Spin-adapted Unitary Transformations Achieve Exact Factorization Via Fermionic Generators and Lie Algebras

Maintaining spin symmetry represents a crucial challenge in generating realistic electronic wavefunctions, yet implementing these transformations on quantum hardware proves difficult due to the complex nature of the underlying fermionic operators. Paarth Jain, Artur F. Izmaylov, and Erik R. Kjellgren, from the University of Toronto and University of Southern Denmark, now present a computationally efficient method for exactly factorizing these spin-adapted transformations, expressing them as ordered products of Pauli operator exponentials. Their approach leverages the small Lie algebra structure inherent in the elementary operators, reformulating the factorization as a low-dimensional optimisation problem, and avoiding complex symbolic calculations. This breakthrough provides a practical strategy for constructing symmetry-conserving quantum circuits, reducing computational cost and ensuring accurate representation of electronic states in molecular simulations.

The research addresses the challenge of implementing transformations on quantum hardware where fermionic generators translate into noncommuting operators, traditionally requiring approximate decompositions. Instead, the team harnessed the properties of Lie algebras, recognizing that double excitation and deexcitation fermionic operators form compact structures, enabling a precise and efficient factorization scheme. This method involves rewriting the exponential of a sum of elementary fermionic generators as an ordered product of exponentials of the algebra’s basis elements, each directly implementable as a quantum circuit.

This approach circumvents complex symbolic manipulations by reformulating the problem as a low-dimensional nonlinear optimization over matrix exponentials within the adjoint representation of the Lie algebra. The number of basis elements within these relevant algebras remains modest, fewer than 150, allowing for efficient numerical factorization. Researchers analyzed the structure of these Lie algebras across different seniority sectors to optimize the factorization process. This numerical treatment achieves high precision while avoiding the computational burden of symbolic complexity, offering a significant advancement over traditional methods. The team demonstrated the factorization for representative spin-adapted double excitation and deexcitation transformations, validating the approach and establishing its potential for constructing compact, symmetry-preserving variational ansätze in quantum chemistry.

Spin-Adapted Generators Reduce VQE Parameters Significantly

Scientists achieved a significant reduction in the number of variational parameters required for accurate convergence in adaptive Variational Quantum Eigensolver (VQE) calculations by implementing spin-adapted fermionic generators. These generators, designed to preserve spin symmetry, demonstrably improve the efficiency of quantum simulations of molecular systems. For the H2O molecule, using the spin-adapted fermionic operator pool required only 54 variational parameters to reach a gradient threshold of 10−5 atomic units, a substantial decrease from the 104 parameters needed with a conventional fermionic operator pool. A similar trend was observed for BeH2, where the spin-adapted approach converged with 55 parameters compared to the 95 parameters required by the standard method.

These results confirm that incorporating total spin symmetry systematically reduces the dimensionality of the variational space needed for accurate convergence. The team demonstrated that the spin-adapted fermionic ansatz effectively spans the configuration-state function (CSF) space, leveraging the fact that the nonrelativistic, spin-free electronic Hamiltonian commutes with the total spin operator. Further investigation into the O2 molecule in its triplet configuration revealed a critical advantage of the spin-adapted approach: it prevents variational collapse to lower-energy singlet states, a common issue with conventional fermionic operators that do not conserve total spin. Specifically, the adaptive VQE calculations for O2, starting from a triplet reference, followed the full configuration interaction (FCI) triplet energy curve across all bond lengths when using the spin-adapted fermionic operator pool. In contrast, the conventional fermionic operator pool led to a collapse to the lower-energy singlet solution, highlighting the ability of spin-adapted generators to maintain spin purity and accurately represent electronic states.

Symmetry Conserving Circuit Factorization via Optimization

This research introduces a new framework for factorizing spin-adapted unitary transformations into ordered products of exponentials, enabling their implementation within compact Lie algebras of symmetrized excitations. By working within the adjoint representation of these algebras, the team reformulated the factorization problem as a low-dimensional nonlinear optimization, achieving numerically exact results without relying on complex symbolic manipulations. This method inherently preserves total spin and particle-number symmetries, offering a significant advantage over approaches that attempt to enforce symmetry through penalty terms or additional qubits. The findings demonstrate a practical strategy for constructing symmetry-conserving circuits, reducing the number of variational parameters required in benchmark calculations while maintaining exact spin symmetry throughout the optimization process. This advancement establishes a general pathway for translating symmetry-adapted unitary transformations into implementable quantum circuits, offering potential for designing new quantum algorithms with greater expressivity and reduced circuit depth. The adjoint-based formulation provides a systematic link between Lie-algebraic structure and variational algorithm design, offering a principled strategy for constructing efficient, symmetry-preserving transformations for quantum simulation.