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arXiv:2609.37101

Minimal building blocks for molecular quantum circuits with exact spin symmetry

Mengwei Liu, Zhenyu Li

quant-phphysics.chem-phSubmitted Sep 29, 2026• Announced Sep 30, 2026• 0 citations
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Abstract

Preserving particle number and spin helps quantum circuits target molecular electronic states, but does not guarantee access to every state with the required quantum numbers. We determine which additional operations, combined with spin-independent orbital rotations connecting all spatial orbitals, generate every real state-space rotation within each complete subspace of fixed particle number $N$, total spin $S$, and spin projection $M_S$. We consider spin-free molecular calculations in real orbitals, without an additional spatial-symmetry restriction, and continuously tunable operations that preserve particle number, full spin symmetry, and real amplitudes. Below the maximal-spin limits set by the electron and hole numbers, repeated singlet-pair transfer between any two fixed spatial orbitals is sufficient. On nontrivial maximal-spin boundaries, pair transfer vanishes, and an additional generator built from one- and two-electron terms is sufficient exactly when its action in the target subspace is not a linear combination of orbital-rotation generators. The minimum number of spatial orbitals needed by one additional generator is two in the interior and three on nontrivial boundaries, even when terms involving more than two electrons are allowed. Both minima are attained using only one- and two-electron terms. The three-orbital optimum rotates two orbitals according to the occupation of a third, and its unitary factors exactly into eight commuting Pauli rotations. Molecular benchmarks show that this operation removes the observed boundary energy-error plateaus, while sparse interior constructions attain the prescribed energy accuracy with fewer compiled CNOT gates in selected fixed-orbital comparisons.

Comments
72 pages, 7 figures; includes 25 pages of Supplementary Information. Data and code: https://doi.org/10.5281/zenodo.22230014

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Preserving particle number and spin helps quantum circuits target molecular electronic states, but does not guarantee access to every state with the required quantum numbers. We determine which additional operations, combined with spin-independent orbital rotations connecting all spatial orbitals, generate every real state-space rotation within each complete subspace of fixed particle number NNN, total spin SSS, and spin projection MSM_SMS​. We consider spin-free molecular calculations in real orbitals, without an additional spatial-symmetry restriction, and continuously tunable operations that preserve particle number, full spin symmetry, and real amplitudes. Below the maximal-spin limits set by the electron and hole numbers, repeated singlet-pair transfer between any two fixed spatial orbitals is sufficient. On nontrivial maximal-spin boundaries, pair transfer vanishes, and an additional generator built from one- and two-electron terms is sufficient exactly when its action in the target subspace is not a linear combination of orbital-rotation generators. The minimum number of spatial orbitals needed by one additional generator is two in the interior and three on nontrivial boundaries, even when terms involving more than two electrons are allowed. Both minima are attained using only one- and two-electron terms. The three-orbital optimum rotates two orbitals according to the occupation of a third, and its unitary factors exactly into eight commuting Pauli rotations. Molecular benchmarks show that this operation removes the observed boundary energy-error plateaus, while sparse interior constructions attain the prescribed energy accuracy with fewer compiled CNOT gates in selected fixed-orbital comparisons.
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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.
Quantum chemistry is a key target for quantum computing, but benchmarking quantum algorithms for large molecular systems remains challenging due to the lack of exactly solvable yet structurally realistic models. In particular, molecular Hamiltonians typically containO(N4)O\left(N^{4}\right)O(N4)Pauli terms, significantly increasing the cost of quantum simulations, while many exactly solvable models, such as the one-dimensional Fermi-Hubbard (1D FH) model, contain onlyO(N)O(N)O(N)terms. In this work, we introduce the orbital-rotated Fermi-Hubbard (ORFH) model as a scalable and exactly solvable benchmarking problem for quantum chemistry algorithms. Starting from the 1D FH model, we apply a spin-involved orbital rotation to construct a Hamiltonian that retains the exact ground-state energy but exhibits a Pauli term count scaling asO(N4)O\left(N^{4}\right)O(N4), similar to real molecular systems. We analyze the ORFH Hamiltonian from multiple perspectives, including operator norm and electronic correlation. We benchmark variational quantum eigensolver (VQE) optimizers and Pauli term grouping methods, and compare their performance with those for hydrogen chains. Furthermore, we show that the ORFH Hamiltonian increases the computational difficulty for classical methods such as the density matrix renormalization group (DMRG), offering a nontrivial benchmark beyond quantum algorithms. Our results demonstrate that the ORFH model provides a versatile and scalable testbed for benchmarking quantum chemistry algorithms under realistic structural conditions, while maintaining exact solvability even at large system sizes.