Incorrect coefficients in Jordan-Wigner Transform

[kvmto]

Description

There is a discrepancy in the Pauli string coefficients between OpenFermion and CUDA-QX.

• The Pauli strings themselves match in both OpenFermion and CUDA-QX for all molecules.
• The coefficients match for H_2 but not for other molecules. For example, mismatches are observed for H_4.
• Some coefficients differ only by sign, suggesting a possible parity flip or phase shift issue.

Steps to Reproduce

1. Setup & Imports

   • Import necessary libraries: CUDA-Q, CUDA-QX, SciPy, PySCF, and NumPy.
   • Define the molecular structure for H_4 using Cartesian coordinates.

2. Compute Molecular Energies

   • FCI Energy (PySCF): Compute exact energy using Full Configuration Interaction (FCI).
   • OpenFermion Energy (CUDA-Q): Generate Hamiltonian, compute lowest eigenvalue.
   • CUDA-QX Energy: Generate Hamiltonian using CASCI, compute lowest eigenvalue.

3. Extract & Compare Pauli Terms

   • Convert Hamiltonians into Pauli string representations with coefficients.
   • Compare OpenFermion and CUDA-QX coefficients.
   • Print mismatches where coefficients differ beyond a tolerance (1e-15).

Purpose: Identify discrepancies between OpenFermion and CUDA-QX Hamiltonians, specifically in Pauli string coefficients.

Expected Behavior

Pauli string coefficients should match between OpenFermion and CUDA-QX for all molecules.

Actual Behavior

For molecules other than H_2, some coefficients do not match. Even when using PySCF-verified integrals, CUDA-QX produces incorrect eigenvalue energies, suggesting the issue lies in the Jordan-Wigner implementation in CUDA-QX.