The method of moments of an absorption line is generalized to allow for a continuous level scheme. The range of problems which can be treated by this generalized method partially overlaps with those encountered by traditional perturbative methods. A test case consisting of an impurity ion that relaxes to conduction-electron bath is studied, and the compatibility of this method with other perturbative methods is demonstrated. It is also argued that this method is applicable to problems where density-matrix expansions are not valid. Further possible applications are discussed.
The Kernel Energy Method (KEM) delivers accurate full molecule energies using less computational resources than standard ab-initio quantum chemical approaches. KEM achieves this efficiency by decomposing a system of atoms into disjoint subsets called kernels. The results of full ab-initio calculations on each individual single kernel and on each double kernel formed by the union of each pair of single kernels are combined in an equation of a form that is specific to KEM to provide an approximation to the full molecule energy. KEM has been demonstrated to give accurate molecular energies over a wide range of systems, chemical methods and basis sets. The efficiency of KEM makes calculations on large systems tractable. It has been shown to be accurate for calculations on large biomolecules including proteins and DNA. KEM has recently been generalized to provide the one-body density matrix for the full molecule. This generalization greatly expands the utility of the method since the density matrix enables simple calculation of the expectation value of any observable associated with a one-body operator. More importantly, the generalization enables results from KEM to be constrained to satisfy essential quantum mechanical conditions. KEM is generalized to density matrices by taking the density matrices obtained from the single and double kernels and adapting them to the full molecule basis. These augmented kernel density matrices are summed according to the standard KEM expansion. This initial matrix does not necessarily correspond to a valid quantum mechanical N-electron wavefunction which is normalized and obeys the Pauli principle. Such a density matrix is called N-representable. N-representability is imposed on the initial matrix by using the Clinton equations, which deliver a single-determinant N-representable one-body density matrix. Single-determinant N-representability is ensured by enforcing the density matrix to be a normalized projector. Such normalized projectors are factorizable into matrices which deliver full molecule molecular orbitals. Although energies have been demonstrated to be accurate in the original energy expansion form of KEM, there is no requirement that the energy corresponds to the expectation value of a valid quantum mechanical wavefunction. Because of this, there is no requirement that the energies satisfy the variational theorem. Imposing single-determinant N-representability on the KEM density matrix by using the Clinton equations recovers the variational bound on the energy obtained from KEM. Recovery of the variational bound by the N-representable KEM density matrix has been demonstrated by calculations on several water clusters. Energies from full molecule restricted Hartree-Fock calculations were compared to energies obtained from the KEM energy expansion and to energies obtained from the KEM density matrix expansion which had N-representability imposed. In each case where the energy expansion gave energies that violated the variational bound the N-representable density matrix gave energies that satisfied the bound and in fact were more accurate than the simple energy expansion results. The effect of imposing N-representability on the KEM density matrix has also been investigated in the context of Kohn-Sham Density Functional Theory (KS/DFT). Calculations on a simple noble gas system and a single water cluster were done. KS/DFT energies were compared to energies from the KEM energy expansion and energies associated with the KEM density matrix expansion on which N-representability was imposed. The energies from the KEM density matrix expansion made N-representable were more accurate than those obtained from straightforward KEM energy expansions in nearly all cases for the noble gas system. For the water cluster the accuracy of the energy obtained from the N-representable KEM density matrix was nearly four times closer to the energy calculated for the full system than the KEM energy expansion result. The Clinton-purified N-representable matrix obtained from the KEM density expansion can be used to calculate expectation values for any one-electron operator. In particular, a KEM density matrix approach can be used in the study of quantum crystallography.
In this paper, we study decoherence in Grover's quantum search algorithm using a perturbative method. We assume that each two-state system (qubit) that belongs to a register suffers a phase-flip error (${\ensuremath{\sigma}}_{z}$ error) with probability
p independently at every step in the algorithm, where $0\ensuremath{\leqslant}p\ensuremath{\leqslant}1$. Considering an
n-qubit density operator to which Grover's iterative operation is applied
M times, we expand it in powers of
2Mnp and derive its matrix element order by order under the large-
n limit. [In this large-
n limit, we assume
p is small enough, so that
2Mnp can take any real positive value or zero. We regard $x\ensuremath{\equiv}2Mnp$ $(\ensuremath{\geqslant}0)$ as a perturbative parameter.] We obtain recurrence relations between terms in the perturbative expansion. By these relations, we compute higher orders of the perturbation efficiently, so that we extend the range of the perturbative parameter that provides a reliable analysis. Calculating the matrix element numerically by this method, we derive the maximum value of the perturbative parameter
x at which the algorithm finds a correct item with a given threshold of probability
Pth or more. (We refer to this maximum value of
x as
xc, a critical point of
x.) We obtain a curve of
xc as a function of
Pth by repeating this numerical calculation for many points of
Pth and find the following facts: a tangent of the obtained curve at
Pth=1 is given by $x=(8∕5)(1\ensuremath{-}{P}_{\mathrm{th}})$, and we have ${x}_{\mathrm{c}}>\ensuremath{-}(8∕5){\mathrm{log}}_{e}\phantom{\rule{0.2em}{0ex}}{P}_{\mathrm{th}}$ near
Pth=0.