Angular momentum orbital space of molecules and extend systems for quantum computing

We developed highly efficient quantum algorithms for simulating the electronic Hamiltonian with spherical harmonics (SH) basis function by leveraging the angular momentum entanglement. For the evaluation of molecular integral using SH basis, our new scheme decomposes the computation into calculations depending solely on rotational property separated from the physical system. The coupling of angular momentum captures this sole dependency on the rotational properties. In quantum mechanics, the entanglement structure between the angular momentum states of an orbital and the surrounding field can be calculated using the Clebsch-Gordan (CG) transformation. The CG selection rule enforces the conservation of angular momentum by determining which angular momentum states are allowed in the resulting composite system. The SH basis is orthogonal and unitary, reducing the basis size needed to diagonalize the Coulomb operator. For the second quantized representation, original fermion ladder operators are projected to the spherical harmonic basis, forming the angular momentum orbital space. Our new approach greatly simplifies the direct simulation of this Hamiltonian using quantum computers with atomic orbital angular momentum block encoding. The encoding speedup for the small molecule is about six orders of magnitude compared to the existing method using Pauli operators.