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Exact Factorization of Unitary Transformations with Spin-Adapted Generators

Paarth Jain, Artur F. Izmaylov, Erik R. Kjellgren

TL;DR

The paper tackles the challenge of preserving spin symmetry in variational quantum algorithms by introducing an exact factorization of spin-adapted unitaries into ordered exponentials of Pauli operators, using compact Lie algebras and the adjoint representation to avoid symbolic BCH expansions. This turns the reparametrization into a low-dimensional nonlinear optimization, yielding precise circuit decompositions with central elements treated as commuting factors. The approach reduces variational parameter counts by roughly a factor of two in adaptive VQE tests (e.g., H$_2$O and BeH$_2$) while maintaining spin quantum numbers, and it prevents spin contamination in spin-sensitive cases like O$_2$ triplet, outperforming penalty or projection methods. Overall, the framework links Lie-algebraic structure to practical, symmetry-preserving quantum circuit design, with potential extensions to higher seniority and spin–orbit spaces for enhanced expressivity at reduced circuit depth.

Abstract

Preserving spin symmetry in variational quantum algorithms is essential for producing physically meaningful electronic wavefunctions. Implementing spin-adapted transformations on quantum hardware, however, is challenging because the corresponding fermionic generators translate into noncommuting Pauli operators. In this work, we introduce an exact and computationally efficient factorization of spin-adapted unitaries derived from fermionic double excitation and deexcitation rotations. These unitaries are expressed as ordered products of exponentials of Pauli operators. Our method exploits the fact that the elementary operators in these generators form small Lie algebras. By working in the adjoint representation of these algebras, we reformulate the factorization problem as a low-dimensional nonlinear optimization over matrix exponentials. This approach enables precise numerical reparametrization of the unitaries without relying on symbolic manipulations. The proposed factorization provides a practical strategy for constructing symmetry-conserving quantum circuits within variational algorithms. It preserves spin symmetry by design, reduces implementation cost, and ensures the accurate representation of electronic states in quantum simulations of molecular systems.

Exact Factorization of Unitary Transformations with Spin-Adapted Generators

TL;DR

The paper tackles the challenge of preserving spin symmetry in variational quantum algorithms by introducing an exact factorization of spin-adapted unitaries into ordered exponentials of Pauli operators, using compact Lie algebras and the adjoint representation to avoid symbolic BCH expansions. This turns the reparametrization into a low-dimensional nonlinear optimization, yielding precise circuit decompositions with central elements treated as commuting factors. The approach reduces variational parameter counts by roughly a factor of two in adaptive VQE tests (e.g., HO and BeH) while maintaining spin quantum numbers, and it prevents spin contamination in spin-sensitive cases like O triplet, outperforming penalty or projection methods. Overall, the framework links Lie-algebraic structure to practical, symmetry-preserving quantum circuit design, with potential extensions to higher seniority and spin–orbit spaces for enhanced expressivity at reduced circuit depth.

Abstract

Preserving spin symmetry in variational quantum algorithms is essential for producing physically meaningful electronic wavefunctions. Implementing spin-adapted transformations on quantum hardware, however, is challenging because the corresponding fermionic generators translate into noncommuting Pauli operators. In this work, we introduce an exact and computationally efficient factorization of spin-adapted unitaries derived from fermionic double excitation and deexcitation rotations. These unitaries are expressed as ordered products of exponentials of Pauli operators. Our method exploits the fact that the elementary operators in these generators form small Lie algebras. By working in the adjoint representation of these algebras, we reformulate the factorization problem as a low-dimensional nonlinear optimization over matrix exponentials. This approach enables precise numerical reparametrization of the unitaries without relying on symbolic manipulations. The proposed factorization provides a practical strategy for constructing symmetry-conserving quantum circuits within variational algorithms. It preserves spin symmetry by design, reduces implementation cost, and ensures the accurate representation of electronic states in quantum simulations of molecular systems.

Paper Structure

This paper contains 12 sections, 56 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Method
  3. Lie algebra framework
  4. Lie algebras
  5. Adjoint representation
  6. Implementation of unitaries
  7. Results
  8. Computational details
  9. Adaptive VQE calculations
  10. Conclusions
  11. Seniority 4, intermediate singlet
  12. Seniority 4, intermediate triplet

Figures (3)

  • Figure 1: Adaptive VQE calculations for $\text{H}_2\text{O}$ with $\theta_\mathrm{HOH}=109^\circ$ and $R_\mathrm{OH}=0.977\ \mathrm{\AA}$. SD denotes the fermionic operator pool, $\text{S}_\text{SA}\text{D}_\text{SA}$ the spin-adapted fermionic operator pool, and $\text{S}_\text{SA}\text{D}_\text{pair}$ the pool of spin-adapted singles and pair doubles.
  • Figure 2: Adaptive VQE calculations for $\text{Be}\text{H}_2$ with $\theta_\mathrm{HBeH}=180^\circ$ and $R_\mathrm{BeH}=1.35\ \mathrm{\AA}$. SD denotes the fermionic operator pool, $\text{S}_\text{SA}\text{D}_\text{SA}$ the spin-adapted fermionic operator pool, and $\text{S}_\text{SA}\text{D}_\text{pair}$ the pool of spin-adapted singles and pair doubles.
  • Figure 3: Adaptive VQE calculations for $\text{O}_2$ triplet. SD denotes the fermionic operator pool, $\text{S}_\text{SA}\text{D}_\text{SA}$ the spin-adapted fermionic operator pool. FCI triplet and singlet reference curves are shown for comparison.