Polynomial Scaling Localized Active Space Unitary Selective Coupled Cluster Singles and Doubles

We present a polynomial-scaling algorithm for the localized active space unitary selective coupled cluster singles and doubles (LAS-USCCSD) method. In this approach, cluster excitations are selected based on a threshold $\mathnormal{\epsilon}$ determined by the absolute gradients of the LAS-UCCSD energy with respect to cluster amplitudes. Using the generalized Wick's theorem for multireference wave functions, we derive the gradient expression as a polynomial function of 1-, 2-, and 3-body reduced density matrices and 1- and 2-electron integrals, valid for any multireference wave function. The resulting gradient implementation exhibits a memory scaling of $\mathcal{O}(N^6)$. The variational quantum eigensolver is used to optimize the selected cluster excitations on a quantum simulator. By plotting the energy error, defined as the difference between the LAS-USCCSD and corresponding CASCI energies, against the inverse cluster amplitude selection threshold ($\mathnormal{\epsilon}^{-1}$) for polyene chains containing 2 to 5 $\pi$-bond units, we establish a relationship between the energy error and the threshold. To further validate the accuracy of LAS-USCCSD, we computed the cis–trans isomerization energy of stilbene (a 20-qubit system) and the magnetic coupling constant of the tris-hydroxo-bridged chromium dimer \ce{[Cr_2(OH)_3(NH_3)_6]^{3+}} (evaluated as both 12- and 20-qubit systems) using the Qiskit-Qulacs simulator. Assessing such examples is important to determine the practical feasibility of quantum simulations for chemically realistic systems. Toward this goal, with this LAS-USCCSD algorithm we estimated the quantum resources required for simulating an active space of (30e,22o) in \ce{[Cr_2(OH)_3(NH_3)_6]^{3+}}, a size that remains beyond the reach of current quantum simulators for accurate treatment.