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Register a new userVariational approach for the electronic structure calculation on the second-order reduced density matrices and the $N$-representability problem
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The reduced-density-matrix method is an promising candidate for the next generation electronic structure calculation method; it is equivalent to solve the Schrodinger equation for the ground state. The number of variables is the same as a four electron system and constant regardless of the electrons in the system. Thus many researchers have been dreaming of a much simpler method for quantum mechanics. In this chapter, we give a overview of the reduced-density matrix method; details of the theories, methods, history, and some new computational results. Typically, the results are comparable to the CCSD(T) which is a sophisticated traditional approach in quantum chemistry.
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The second-order reduced density matrix method (the RDM method) has performed well in determining energies and properties of atomic and molecular systems, achieving coupled-cluster singles and doubles with perturbative triples (CC SD(T)) accuracy without using the wave-function. One question that arises is how well does the RDM method perform with the same conditions that result in CCSD(T) accuracy in the strong correlation limit. The simplest and a theoretically important model for strongly correlated electronic systems is the Hubbard model. In this paper, we establish the utility of the RDM method when employing the $P$, $Q$, $G$, $T1$ and $T2^prime$ conditions in the two-dimension al Hubbard model case and we conduct a thorough study applying the $4times 4$ Hubbard model employing a coefficients. Within the Hubbard Hamilt onian we found that even in the intermediate setting, where $U/t$ is between 4 and 10, the $P$, $Q$, $G$, $T1$ and $T2^prime$ conditions re produced good ground state energies.
Variational calculation of the ground state energy and its properties using the second-order reduced density matrix (2-RDM) is a promising approach for quantum chemistry. A major obstacle with this approach is that the $N$-representability conditions are too difficult in general. Therefore, we usually employ some approximations such as the $P$, $Q$, $G$, $T1$ and $T2^prime$ conditions, for realistic calculations. The results of using these approximations and conditions in 2-RDM are comparable to those of CCSD(T). However, these conditions do not incorporate an important property; size-consistency. Size-consistency requires that energies $E(A)$, $E(B)$ and $E(A...B)$ for two infinitely separated systems $A$, $B$, and their respective combined system $A...B$, to satisfy $E(A...B) = E(A) + E(B)$. In this study, we show that the size-consistency can be satisfied if 2-RDM satisfies the following conditions: (i) 2-RDM is unitary invariant diagonal $N$-representable; (ii) 2-RDM corresponding to each subsystem is the eigenstate of the number of corresponding electrons; and (iii) 2-RDM satisfies at least one of the $ P$, $Q$, $G$, $T1$ and $T2^prime$ conditions.
By combining a parameterized Hermitian matrix, the realignment matrix of the bipartite density matrix $rho$ and the vectorization of its reduced density matrices, we present a family of separability criteria, which are stronger than the computable cross norm or realignment (CCNR) criterion. With linear contraction methods, the proposed criteria can be used to detect the multipartite entangled states that are biseparable under any bipartite partitions. Moreover, we show by examples that the presented multipartite separability criteria can be more efficient than the corresponding multipartite realignment criterion based on CCNR, multipartite correlation tensor criterion and multipartite covariance matrix criterion.
We present a method for the calculation of photoemission spectra in terms of reduced density matrices. We start from the spectral representation of the one-body Greens function G, whose imaginary part is related to photoemission spectra, and we introduce a frequency-dependent effective energy that accounts for all the poles of G. Simple approximations to this effective energy give accurate spectra in model systems in the weak as well as strong correlation regime. In real systems reduced density matrices can be obtained from reduced density-matrix functional theory. Here we use this approach to calculate the photoemission spectrum of bulk NiO: our method yields a qualitatively correct picture both in the antiferromagnetic and paramagnetic phases, contrary to mean-field methods, in which the paramagnet is a metal.
Recently, an adaptive variational algorithm termed Adaptive Derivative-Assembled Pseudo-Trotter ansatz Variational Quantum Eigensolver (ADAPT-VQE) has been proposed by Grimsley et al. (Nat. Commun. 10, 3007) while the number of measurements required to perform this algorithm scales O(N^8). In this work, we present an efficient adaptive variational quantum solver of the Schrodinger equation based on ADAPT-VQE together with the reduced density matrix reconstruction approach, which reduces the number of measurements from O(N^8) to O(N^4). This new algorithm is quite suitable for quantum simulations of chemical systems on near-term noisy intermediate-scale hardware due to low circuit complexity and reduced measurement. Numerical benchmark calculations for small molecules demonstrate that this new algorithm provides an accurate description of the ground-state potential energy curves. In addition, we generalize this new algorithm for excited states with the variational quantum deflation approach and achieve the same accuracy as ground-state simulations.
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