arXiv subjects

Fabian M. Faulstich

Publications and source records attributed to Fabian M. Faulstich.

24 records · Page 2Linked to original sources

arXiv2020

Discontinuous Galerkin method with Voronoi partitioning for Quantum Simulation of Chemistry

Molecular orbitals based on the linear combination of Gaussian type orbitals are arguably the most employed discretization in quantum chemistry simulations, both on quantum and classical devices. To circumvent a potentially dense two-body interaction tensor and obtain lower asymptotic costs for quantum simulations of chemistry, the discontinuous Galerkin (DG) procedure using a rectangular partitioning strategy was recently piloted [McClean et al, New J. Phys. 22, 093015, 2020]. The DG approach interpolates in a controllable way between a compact description of the two-body interaction tensor through molecular orbitals and a diagonal characterization through primitive basis sets, such as a planewave dual basis set. The DG procedure gives rise to a block-diagonal representation of the two-body interaction with reduced number of two-electron repulsion integrals, which in turn reduces the cost of quantum simulations. In the present work we extend this approach to be applicable to molecular and crystalline systems of arbitrary geometry. We take advantage of the flexibility of the planewave dual basis set, and combine the discontinuous Galerkin procedure with a general partitioning strategy based on the Voronoi decomposition. We numerically investigate the performance, at the mean-field and correlated levels, with quasi-1D, 2D and 3D partitions using hydrogen chains, H$_4$, CH$_4$ as examples, respectively. We also apply the method to graphene as a prototypical example of crystalline systems.

physics.comp-ph↗

arXiv2019

Analysis of The Tailored Coupled-Cluster Method in Quantum Chemistry

In quantum chemistry, one of the most important challenges is the static correlation problem when solving the electronic Schrödinger equation for molecules in the Born--Oppenheimer approximation. In this article, we analyze the tailored coupled-cluster method (TCC), one particular and promising method for treating molecular electronic-structure problems with static correlation. The TCC method combines the single-reference coupled-cluster (CC) approach with an approximate reference calculation in a subspace [complete active space (CAS)] of the considered Hilbert space that covers the static correlation. A one-particle spectral gap assumption is introduced, separating the CAS from the remaining Hilbert space. This replaces the nonexisting or nearly nonexisting gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital usually encountered in standard single-reference quantum chemistry. The analysis covers, in particular, CC methods tailored by tensor-network states (TNS-TCC methods). The problem is formulated in a nonlinear functional analysis framework, and, under certain conditions such as the aforementioned gap, local uniqueness and existence are proved using Zarantonello's lemma. From the Aubin--Nitsche-duality method, a quadratic error bound valid for TNS-TCC methods is derived, e.g., for linear-tensor-network TCC schemes using the density matrix renormalization group method.

math.NA↗

arXiv2019

Discontinuous Galerkin discretization for quantum simulation of chemistry

Methods for electronic structure based on Gaussian and molecular orbital discretizations offer a well established, compact representation that forms much of the foundation of correlated quantum chemistry calculations on both classical and quantum computers. Despite their ability to describe essential physics with relatively few basis functions, these representations can suffer from a quartic growth of the number of integrals. Recent results have shown that, for some quantum and classical algorithms, moving to representations with diagonal two-body operators can result in dramatically lower asymptotic costs, even if the number of functions required increases significantly. We introduce a way to interpolate between the two regimes in a systematic and controllable manner, such that the number of functions is minimized while maintaining a block diagonal structure of the two-body operator and desirable properties of an original, primitive basis. Techniques are analyzed for leveraging the structure of this new representation on quantum computers. Empirical results for hydrogen chains suggest a scaling improvement from $O(N^{4.5})$ in molecular orbital representations to $O(N^{2.6})$ in our representation for quantum evolution in a fault-tolerant setting, and exhibit a constant factor crossover at 15 to 20 atoms. Moreover, we test these methods using modern density matrix renormalization group methods classically, and achieve excellent accuracy with respect to the complete basis set limit with a speedup of 1-2 orders of magnitude with respect to using the primitive or Gaussian basis sets alone. These results suggest our representation provides significant cost reductions while maintaining accuracy relative to molecular orbital or strictly diagonal approaches for modest-sized systems in both classical and quantum computation for correlated systems.

quant-ph↗

arXiv2018

The Coupled-Cluster Formalism - a Mathematical Perspective

The Coupled-Cluster theory is one of the most successful high precision methods used to solve the stationary Schrödinger equation. In this article, we address the mathematical foundation of this theory with focus on the advances made in the past decade. Rather than solely relying on spectral gap assumptions (non-degeneracy of the ground state), we highlight the importance of coercivity assumptions - Gårding type inequalities - for the local uniqueness of the Coupled-Cluster solution. Based on local strong monotonicity, different sufficient conditions for a local unique solution are suggested. One of the criteria assumes the relative smallness of the total cluster amplitudes (after possibly removing the single amplitudes) compared to the Gårding constants. In the extended Coupled-Cluster theory the Lagrange multipliers are wave function parameters and, by means of the bivariational principle, we here derive a connection between the exact cluster amplitudes and the Lagrange multipliers. This relation might prove useful when determining the quality of a Coupled-Cluster solution. Furthermore, the use of an Aubin-Nitsche duality type method in different Coupled-Cluster approaches is discussed and contrasted with the bivariational principle.

physics.chem-ph↗

arXiv2018

Numerical and Theoretical Aspects of the DMRG-TCC Method Exemplified by the Nitrogen Dimer

In this article, we investigate the numerical and theoretical aspects of the coupled-cluster method tailored by matrix-product states. We investigate chemical properties of the used method, such as energy size extensivity and the equivalence of linked and unlinked formulation. The existing mathematical analysis is here elaborated in a quantum chemical framework. In particular, we highlight the use of a so-called CAS-ext gap describing the basis splitting between the complete active space and the external part. Moreover, the behavior of the energy error as a function of the optimal basis splitting is discussed. We show numerical investigations on the robustness with respect to the bond dimensions of the single orbital entropy and the mutual information, which are quantities that are used to choose the complete active space. Furthermore, we extend the mathematical analysis with a numerical study on the complete active space dependence of the error.

physics.chem-ph↗

arXiv2017

Unraveling Mirror Properties in Time-Delayed Quantum Feedback Scenarios

We derive in the Heisenberg picture a widely used phenomenological coupling element to treat feedback effects in quantum optical platforms. Our derivation is based on a microscopic Hamiltonian, which describes the mirror-emitter dynamics based on a dielectric, a mediating fully quantized electromagnetic field, and a single two-level system in front of the dielectric. The dielectric is modeled as a a system of identical two-state atoms. The Heisenberg equation yields a system of describing differential operator equations, which we solve in the Weisskopf-Wigner limit. Due to a finite round-trip time between emitter and dielectric, we yield delay differential operator equations. Our derivation motivates and justifies the typical phenomenological assumed coupling element and allows, furthermore, a generalization to a variety of mirrors, such as dissipative mirrors or mirrors with gain dynamics.

quant-ph↗