the ground state energies of the anharmonic oscillator using only the leading large-order coefficient $b_0$ (apart from the trivial expansion coefficent $a_0=1/2$). The errors are less than 0.5 for all g. The a…
We present a physically motivated variational wave function for the ground state of the asymmetric quantum Rabi model (AQRM). The wave function is a weighted superposition of squeezed coherent states entangled with non-orthogonal qubit states, and relies only on three variational parameters in the regimes of interest where the squeezing effect becomes negligible. The variational expansion describes the ground state remarkably well in almost all parameter regimes, especially with arbitrary bias. We use the variational result to calculate various relevant physical observables of the ground state…
As an application of a recently developed variational perturbation theory we find the first 22 terms of the convergent strong-coupling series expansion for the ground state energy of the quartic anharmonic oscillator.
Recently-developed variational perturbation expansions converge exponentially fast for positive coupling constants. They do not, however, possess the correct left-hand cut in the complex coupling constant plane, implying a wrong large-order behavior of their Taylor expansion coefficients. We correct this deficiency and present a method of resumming divergent series with their proper large-order behavior. For a given set of expansion coefficients, knowledge of the large-order behavior considerably improves the quality of the approximation.
In this paper we introduce a generalization to the algebraic Bender-Wu recursion relation for the eigenvalues and the eigenfunctions of the anharmonic oscillator. We extend this well known formalism to the time-dependent quantum statistical Schroedinger equation, thus obtaining the imaginary-time evolution amplitude by solving a recursive set of ordinary differential equations. This approach enables us to evaluate global and local quantum statistical quantities of the anharmonic oscillator to much higher orders than by evaluating Feynman diagrams. We probe our perturbative results by deriving …
We prove a powerful scaling property for the extremality condition in the recently developed variational perturbation theory which converts divergent perturbation expansions into exponentially fast convergent ones. The proof is given for the energy eigenvalues of an anharmonic oscillator with an arbitrary $x^p$-potential. The scaling property greatly increases the accuracy of the results.
Variational quantum algorithms are considered to be appealing applications of near-term quantum computers. However, it has been unclear whether they can outperform classical algorithms or not. To reveal their limitations, we must seek a technique to benchmark them on large scale problems. Here, we propose a perturbative approach for efficient benchmarking of variational quantum algorithms. The proposed technique performs perturbative expansion of a circuit consisting of Clifford and Pauli rotation gates, which is enabled by exploiting the classical simulatability of Clifford circuits. Our meth…
In the study of open quantum systems, the polaron transformation has recently attracted a renewed interest as it offers the possibility to explore the strong system-bath coupling regime. Despite this interest, a clear and unambiguous analysis of the regimes of validity of the polaron transformation is still lacking. Here we provide such a benchmark, comparing second order perturbation theory results in the original untransformed frame, the polaron frame and the variational extension with numerically exact path integral calculations of the equilibrium reduced density matrix. Equilibrium quantit…
A new variational perturbation theory is developed based on the $q-$deformed oscillator. It is shown that the new variational perturbation method provides 200 or 10 times better accuracy for the ground state energy of anharmonic oscillator than the Gaussian and the improved Gaussian approximation, respectively.
We discuss two simple variational approaches to quantum wells. The trial harmonic functions analyzed in an earlier paper give reasonable results for all well depths and are particularly suitable for deep wells. On the other hand, the exponential functions proposed here are preferable for shallow wells. We compare the shallow-well expansions for both kind of functions and show that they do not exhibit the cubic term appearing in the exact series. It is also shown that the deep-well expansion for the harmonic functions agree with the first terms of perturbation theory.
A new recursion procedure for deriving renormalized perturbation expansions for the one-dimensional anharmonic oscillator is offered. Based upon the $\hbar$-expansions and suitable quantization conditions, the recursion formulae obtained have the same simple form both for ground and excited states and can be easily applied to any renormalization scheme. As an example, the renormalized expansions for the sextic anharmonic oscillator are considered.
The newly developed single trajectory quadrature method is applied to a two-dimensional example. The results based on different versions of new perturbation expansion and the new Green's function deduced from this method are compared to each other, also compared to the result from the traditional perturbation theory. As the first application to higher-dimensional non-separable potential the obtained result further confirms the applicability and potential of this new method.
The explicit semiclassical treatment of logarithmic perturbation theory for the bound-state problem within the framework of the radial Klein-Gordon equation with attractive real-analytic screened Coulomb potentials, contained time-component of a Lorentz four-vector and a Lorentz-scalar term, is developed. Based upon $\hbar$-expansions and suitable quantization conditions a new procedure for deriving perturbation expansions is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and excited states have been obtained. As…
The variational perturbation theory for wave functions, which has been shown to work well for bound states of the anharmonic oscillator, is applied to resonance states of the anharmonic oscillator with negative coupling constant. We obtain uniformly accurate wave functions starting from the bound states.
