Extension of the Jordan-Wigner mapping to nonorthogonal spin orbitals for quantum computing application to valence bond approaches

TL;DR

This work tackles the challenge of encoding nonorthogonal spin orbitals for quantum computing of valence-bond–type wavefunctions. It introduces a Jordan-Wigner–like mapping adapted to nonorthogonal orbitals, including a biorthogonal variant to control Pauli-string growth, and implements the approach in PennyLane. Validation against Löwdin rules for nonorthogonal Slater determinants demonstrates exact agreement, and preliminary VB-like H$_4$ tests yield chemically meaningful coefficients and weights. The results establish a feasible path toward VB-inspired quantum algorithms on near-term devices, pending development of an interpretable and efficient ansatz for larger systems.

Abstract

Quantum computing offers a promising platform to address the computational challenges inherent in quantum chemistry, and particularly in valence bond (VB) methods, which are chemically appealing but suffer from high computational cost due to the use of nonorthogonal orbitals. While various fermionic-to-spin mappings exist for orthonormal spin orbitals, such as the widely used Jordan-Wigner transformations, an analogous framework for nonorthogonal spin orbitals remains undeveloped. In this work, we propose an alternative Jordan-Wigner-type mapping tailored for the nonorthogonal case, with the goal of enabling efficient quantum simulations of VB-type wavefunctions. Our approach paves the way towards the development of chemically interpretable and computationally feasible valence bond algorithms on near-term quantum devices. An initial theoretical analysis and a preliminary application demonstrate the feasibility of this encoding and its potential for extending the applicability of VB methods to larger and more complex systems.

Figures (1)

• Figure 1: H$_4$ and its possible spin-coupling modes in the singlet state, with blue lines indicating the spin-couplings between two electrons. The $R_1$ and $R_2$ parameters for the definition of the adopted geometry are also indicated.