arXiv subjects

Aleksei V. Ivanov

Publications and source records attributed to Aleksei V. Ivanov.

16 recordsLinked to original sources

arXiv2025

A Monte Carlo approach to bound Trotter error

Trotter product formulas are a natural and powerful approach to perform quantum simulation. However, the error analysis of product formulas is challenging, and their cost is often overestimated. It is established that Trotter error can be bounded in terms of spectral norms of nested commutators of the Hamiltonian partitions [Childs et al., Phys. Rev. X 11, 011020], but evaluating these expressions is challenging, often achieved by repeated application of the triangle inequality, significantly loosening the bound. Here, we show that the spectral norm of an operator can be upper bounded by the spectral norm of an equivalent sign-problem-free operator, which can be calculated efficiently to large system sizes using projector Monte Carlo simulation. For a range of Hamiltonians and considering second-order formulas, we demonstrate that this Monte Carlo-based bound is often extremely tight, and even exact in some instances. For the uniform electron gas we reduce the cost of performing Trotterization from the literature by an order of magnitude. For the Pariser-Parr-Pople model for linear acene molecules, which has $\mathcal{O}(N^2)$ long-range interaction terms, we show that it suffices to use $\mathcal{O}(N^{0.57})$ Trotter steps and circuit depth $\mathcal{O}(N^{1.57})$ to implement Hamiltonian simulation. We hope that this approach will lead to a better understanding of the potential accuracy of Trotterization in a range of important applications.

quant-ph↗

arXiv2025

Quantum Computation of Electronic Structure with Projector Augmented-Wave Method and Plane Wave Basis Set

Quantum simulation of materials is a promising application area of quantum computers. To practically realize this promise, we must reduce quantum resources while maintaining accuracy. In electronic structure calculations on classical computers, resource reduction has been achieved by using the projector augmented-wave method (PAW) and plane wave basis sets. However, the PAW method generalized for many-body states introduces non-orthogonality effects which impede its direct application to quantum computing. In this work, we develop a unitary variant of the PAW (UPAW) that preserves the orthogonality constraints. We provide a linear-combination-of-unitaries decomposition of the UPAW Hamiltonian to enable ground state estimation using qubitized quantum phase estimation. Additionally, we further improve algorithmic efficiency by extending classical down-sampling techniques into the quantum setting. We then estimate quantum resources for crystalline solids to estimate the energy within chemical accuracy with respect to the full basis set limit, and also consider a supercell approach which is more suitable for calculations of defect states. We provide the quantum resources for energy estimation of a nitrogen-vacancy defect centre in diamond which is a challenging system for classical algorithms and a quintessential problem in the studies of quantum point defects.

quant-ph↗

arXiv2025

Quantum Simulations of Chemistry in First Quantization with any Basis Set

Quantum computation of the energy of molecules and materials is one of the most promising applications of fault-tolerant quantum computers. Practical applications require development of quantum algorithms with reduced resource requirements. Previous work has mainly focused on quantum algorithms where the Hamiltonian is represented in second quantization with compact basis sets while existing methods in first quantization are limited to a grid-based basis. In this work, we present a new method to solve the generic ground-state chemistry problem in first quantization using any basis set. We achieve asymptotic speedup in Toffoli count for molecular orbitals, and orders of magnitude improvement using dual plane waves as compared to the second quantization counterparts. In some instances, our approach provides similar or even lower resources compared to previous first quantization plane wave algorithms that, unlike our approach, avoids the loading of the classical data. The developed methodology can be applied to variety of applications, where the matrix elements of a first quantized Hamiltonian lack simple circuit representation.

quant-ph↗

arXiv2025

Pauli Decomposition via the Fast Walsh-Hadamard Transform

The decomposition of a square matrix into a sum of Pauli strings is a classical pre-processing step required to realize many quantum algorithms. Such a decomposition requires significant computational resources for large matrices. We present an exact and explicit formula for the Pauli string coefficients which inspires an efficient algorithm to compute them. More specifically, we show that up to a permutation of the matrix elements, the decomposition coefficients are related to the original matrix by a multiplication of a generalised Hadamard matrix. This allows one to use the Fast Walsh-Hadamard transform and calculate all Pauli decomposition coefficients in $\mathcal{O}(N^2\log N)$ time and using $\mathcal{O}(1)$ additional memory, for an $N\times N$ matrix. A numerical implementation of our equation outperforms currently available solutions.

