Slater-Condon rules are at the heart of any quantum chemistry method as they allow to simplify
3N-dimensional integrals as sums of 3- or 6-dimensional integrals. In this paper, we propose an efficient implementation of those rules in order to identify very rapidly which integrals are involved in a matrix element expressed in the determinant basis set. This implementation takes advantage of the bit manipulation instructions on x86 architectures that were introduced in 2008 with the SSE4.2 instruction set. Finding which spin-orbitals are involved in the calculation of a matrix element doesn't depend on the number of electrons of the system.
We demonstrated in the previous chapter (Chapter 1) that the evaluation of matrix elements between states represented by the Slater determinant can be evaluated unambiguously using Slater’s rule [1, 28, 40]. While the determinantal representation and the evaluation matrix elements between the states are quite straightforward, this procedure can eventually be ineffective and cumbersome with increasing size of the system. The occupation number formalism or second quantization technique introduced by Dirac in quantum field theory, in this context has been found to be quite formidable and elegant. This formalism not only allows us to represent the states and the operators in a compact and convenient form but also enables us to deal with systems containing variable numbers of particles without explicitly specifying the number of particles. It implicitly assumes the existence of an unspecified number of functions in a single-particle basis. These one-particle functions, which constitute the Hilbert space for N particles, are generally chosen to be spinorbital because of their orthogonality. In the case of fermions, the many-particle basis is the anti-symmetric product of N one-particle basis functions (Slater determinant). The vector space consisting of all possible anti-symmetric Hilbert spaces is referred to as the Fock space, in which the operators and functions are represented without specifying the number of particles.
Learning compact, interpretable descriptions of quantum many-body states is an important task in quantum science. We study the task of learning the Slater determinant with maximum fidelity to an arbitrary fermionic many-body state, with motivation from both Hartree-Fock methods and agnostic tomography. Given an
n-fermion wavefunction built from
m fermionic modes, we provide classical and quantum algorithms returning a Slater determinant with fidelity within
ε of maximal in time
mpoly(n,1/ε). We prove matching hardness lower bounds, assuming standard complexity conjectures, along some parameter axes. Given access to quantum copies, we prove this can be accomplished with
poly(m,n,1/ε) copies of
ρ. We also show that above a fidelity of
2/3 any stationary point is the unique global maximum while below
2/3 the optimization landscape can have spurious stationary points, and hence
2/3 marks a transition point in the optimization landscape for this problem. We apply the algorithm to the Fermi-Hubbard model, extracting the closest Slater determinant from neural quantum state solutions. Together, our results provide algorithmic tools with provable guarantees in understanding fermionic many-body systems with classical or quantum simulation.
The matrix element of a general many-body operator between two Slater determinants is calculated explicitly. For this, a split-and-pair method is introduced that provides a convenient expression of Wick's theorem and simplifies many-body calculations. The same method is used to determine the generating function of the matrix elements of many-body operators. The split-and-pair method allows also for the diagonalization of the density correlation operators $:n({x}_{1})\ensuremath{\cdots}n({x}_{k}):$, where $n(x)={\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(x)\ensuremath{\psi}(x)$ is the density operator. The relation between the split-and-pair method and quantum group theory is clarified.
Slater determinants are product states of filled quantum fermionic orbitals. When they are expressed in a configuration space basis chosen a priori, their entanglement is bound and controlled. This suggests that an exact representation of Slater determinants as finitely-correlated states is possible. In this paper we analyze this issue and provide an exact Matrix Product representation for Slater determinant states. We also argue possible meaningful extensions that embed more complex configuration interaction states into the description.
Slater determinants are product states of filled quantum fermionic orbitals. When they are expressed in a configuration space basis chosen a priori, their entanglement is bound and controlled. This suggests that an exact representation of Slater determinants as finitely-correlated states is possible. In this paper we analyze this issue and provide an exact Matrix Product representation for Slater determinant states. We also argue possible meaningful extensions that embed more complex configuration interaction states into the description.