We restate the adiabatic elimination approximation as the first term in a singular perturbation expansion. We use the invariant manifold formalism for singular perturbations in dynamical systems to identify systematic improvements on adiabatic elimination, connecting with well established quantum mechanical perturbation methods. We prove convergence of the expansions when energy scales are well separated. We state and solve the problem of hermiticity of improved effective hamiltonians.
We propose a variational perturbation method based on the observation that eigenvalues of each parity sector of both the anharmonic and double-well oscillators are approximately equi-distanced. The generalized deformed algebra satisfied by the invariant operators of the systems provides well defined Hilbert spaces to both of the oscillators. There appears a natural expansion parameter defined by the ratios of three distance scales of the trial wavefunctions. The energies of the ground state and the first order excited state, in the $0^{th}$ order variational approximation, are obtained with er…
We study properties of eigenfunctions of perturbed systems, given on the eigenbases of unperturbed, integrable systems. For a given pair of perturbed and unperturbed systems, with respect to the energy of each perturbed state, the unperturbed basis states can be divided into two groups: one in the classically-allowed region and the other in the classically-forbidden region; correspondingly, the eigenfunction of the perturbed state is also divided into two parts. In the semiclassical limit, it is shown that, making use of components of the eigenfunction in its classically-allowed region, its co…
We compute the shift of the transition temperature for a homogenous weakly interacting Bose gas in leading order in the scattering length a for given particle density n. Using variational perturbation theory through six loops in a classical three-dimensional scalar field theory, we obtain Delta T_c/T_c = 1.25+/-0.13 a n^(1/3), in agreement with recent Monte-Carlo results.
The Quantum Fisher Information matrix (QFIM) is a central metric in promising algorithms, such as Quantum Natural Gradient Descent and Variational Quantum Imaginary Time Evolution. Computing the full QFIM for a model with $d$ parameters, however, is computationally expensive and generally requires $\mathcal{O}(d^2)$ function evaluations. To remedy these increasing costs in high-dimensional parameter spaces, we propose using simultaneous perturbation stochastic approximation techniques to approximate the QFIM at a constant cost. We present the resulting algorithm and successfully apply it to pr…
The shift of the Bose-Einstein condensation temperature for a homogenous weakly interacting Bose gas in leading order in the scattering length `a' is computed for given particle density `n.' Variational perturbation theory is used to resum the corresponding perturbative series for Delta<phi^2>/Nu in a classical three-dimensional scalar field theory with coupling `u' and where the physical case of N=2 field components is generalized to arbitrary N. Our results for N=1,2,4 are in agreement with recent Monte-Carlo simulations; for N=2, we obtain Delta T_c/T_c = 1.27 +/- 0.11 a n^(1/3). We use sev…
Covariant perturbation expansion is an important method in quantum field theory. In this paper an expansion up to arbitrary order for off-diagonal heat kernels in flat space based on the covariant perturbation expansion is given. In literature, only diagonal heat kernels are calculated based on the covariant perturbation expansion.
We present a simple operator perturbation theory in the backward Heisenberg picture. Compared with the well-known Heisenberg picture, the revised picture is based on the backward time instead of the forward time. The unique feature of the uncommon picture is that the perturbation expansion becomes very simple and the famous Dyson expansion is not directly involved. Its relationship with the perturbation expansion for the density operator developed by Kubo is also discussed.
The exchange contribution to the energy of the hydrogen atom interacting with a proton is calculated from the polarization expansion of the wave function using the conventional surface-integral formula and two formulas involving volume integrals: the formula of the symmetry-adapted perturbation theory (SAPT) and the variational formula recommended by us. At large internuclear distances $R$, all three formulas yield the correct expression $-(2/e)Re^{-R}$, but approximate it with very different convergence rates. In the case of the SAPT formula, the convergence is geometric with the error fallin…
This work establishes a connection between canonical perturbation series in quantum mechanics and a Kato expansion for the resolvent of the Liouville superoperator. Our approach leads to an explicit expression for a generator of a block-diagonalizing Dyson's ordered exponential in arbitrary perturbation order. Unitary intertwining of perturbed and unperturbed averaging superprojectors allows for a description of ambiguities in the generator and block-diagonalized Hamiltonian. We compare the efficiency of the corresponding computational algorithm with the efficiencies of the Van Vleck and Magnu…
The Complex Kohn variational method for electron-polyatomic molecule scattering is formulated using an overset grid representation of the scattering wave function. The overset grid consists of a central grid and multiple dense, atom-centered subgrids that allow the simultaneous spherical expansions of the wave function about multiple centers. Scattering boundary conditions are enforced by using a basis formed by the repeated application of the free particle Green's function and potential, $\hat{G}^+_0\hat{V}$ on the overset grid in a "Born-Arnoldi" solution of the working equations. The theory…