quant-ph↗

arXiv2024

GPAW: An open Python package for electronic-structure calculations

We review the GPAW open-source Python package for electronic structure calculations. GPAW is based on the projector-augmented wave method and can solve the self-consistent density functional theory (DFT) equations using three different wave-function representations, namely real-space grids, plane waves, and numerical atomic orbitals. The three representations are complementary and mutually independent and can be connected by transformations via the real-space grid. This multi-basis feature renders GPAW highly versatile and unique among similar codes. By virtue of its modular structure, the GPAW code constitutes an ideal platform for implementation of new features and methodologies. Moreover, it is well integrated with the Atomic Simulation Environment (ASE) providing a flexible and dynamic user interface. In addition to ground-state DFT calculations, GPAW supports many-body GW band structures, optical excitations from the Bethe-Salpeter Equation (BSE), variational calculations of excited states in molecules and solids via direct optimization, and real-time propagation of the Kohn-Sham equations within time-dependent DFT. A range of more advanced methods to describe magnetic excitations and non-collinear magnetism in solids are also now available. In addition, GPAW can calculate non-linear optical tensors of solids, charged crystal point defects, and much more. Recently, support of GPU acceleration has been achieved with minor modifications of the GPAW code thanks to the CuPy library. We end the review with an outlook describing some future plans for GPAW.

cond-mat.mtrl-sci↗

arXiv2023

Electronic excitations of the charged nitrogen-vacancy center in diamond obtained using time-independent variational density functional calculations

Elucidation of the mechanism for optical spin initialization of point defects in solids in the context of quantum applications requires an accurate description of the excited electronic states involved. While variational density functional calculations have been successful in describing the ground state of a great variety of systems, doubts have been expressed in the literature regarding the ability of such calculations to describe electronic excitations of point defects. A direct orbital optimization method is used here to perform time-independent, variational density functional calculations of a prototypical defect, the negatively charged nitrogen-vacancy center in diamond. The calculations include up to 511 atoms subject to periodic boundary conditions and the excited state calculations require similar computational effort as ground state calculations. Contrary to some previous reports, the use of local and semilocal density functionals gives the correct ordering of the low-lying triplet and singlet states, namely ${}^{3}A_2 < {}^{1}E < {}^{1}A_1 < {}^{3}E$. Furthermore, the more advanced meta generalized gradient approximation functionals give results that are in remarkably good agreement with high-level, many-body calculations as well as available experimental estimates, even for the excited singlet state which is often referred to as having multireference character. The lowering of the energy in the triplet excited state as the atom coordinates are optimized in accordance with analytical forces is also close to the experimental estimate and the resulting zero-phonon line triplet excitation energy is underestimated by only 0.15 eV. The approach used here is found to be a promising tool for studying electronic excitations of point defects in, for example, systems relevant for quantum technologies.

cond-mat.mtrl-sci↗

arXiv2023

Quantum Computation for Periodic Solids in Second Quantization

In this work, we present a quantum algorithm for ground-state energy calculations of periodic solids on error-corrected quantum computers. The algorithm is based on the sparse qubitization approach in second quantization and developed for Bloch and Wannier basis sets. We show that Wannier functions require less computational resources with respect to Bloch functions because: (i) the L$_1$ norm of the Hamiltonian is considerably lower and (ii) the translational symmetry of Wannier functions can be exploited in order to reduce the amount of classical data that must be loaded into the quantum computer. The resource requirements of the quantum algorithm are estimated for periodic solids such as NiO and PdO. These transition metal oxides are industrially relevant for their catalytic properties. We find that ground-state energy estimation of Hamiltonians approximated using 200--900 spin orbitals requires {\it ca.}~$10{}^{10}$--$10^{12}$ T gates and up to $3\cdot10^8$ physical qubits for a physical error rate of $0.1\%$.

quant-ph↗

arXiv2023

Variational Density Functional Calculations of Excited States: Conical Intersection and Avoided Crossing in Ethylene Bond Twisting

Theoretical studies of photochemical processes require a description of the energy surfaces of excited electronic states, especially near degeneracies, where transitions between states are most likely. Systems relevant to photochemical applications are typically too large for high-level multireference methods, and while time-dependent density functional theory (TDDFT) is efficient, it can fail to provide the required accuracy. A variational, time-independent density functional approach is applied to the twisting of the double bond and pyramidal distortion in ethylene, the quintessential model for photochemical studies. By allowing for symmetry breaking, the calculated energy surfaces exhibit the correct topology around the twisted-pyramidalized conical intersection even when using a semilocal functional approximation, and by including explicit self-interaction correction, the torsional energy curves are in close agreement with published multireference results. The findings of the present work point to the possibility of using a single determinant time-independent density functional approach to simulate nonadiabatic dynamics, even for large systems where multireference methods are impractical and TDDFT is often not accurate enough.

physics.chem-ph↗

arXiv2021

Chemical bonding theories as guides for self-interaction corrected solutions: multiple local minima and symmetry breaking

