Quantum Equation Of Motion Research Articles

Estimating Molecular Thermal Averages with the Quantum Equation of Motion and Informationally Complete Measurements.

By leveraging the Variational Quantum Eigensolver (VQE), the "quantum equation of motion" (qEOM) method established itself as a promising tool for quantum chemistry on near-term quantum computers and has been used extensively to estimate molecular excited states. Here, we explore a novel application of this method, employing it to compute thermal averages of quantum systems, specifically molecules like ethylene and butadiene. A drawback of qEOM is that it requires measuring the expectation values of a large number of observables on the ground state of the system, and the number of necessary measurements can become a bottleneck of the method. In this work, we focus on measurements through informationally complete positive operator-valued measures (IC-POVMs) to achieve a reduction in the measurement overheads by estimating different observables of interest through the measurement of a single set of POVMs. We show with numerical simulations that the qEOM combined with IC-POVM measurements ensures satisfactory accuracy in the reconstruction of the thermal state with a reasonable number of shots.

Quantum Equation of Motion with Orbital Optimization for Computing Molecular Properties in Near-Term Quantum Computing.

Determining the properties of molecules and materials is one of the premier applications of quantum computing. A major question in the field is how to use imperfect near-term quantum computers to solve problems of practical value. Inspired by the recently developed variants of the quantum counterpart of the equation-of-motion (qEOM) approach and the orbital-optimized variational quantum eigensolver (oo-VQE), we present a quantum algorithm (oo-VQE-qEOM) for the calculation of molecular properties by computing expectation values on a quantum computer. We perform noise-free quantum simulations of BeH2 in the series of STO-3G/6-31G/6-31G* basis sets and of H4 and H2O in 6-31G using an active space of four electrons and four spatial orbitals (8 qubits) to evaluate excitation energies, electronic absorption, and, for twisted H4, circular dichroism spectra. We demonstrate that the proposed algorithm can reproduce the results of conventional classical CASSCF calculations for these molecular systems.

Quantum Simulation of Molecular Electronic States with a Transcorrelated Hamiltonian: Higher Accuracy with Fewer Qubits.

Simulation of electronic structure is one of the most promising applications on noisy intermediate-scale quantum (NISQ) era devices. However, NISQ devices suffer from a number of challenges like limited qubit connectivity, short coherence times, and sizable gate error rates. Thus, desired quantum algorithms should require shallow circuit depths and low qubit counts to take advantage of these devices. Here, we attempt to reduce quantum resource requirements for molecular simulations on a quantum computer while maintaining the desired accuracy with the help of classical quantum chemical theories of canonical transformation and explicit correlation. In this work, compact ab initio Hamiltonians are generated classically, in the second quantized form, through an approximate similarity transformation of the Hamiltonian with (a) an explicitly correlated two-body unitary operator with generalized pair excitations that remove the Coulombic electron-electron singularities from the Hamiltonian and (b) a unitary one-body operator to efficiently capture the orbital relaxation effects required for accurate description of the excited states. The resulting transcorrelated Hamiltonians are able to describe both the ground and the excited states of molecular systems in a balanced manner. Using the variational quantum eigensolver (VQE) method based on the unitary coupled cluster with singles and doubles (UCCSD) ansatz and only a minimal basis set (ANO-RCC-MB), we demonstrate that the transcorrelated Hamiltonians can produce ground state energies comparable to the reference CCSD energies with the much larger cc-pVTZ basis set. This leads to a reduction in the number of required CNOT gates by more than 3 orders of magnitude for the chemical species studied in this work. Furthermore, using the quantum equation of motion (qEOM) formalism in conjunction with the transcorrelated Hamiltonian, we are able to reduce the deviations in the excitation energies from the reference EOM-CCSD/cc-pVTZ values by an order of magnitude. The transcorrelated Hamiltonians developed here are Hermitian and contain only one- and two-body interaction terms and thus can be easily combined with any quantum algorithm for accurate electronic structure simulations.

