Second Quantization-Based Methods in Quantum Chemistry

Noisy intermediate-scale quantum algorithms

“…There are a number of well-known mappings that allow the description of a fermionic system by the standard model of QC. They are the Jordan-Wigner transformation (Jordan and Wigner, 1928), Bravyi-Kitaev transformation (Bravyi and Kitaev, 2002) and Ball-Verstraete-Cirac transformation (Ball, 2005;Verstraete and Cirac, 2005). More advanced mappings using the interaction graph of the Hamiltonian (Setia and Whitfield, 2018) or customized quasi-local encodings (Chien and Whitfield, 2020) have been introduced as well.…”

“…We refer to (Herbst, 2018;Kottmann, 2018;Rohwedder, 2010) and the textbook (Yserentant, 2010) for the direct construction and discretization of this continous Hilbert spaces. A more compact, but formally equivalent, definition is offered through second quantization by introducing the abstract anti-commuting field operators ψ † (x) and ψ (x) that create and annihilate electrons at spin-coordinate x k ∈ R 3 × ± 1 2 (Jordan and Klein, 1927;Jørgensen, 2012;Surján, 2012). The electronic Hamiltonian can then be written as…”

Benchmarking the Variational Reduced Density Matrix Theory in the Doubly Occupied Configuration Interaction Space with Integrable Pairing Models

“…In second quantization, an N -particle Hamiltonian with pairwise interactions can be written as where t and V are the one-body energy and the two-body interaction terms, respectively. c i † and c j are the standard fermion creation and annihilation operators in a given orthonormal single-particle basis { i , j , k , l , ...}.…”

A Quantum Computing View on Unitary Coupled Cluster Theory

“…Throughout this review, we assume the reader is familiar with second quantisation [28][29][30][31], the Hartree-Fock (HF) method [31,32], and passing knowledge of the Full Configuration Interaction (FCI) [31,32] method for single-reference states in quantum chemistry. Additionally, we assume the reader is at least at the level of the well-versed beginner in their understanding of quantum computation, perhaps at the level of a beginning graduate student or an advanced undergraduate [33].…”

“…An advantage with variational over projected methods are they furnish an upper bound to the FCI energy, and of that reason we expect better agreement with FCI results. The deviation from the variational and projected energy, however, may be small and of little practical consequence -often, this has been observed in the weakly correlated regime [28]. If the single-reference state recovers mostly of the ground state then the CC methods would differ only slightly.…”