From Wavefunction Regularity to Eigenvector Conditioning: Accuracy--Cost Trade-offs of Quantum Algorithms for Non-Hermitian Transcorrelated Hamiltonians

arXiv (Cornell University)

Abstract

The electron--electron Coulomb singularity forces cusp structure in electronic wavefunctions. This cusp structure limits wavefunction regularity and slows the convergence of expansions in smooth bases, requiring large basis sets to reach a target accuracy. The transcorrelated (TC) method incorporates this short-range behavior into the Hamiltonian, thereby improving the regularity of the transformed wavefunction and accelerating convergence toward the complete-basis-set limit. The resulting Hamiltonian is non-Hermitian and non-normal, which limits the direct applicability of standard quantum algorithms. For such non-normal operators, the complexity of several recent quantum algorithms depends explicitly on quantities related to eigenvector conditioning and eigenvalue sensitivity. Understanding these quantities is essential for assessing both algorithmic complexity and quantum resource requirements. In this work, we address these issues in two stages. We first study a solvable model, which provides a controlled setting for isolating how transcorrelation affects basis convergence and eigenvector conditioning. Within this setting, we investigate two distinct one-parameter Jastrow correlators. We derive the asymptotic energy-error scaling: $N^{-1/2}$ for the bare projection and $N^{-3/2}$ for both TC correlators at generic fixed parameters, with $N$denoting the basis size. Furthermore, we show that this faster convergence can yield a polynomial improvement in the asymptotic query-complexity upper bound. We then extend the analysis to second-quantized electronic-structure Hamiltonians and use particle-number-sector diagnostics to distinguish target-sector from auxiliary-sector conditioning. Our work provides a framework for understanding the central accuracy--cost trade-off in TC Hamiltonians.

Authors 6

1. Cheng-Lin Hong

   Helmholtz-Zentrum Dresden-Rossendorf · Center for Advanced Systems Understanding

   Affiliation as printed

   Center for Advanced Systems Understanding (CASUS) , 02826 Görlitz , Germany

   Helmholtz-Zentrum Dresden-Rossendorf (HZDR) , 01328 Dresden , Germany

2. Emiel Koridon Aachen $⟨aQa^L⟩$ Applied Quantum Algorithms

   Leiden University · Covestro (Germany)

   Affiliation as printed

   Covestro Deutschland AG , Leverkusen , Nordrhein-Westfalen 51373 , Germany

   ⟨aQa L ⟩ Applied Quantum Algorithms , Leiden University , the Netherlands

3. Paul K. Faehrmann

   Freie Universität Berlin

   Affiliation as printed

   Dahlem Center for Complex Quantum Systems , Freie Universität Berlin , 14195 Berlin , Germany

   Walrus Computing , Munich , Germany

4. Thomas D. Kühne
5. Stefano Polla Aachen $⟨aQa^L⟩$ Applied Quantum Algorithms

   Leiden University · QuSoft · University of Amsterdam

   Affiliation as printed

   QuSoft, HIMS & IvI , University of Amsterdam , the Netherlands

   ⟨aQa L ⟩ Applied Quantum Algorithms , Leiden University , the Netherlands

6. Werner Dobrautz