Fermi--Löwdin orbitals (FLO) are a special set of localized orbitals, which have become commonly used in combination with the Perdew--Zunger self-interaction correction (SIC) in the FLO-SIC method. The FLOs are obtained for a set of occupied orbitals by specifying a classical position for each electron. These positions are known as Fermi-orbital descriptors (FODs), and they have a clear relation to chemical bonding. In this study, we show how FLOs and FODs can be used to initialize, interpret and justify SIC solutions in a common chemical picture, both within FLO-SIC and in traditional variational SIC, and to locate distinct local minima in either of these approaches. We demonstrate that FLOs based on Lewis' theory lead to symmetry breaking for benzene -- the electron density is found to break symmetry already at the symmetric molecular structure -- while ones from Linnett's double-quartet theory reproduce symmetric electron densities and molecular geometries. Introducing a benchmark set of 16 planar, cyclic molecules, we show that using Lewis' theory as the starting point can lead to artifactual dipole moments of up to 1 Debye, while Linnett SIC dipole moments are in better agreement with experimental values. We suggest using the dipole moment as a diagnostic of symmetry breaking in SIC and monitoring it in all SIC calculations. We show that Linnett structures can often be seen as superpositions of Lewis structures and propose Linnett structures as a simple way to describe aromatic systems in SIC with reduced symmetry breaking. The role of hovering FODs is also briefly discussed.

physics.comp-ph↗

arXiv2021

Direct Energy Minimization Based on Exponential Transformation in Density Functional Calculations of Finite and Extended Systems

The energy minimization involved in density functional calculations of electronic systems can be carried out using an exponential transformation that preserves the orthonormality of the orbitals. The energy of the system is then represented as a function of the elements of a skew-Hermitian matrix that can be optimized directly using unconstrained minimization methods. An implementation based on the limited memory Broyden-Fletcher-Goldfarb-Shanno approach with inexact line search and a preconditioner is presented and the performance compared with that of the commonly used self-consistent field approach. Results are presented for the G2 set of 148 molecules, liquid water configurations with up to 576 molecules and some insulating crystals. A general preconditioner is presented that is applicable to systems with fractional orbital occupation as is, for example, needed in the k-point sampling for periodic systems. This exponential transformation direct minimization approach is found to outperform the standard implementation of the self-consistent field approach in that all the calculations converge with the same set of parameter values and it requires less computational effort on average. The formulation of the exponential transformation and the gradients of the energy presented here are quite general and can be applied to energy functionals that are not unitary invariant such as self-interaction corrected functionals.

physics.comp-ph↗

arXiv2021

Method for Calculating Excited Electronic States Using Density Functionals and Direct Orbital Optimization with Real Space Grid or Plane Wave Basis Set

A direct orbital optimization method is presented for density functional calculations of excited electronic states using either a real space grid or a plane wave basis set. The method is variational, provides atomic forces in the excited states, and can be applied to Kohn-Sham (KS) functionals as well as orbital-density dependent functionals (ODD) including explicit self-interaction correction. The implementation for KS functionals involves two nested loops: (1) An inner loop for finding a stationary point in a subspace spanned by the occupied and a few virtual orbitals corresponding to the excited state; (2) an outer loop for minimizing the energy in a tangential direction in the space of the orbitals. For ODD functionals, a third loop is used to find the unitary transformation that minimizes the energy functional among occupied orbitals only. Combined with the maximum overlap method, the algorithm converges in challenging cases where conventional self-consistent field algorithms tend to fail. The benchmark tests presented include two charge-transfer excitations in nitrobenzene and an excitation of CO to degenerate $π^\ast$ orbitals where the importance of complex orbitals is illustrated. An application of the method to several metal-to-ligand charge-transfer and metal-centred excited states of an Fe$^{\rm II}$ photosensitizer complex is described and the results compared to reported experimental estimates. The method is also used to study the effect of Perdew-Zunger self-interaction correction on valence and Rydberg excited states of several molecules, both singlet and triplet states, and the performance compared to semilocal and hybrid functionals.

physics.comp-ph↗

arXiv2021

Mn Dimer can be Described Accurately with Density Functional Calculations when Self-interaction Correction is Applied

Qualitatively incorrect results are obtained for the Mn dimer in density functional theory calculations using the generalized gradient approximation (GGA) and similar results are obtained from local density and meta-GGA functionals. The coupling is predicted to be ferromagnetic rather than antiferromagnetic and the bond between the atoms is predicted to be an order of magnitude too strong and about an Ångstrøm too short. Explicit, self-interaction correction (SIC) applied to a commonly used GGA energy functional, however, provides close agreement with both experimental data and high-level, multi-reference wave function calculations. These results show that the failure is not due to strong correlation but rather the single electron self-interaction that is necessarily introduced in estimates of the classical Coulomb and exchange-correlation energy when only the total electron density is used as input. The corrected functional depends explicitly on the orbital densities and can, therefore, avoid the introduction of self-Coulomb interaction. The error arises because of over-stabilization of bonding $d$-states in the minority spin channel resulting from an overestimate of the $d$-electron self-interaction in the semi-local exchange-correlation functionals. Since the computational effort in the self-interaction corrected calculations scales with system size in the same way as for regular semi-local functional calculations, this approach provides a way to calculate properties of Mn nanoclusters as well as biomolecules and extended solids where Mn dimers and larger cluster are present, while multi-reference wave function calculations can only be applied to small systems.

