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Here we present an efficient and numerically stable procedure for compressinga correlation matrix into a set of local unitary single-particle gates, whichleads to a very efficient way of forming the matrix product state (MPS)approximation of a pure fermionic Gaussian state, such as the ground state of aquadratic Hamiltonian. The procedure involves successively diagonalizingsubblocks of the correlation matrix to isolate local states which are purelyoccupied or unoccupied. A small number of nearest neighbor unitary gatesisolates each local state. The MPS of this state is formed by applying themany-body version of these gates to a product state. We treat the simple caseof compressing the correlation matrix of spinless free fermions with definiteparticle number in detail, though the procedure is easily extended to fermionswith spin and more general BCS states (utilizing the formalism of Majoranamodes). We also present a DMRG-like algorithm to obtain the compressedcorrelation matrix directly from a hopping Hamiltonian. In addition, we discussa slight variation of the procedure which leads to a simple construction of themultiscale entanglement renormalization ansatz (MERA) of a fermionic Gaussianstate, and present a simple picture of orthogonal wavelet transforms in termsof the gate structure we present in this paper. As a simple demonstration weanalyze the Su-Schrieffer-Heeger model (free fermions on a 1D lattice withstaggered hopping amplitudes).

Compression of Correlation Matrices and an Efficient Method for Forming Matrix Product States of Fermionic Gaussian States