Quantum corrections to the classical model of the atom-field system

The nonlinear-oscillating system in action-angle variables is characterized by the dependence of frequency of oscillation ω(I) on action I. Periodic perturbation is capable of realizing in the system a stable nonlinear resonance at which the action I adapts to the resonance condition ω(I(0))≃ω, that is, "sticking" in the resonance frequency. For a particular physical problem there may be a case when I≫ℏ is the classical quantity, whereas its correction ΔI≃ℏ is the quantum quantity. Naturally, dynamics of ΔI is described by the quantum equation of motion. In particular, in the moderate nonlinearity approximation ɛ≪(dω/dI)(I/ω)≪1/ɛ, where ɛ is the small parameter, the description of quantum state is reduced to the solution of the Mathieu-Schrödinger equation. The state formed as a result of sticking in resonance is an eigenstate of the operator ΔI that does not commute with the Hamiltonian H. Expanding the eigenstate wave functions in Hamiltonian eigenfunctions, one can obtain a probability distribution of energy level population. Thus, an inverse level population for times lower than the relaxation time can be obtained.

A remark on renormalization of the quantum equation of motion for the matrix sigma model

The infinite part of a 1-prong vertex for the matrix σ-model is calculated in the background field formalism in one-loop approach. It is shown that renormalizations of the quantum equation of motion and effective action coincide if an additional factor is introduced. Bibliography: 7 titles.

Renormalization of the quantum equation of motion for Yang-Mills fields in the background formalism

The aim of this paper is to compare the renormalizations of the infinite parts of the effective action and quantum equation of motion in gauge fields theory with structural group SU (N) in the one-loop approach and to show that they do not coincide. To this end, we use the background formalism, in which a gauge field is decomposed into the sum of a classical (background) field and a quantum field: A = B + gQ. The appearance of an additional factor, which provides the recovery of the equality of renormalizations of equations and action, is explained by comparison with results of the standard renormalization theory of Yang-Mills fields. Bibliography: 7 titles.

Effective action and the quantum equation of motion

We carefully analyse the use of the effective action in dynamical problems, in particular the conditions under which the equation $\frac{\delta \Ga} {\delta \phi}=0$ can be used as a quantum equation of motion, and the relation between the asymptotic states involved in the definition of $\Ga$ and the initial state of the system. By considering the quantum mechanical example of a double-well potential, where we can get exact results for the time evolution of the system, we show that an approximation to the effective potential in the quantum equation of motion that correctly describes the dynamical evolution of the system is obtained with the help of the wilsonian RG equation (already at the lowest order of the derivative expansion), while the commonly used one-loop effective potential fails to reproduce the exact results.

Projection Operator Approach to Langevin Equations in 4 Theory

We apply the projection operator method (POM) to $\phi^4$ theory and derive both quantum and semiclassical equations of motion for the soft modes. These equations have no time-convolution integral term, in sharp contrast with other well-known results obtained using the influence functional method (IFM) and the closed time path method (CTP). However, except for the fluctuation force field terms, these equations are similar to the corresponding equations obtained using IFM with the linear harmonic approximation, which was introduced to remove the time-convolution integral. The quantum equation of motion in POM can be regarded as a kind of quantum Langevin equation in which the fluctuation force field is given in terms of the operators of the hard modes. These operators are then replaced with c-numbers using a certain procedure to obtain a semiclassical Langevin equation. It is pointed out that there are significant differences between the fluctuation force fields introduced in this paper and those introduced in IFM. The arbitrariness of the definition of the fluctuation force field in IFM is also discussed.