physics.chem-ph↗

arXiv2020

Fast and Robust Algorithm for the Energy Minimization of Spin Systems Applied in an Analysis of High Temperature Spin Configurations in Terms of Skyrmion Density

An algorithm for the minimization of the energy of magnetic systems is presented and applied to the analysis of thermal configurations of a ferromagnet to identify inherent structures, i.e. the nearest local energy minima, as a function of temperature. Over a rather narrow temperature interval, skyrmions appear and reach a high temperature limit for the skyrmion density. In addition, the performance of the algorithm is further demonstrated in a self-consistent field calculation of a skyrmion in an itinerant magnet. The algorithm is based on a geometric approach in which the curvature of the spherical domain is taken into account and as a result the length of the magnetic moments is preserved in every iteration. In the limit of infinitesimal rotations, the minimization path coincides with that obtained using damped spin dynamics while the use of limited-memory quasi-newton minimization algorithms, such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm, significantly accelerates the convergence.

physics.comp-ph↗

arXiv2020

Efficient Optimization Method for Finding Minimum Energy Paths of Magnetic Transitions

Efficient algorithms for the calculation of minimum energy paths of magnetic transitions are implemented within the geodesic nudged elastic band (GNEB) approach. While an objective function is not available for GNEB and a traditional line search can, therefore, not be performed, the use of limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) and conjugate gradient algorithms in conjunction with orthogonal spin optimization (OSO) approach is shown to greatly outperform the previously used velocity projection and dissipative Landau-Lifschitz dynamics optimization methods. The implementation makes use of energy weighted springs for the distribution of the discretization points along the path and this is found to improve performance significantly. The various methods are applied to several test problems using a Heisenberg-type Hamiltonian, extended in some cases to include Dzyaloshinskii-Moriya and exchange interactions beyond nearest neighbours. Minimum energy paths are found for magnetization reversals in a nano-island, collapse of skyrmions in two-dimensional layers and annihilation of a chiral bobber near the surface of a three-dimensional magnet. The LBFGS-OSO method is found to outperform the dynamics based approaches by up to a factor of 8 in some cases.

physics.comp-ph↗

arXiv2020

Variational density functional calculations of excited states via direct optimization

The development of variational density functional theory approaches to excited electronic states is impeded by limitations of the commonly used self-consistent field (SCF) procedure. A method based on a direct optimization approach as well as the maximum overlap method is presented and the performance compared with previously proposed SCF strategies. Excited-state solutions correspond to saddle points of the energy as a function of the electronic degrees of freedom. The approach presented here makes use of a preconditioner determined with the help of the maximum overlap method to guide the convergence on a target nth-order saddle point. The method is found to be more robust and to converge faster than previously proposed SCF approaches for a set of 89 excited states of molecules. A limited-memory formulation of the symmetric rank-one method for updating the inverse Hessian is found to give the best performance. A conical intersection for the carbon monoxide molecule is calculated without resorting to fractional occupation numbers. Calculations on excited states of the hydrogen atom and a doubly excited state of the dihydrogen molecule using a self-interaction corrected functional are presented. For these systems, the self-interaction correction is found to improve the accuracy of density functional calculations of excited states.

physics.chem-ph↗

arXiv2019

Multiple minimum energy paths and scenarios of unwinding transitions in chiral nematic liquid crystals

We apply the minimum energy paths (MEPs) approach to study the helix unwinding transition in chiral nematic liquid crystals. A mechanism of the transition is determined by a MEP passing through a first order saddle point on the free energy surface. The energy difference between the saddle point and the initial state gives the energy barrier of the transition. Two starting approximations for the paths are used to find the MEPs representing different transition scenarios: (a) the director slippage approximation with in-plane helical structures; and (b) the anchoring breaking approximation that involves the structures with profound out-of-plane director deviations. It is shown that, at sufficiently low voltages, the unwinding transition is solely governed by the director slippage mechanism with the planar saddle point structures. When the applied voltage exceeds its critical value below the threshold of the Freedericksz transition, the additional scenario through the anchoring breaking transitions is found to come into play. For these transitions, the saddle point structure is characterized by out-of-plane deformations localized near the bounding surface. The energy barriers for different paths of transitions are computed as a function of the voltage and the anchoring energy strengths.

cond-mat.soft↗