Quantum simulations of molecular systems with intrinsic atomic orbitals

Quantum simulations of molecular systems on quantum computers often employ minimal basis sets of Gaussian orbitals. In comparison with more realistic basis sets, quantum simulations employing minimal basis sets require fewer qubits and quantum gates but yield results of lower accuracy. A natural strategy to achieve more accurate results is to increase the basis set size, which, in turn, requires increasing the number of qubits and quantum gates. Here we explore the use of intrinsic atomic orbitals in quantum simulations of molecules to improve the accuracy of energies and properties at the same computational cost required by a minimal basis. We investigate ground-state energies and one- and two-body density operators in the framework of the variational quantum eigensolver employing and comparing different Ansätze. We also demonstrate the use of this approach in the calculation of ground- and excited-state energies of small molecules by a combination of quantum algorithms using IBM quantum computers.

Figure 1

Ground-state potential-energy curve of ${\mathrm{NH}}_3$ along the ${\mathrm{NH}}_3\to {\mathrm{NH}}_2 +$ H reaction path, at STO-6G (top) and IAO-aug-cc-pVQZ (bottom) level using RHF (dashed light blue line), FCI (dark blue dashed-dot line), VQE with $R_y$ (red circles), $\mathrm{SO}\left(4\right)$ (dark orange diamonds), and q-UCCSD Ansätze (orange triangles). $d$ indicates the depth of the Ansatz.

Figure 2

Ground-state potential-energy surface of ${\mathrm{H}}_2$ calculated using VQE and FCI with an IAO/aug-cc-pVTZ basis (orange triangles and dashed-dot-dot lines) and VQSE and FCI with full aug-cc-pVTZ basis (dark blue circles and dashed line). Statistical uncertainties on the lowest eigenvalue were obtained by sampling the active-space density matrices ten times, repeating the embedding, contraction, and diagonalization procedures for each sample, and collecting statistics with standard procedures.

Figure 3

Left: HONO, top and LUNO, bottom natural orbitals from an MP2 calculation using the IAO/aug-cc-pVQZ basis. Right: quantum circuit for the VQE calculations with SO(4) Ansatz in the HONO-LUNO subspace. $U_3$ denotes an SU(2) gate, parametrized by three Euler angles. ${\sigma }^{\mu }$ and ${\sigma }^{\nu }$ are Pauli operators appearing in the qubit representation of the active space Hamiltonian, $\widehat{H}={\sum }_{\mu \nu =0}^3{\eta }_{\mu \nu }{\sigma }^{\mu }\otimes {\sigma }^{\nu }$.

Figure 4

Top: ground-state potential-energy surface of ${\mathrm{NH}}_3$ along the ${\mathrm{NH}}_3\to {\mathrm{NH}}_2 +$ H reaction path, using a HONO-LUNO active space based on an MP2 calculation at IAO/aug-cc-pVQZ level from RHF (dashed light blue line), FCI (blue dashed-dot line) and VQE with depth-1 SO(4) Ansatz (orange circles). Bottom: deviation of RHF and VQE energies from FCI.

Figure 5

Expectation values of the total spin operator (top) and purity of the VQE density operator (bottom) as a function of the reaction coordinate $R$, evaluated over the VQE wave function with the SO(4) Ansatz. Orange and blue symbols denote hardware calculations carried out on ibmq$\_$rome and on a a classical simulator with noise model from ibmq$\_$manila, respectively.

Figure 6

Spin-resolved one-body density matrix ${\rho }^{(\uparrow )}$ for ${\mathrm{NH}}_3$ in a HONO-LUNO space from VQE with the SO(4) Ansatz as a function of N-H distance (left to right, top to bottom).

Figure 7

Spin-resolved two-body density matrix ${\rho }^{(\uparrow ,\downarrow )}$ for ${\mathrm{NH}}_3$ in a HONO-LUNO space, from VQE with the SO(4) Ansatz as a function of N-H distance (left to right, top to bottom). Numbers 0–3 denote indices (0,0), (0,1), (1,0), and (1,1), respectively.

Figure 8

Ground-state potential-energy surface of ${\mathrm{NH}}_3$ along the ${\mathrm{NH}}_3\to {\mathrm{NH}}_2 +$ H reaction path using quantum imaginary-time evolution without (orange circles) and with (red triangles) readout error mitigation on ibmq$\_$vigo and ibmq$\_$london, respectively.

Figure 9

CCSD total (top), correlation (middle), and binding (bottom) energies of ${\mathrm{NH}}_3$ along the ${\mathrm{NH}}_3\to {\mathrm{NH}}_2 +$ H dissociation path from CCSD/aug-cc-pVQZ (blue crosses), low-energy Hartree-Fock orbitals (red squares), low-energy CASSCF orbitals (dark orange circles), high-occupancy MP2 natural orbitals (orange triangles), and IAO/aug-cc-pVQZ (light blue diamonds).

Figure 10

Same as Fig. 9 but for cc-pVTZ.

Figure 11

Excited-state energies (brown-red, dark-orange, and orange for the first, second, and third excited states) of ${\mathrm{NH}}_3$ along the ${\mathrm{NH}}_3\to {\mathrm{NH}}_2 +$ H reaction path using FCI (lines) and qEOM (symbols), on the $\mathrm{ibmq}\_\mathrm{rome}$ IBM quantum hardware. $E_n\left(M\right)$ indicates the $n\mathrm{th}$ excited state obtained with method $M$. The blue dashed line indicates the energy calculated using the VQE on hardware.

Figure 12

Determinant of the qEOM metric matrix $G$ as a function of reaction coordinate, measured on $\mathrm{ibmq}\_\mathrm{rome}$ (dotted, blue circles). The gray dashed line highlights $\det \left(G\right)=0$.

Figure 13

Analytical gradient evaluation (top) and line search (bottom) in the gradient-descent optimization of VQE parameters with the SO(4) Ansatz.

Figure 14

Fidelity between the VQE density operator and the projector onto the RHF state as a function of reaction coordinate. The gray dotted line indicates the ideal result of a state vector simulation. The blue circles are the results of the experiment on ibmq$\_$rome.