Progress toward larger molecular simulation on a quantum computer: Simulating a system with up to 28 qubits accelerated by point-group symmetry

The exact evaluation of the molecular ground state in quantum chemistry requires an exponentially increasing computational cost. Quantum computation is a promising way to overcome the exponential problem using polynomial-time quantum algorithms. A quantum-classical hybrid optimization scheme known as the variational quantum eigensolver is preferred for noisy intermediate-scale quantum devices. However, the circuit depth becomes one of the bottlenecks of its application to large molecules of more than 20 qubits. In this work, we employ point-group symmetry to reduce the number of operators in constructing ansatz so as to achieve a more compact quantum circuit. We illustrate this methodology with a series of molecules ranging from LiH (12 qubits) to ${\mathrm{C}}_2{\mathrm{H}}_4$ (28 qubits). A significant reduction of up to 82% of the operator numbers is reached on ${\mathrm{C}}_2{\mathrm{H}}_4$. This also sheds light onto further work in this direction to construct even shallower ansatz with enough expressive power and simulate even larger scale systems.

Figure 1

Parameter reduction of ${\text{BeH}}_2$ under various point groups. The orientation of the symmetry elements changes the ratio of reduction slightly. The orange line is the hypothetical situation that the number of configurations belonging to each irrep is equal.

Figure 2

The potential energy curve of the ammonia flipping. (a) Potential energy surface calculated by FCI, CCSD, and the symmetry-reduced UCCSD during ammonia flipping. (b) Energy deviation compared with FCI energy. The flipping process is described by the angle of the $z$ axis, N atom, and H atom. The unit of the error is millihartree.

Figure 3

The convergence process of SymUCCSD for ${\text{C}}_2{\text{H}}_4$. The green line denotes the energy deviation of SymUCCSD compared to FCI energy vs iteration numbers. The orange line denotes the energy deviation derived from CCSD vs FCI and the green line indicates chemical at the level of STO-3G with reference to FCI results (0.0016 hartree).

Figure 4

The character table of the $D_{2h}$ group for ${\text{BeH}}_2$.

Figure 5

The irrep of the $D_{2h}$ point group for a ${\text{BeH}}_2$ molecule. (a) The product table for the irreducible representations of the $D_{2h}$ group. (b) The electron configuration diagram of ${\mathrm{BeH}}_2$ of the reference state, i.e., the Hartree-Fock state with irrep $A_g$. (c) An example of coupled-cluster terms that correspond to a configuration with different irrep $B_{3g}$ compares to the reference state. This term is a single excitation concerning the reference term. (d) An example of coupled-cluster terms that correspond to a configuration with the same irrep $A_g$ as the reference state. Note that it is conventional to label the irrep of the molecular orbital in lower case like $a_g$ and the irrep of the molecular state in upper case like $A_g$.

Figure 6

The symmetry elements of ${\text{BeH}}_2$ in the $D_{2h}$ point group. Rotation axes $C_2$, reflection mirrors $\sigma$, and inversion center $i$ are depicted in the figure.

Figure 7

The isosurface of the wavefunction of the molecular orbital. (Isovalue is 0.03 arbitrary units.)

Figure 8

The number of parameters for a series of molecules before and after the reduction by point-group symmetry. We observed that the ratio of the necessary parameters decreases slightly as the size of the system increases, ranging from 25.6% of ${\mathrm{BeH}}_2$, to 20.3% of ${\mathrm{C}}_2{\mathrm{H}}_2$, and to $17.9\%$ of ${\mathrm{C}}_2{\mathrm{H}}_4$, all of which are of the same $D_{2h}$ symmetry, indicating better efficiency for larger molecules.

Figure 9

The numerical simulation of an ${\text{H}}_4$ molecule with uncertainties. (a) The energy vs the number of shots without introducing noise model. (b) The difference between energies calculated by SymUCCSD and UCCSD ansatzes. (c) The energy vs the depolarization probability. The depolarizing noise model provided by qiskit [79] is used. The quantum error mitigation (QEM) is performed using zero-noise extrapolation based on a linear fit implemented with the mitiq [80] package. The magnitude of the standard deviation is tens of millihartree, which makes the error bar look narrow due to the large scaling of the $y$ axis. (d) The difference between energies calculated by SymUCCSD and UCCSD ansatzes.