Time dependence of the average charge and current in a dissipative mesoscopic circuit

Taking into consideration the interactions between electrons and phonons, we have studied the temporal evolution of the average charge and current in a dissipative mesoscopic RLC circuit. Our results show that a mesoscopic RLC circuit can be treated as an interactive system between an electromagnetic harmonic oscillator and many lattice harmonic oscillators; this is called the bathing of the harmonic oscillators. The results also show that the quantum equation of motion of the linear mesoscopic RLC circuit is identical in form to its classical equation of motion, the only difference between them being their respective meanings. In order to thoroughly study the quantum properties of a dissipative mesoscopic circuit, we have to consider not only the electromagnetic energy of the circuit, but also the crystal lattice vibration energy and the interactive energy between electrons and phonons.

Violation of the correspondence principle: Breakdown of the Bohm-Newton trajectory correspondence in a macroscopic system

The trajectory predicted by Bohm's quantum equation of motion, using an initially angle-localized minimum-uncertainty Gaussian wave packet, is compared with the trajectory predicted by Newton's classical equation of motion for a macroscopic periodically kicked pendulum system. We find that, in violation of the strong form of the correspondence principle, the Bohmian trajectory does not, in general, agree or correspond with the Newtonian trajectory for all times. The Bohm-Newton trajectory correspondence holds as long as the wave packet remains localized in angle. Delocalization of the wave packet in angle triggers the breakdown of the correspondence. The breakdown occurs exponentially quickly if either the Newtonian trajectory that initially coincides with the Bohmian trajectory or one of its classical neighbors is chaotic. In contrast, the breakdown occurs much slower if neither the Newtonian trajectory nor its neighbors are chaotic. These behaviors of the Bohm-Newton trajectory correspondence are generic, or universal, for the class of macroscopic Hamiltonian dynamical systems with bound coordinate space since the kick pendulum is a prototypical member. We predict that neither Bohm's equation of motion nor Newton's correctly predicts the experimentally observed and measured trajectory of a macroscopic system after the breakdown of the correspondence, implying that the all-times agreement required by the strong form of the correspondence principle is unnecessary.

Interpretations of Quantum Field Theory

In this paper we critically review the various attempts that have been made to understand quantum field theory. We focus on Teller's (1990) harmonic oscillator interpretation, and Bohm et al.'s (1987) causal interpretation. The former unabashedly aims to be a purely heuristic account, but we show that it is only interestingly applicable to the free bosonic field. Along the way we suggest alternative models. Bohm's interpretation provides an ontology for the theory—a classical field, with a quantum equation of motion. This too has problems; it is not Lorentz invariant.

Nuclear dynamics in the Wigner representation

The quantum equation of motion of the density operator in the Wigner representation is solved using a stochastic approach. Nuclear ground states and an asymmetric nucleus-nucleus collision below the Coulomb barrier are studied. Quantum effects are shown to cause significant differences in comparison to results obtained from the classical Vlasov equation.

Correlation function on quantum phase space

We show that a quantum equation of motion and the correct expectation value of an observable can be defined on quantum phase space using a two-point correlation function.

Nonrelativistic reduction and interpretation of the Klein-Gordon equation of tachyons

A one-dimensional tachyon Klein-Gordon equation is reduced to a nonrelativistic-tachyon equation of motion. The interpretation of this reduced equation leads to the following conclusions: 1) tachyons can be localized in time instead of in space as compared with bradyons, 2) space representation and momentum representation of bradyonic quantum equation of motion are replaced by time representation and energy representation in tachyon quantum equation of motion and 3) with the aid of these results, it has been found that the solutions of the tachyon Klein-Gordon equation of motion form a complete set.

Radiative reaction and a possible theory for the muon

In classical electron theory, the equation of motion with radiative reaction features a higher order differential structure and a characteristic length. Consideration is given to an analogous quantum equation of motion, a new equation that features a higher order differential structure and the characteristic length of classical theory. In addition to the wave function for an ordinary electron, the wave function for a muon, an electron quantum state that is « excited » dynamically by radiative reaction, appears as a solution to this new quantum equation of motion. It is shown that interaction theory based on this electron quantum equation is compatible with conventional quantum electrodynamics, generalized to include the muon.