Inversion symmetry of singular values and a new orbital ordering method\n in tensor train approximations for quantum chemistry

2020/02/18 by Mi-Song Dupuy, Gero Friesecke, Dupuy, Mi-Song +1
Computer Science · Mathematics · Physics and Astronomy · #Bounded function #Density matrix renormalization group #FOS: Mathematics #FOS: Physical sciences #Geometry #Mathematical analysis #Mathematical physics #Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Physics #Quantum #Quantum many-body systems #Quantum mechanics #Renormalization group #Singular value #Strongly Correlated Electrons (cond-mat.str-el) #Superposition principle #Tensor (intrinsic definition) #Tensor decomposition and applications #Wave function #cond-mat.str-el #cs.NA #math.NA

paper · pdf · doi:10.48550/arxiv.2002.07459

arxiv created 2020/02/18 · openalex publication_date 2020/02/18 · arxiv updated 2020/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

The tensor train approximation of electronic wave functions lies at the core\nof the QC-DMRG (Quantum Chemistry Density Matrix Renormalization Group) method,\na recent state-of-the-art method for numerically solving the N-electron\nSchr "odinger equation. It is well known that the accuracy of TT approximations\nis governed by the tail of the associated singular values, which in turn\nstrongly depends on the ordering of the one-body basis.\n Here we find that the singular values s₁\≥ s₂\≥ ... \≥ s_d of tensors\nrepresenting ground states of noninteracting Hamiltonians possess a surprising\ninversion symmetry, s₁s_d=s₂s_d-1=s₃s_d-2=..., thus reducing the tail\nbehaviour to a single hidden invariant, which moreover depends explicitly on\nthe ordering of the basis. For correlated wavefunctions, we find that the tail\nis upper bounded by a suitable superposition of the invariants. Optimizing the\ninvariants or their superposition thus provides a new ordering scheme for\nQC-DMRG. Numerical tests on simple examples, i.e. linear combinations of a few\nSlater determinants, show that the new scheme reduces the tail of the singular\nvalues by several orders of magnitudes over existing methods, including the\nwidely used Fiedler order.\n

Citations

• Ab initio quantum chemistry using the density matrix renormalization group
• Density-matrix algorithms for quantum renormalization groups
• Equivalence of the variational matrix product method and the density matrix renormalization group applied to spin chains
• A tutorial on spectral clustering
• Improving Vocabulary Knowledge and Reading Attitudes in 4th Grade Students Through Direct Vocabulary Instruction.

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