Considerations for evaluating thermodynamic properties with hybrid quantum-classical computing work flows

Quantum chemistry applications on quantum computers currently rely heavily on the variational quantum eigensolver (VQE) algorithm. This hybrid quantum-classical algorithm aims at finding ground-state solutions of molecular systems based on the variational principle. VQE calculations can be systematically implemented for perturbations to each molecular degree of freedom, generating a Born-Oppenheimer potential-energy surface (PES) for the molecule. The PES can then be used to derive thermodynamic properties, which are often desirable for applications in chemical engineering and materials design. It is clear from this process that quantum chemistry applications contain a substantial classical computing component in addition to steps that can be performed using a quantum computer. In order to design efficient work flows that take full advantage of each hardware type, it is critical to consider the entire process so that the high-accuracy electronic energies possible from quantum computing are not squandered in the process of calculating thermodynamic properties. We present a summary of the hybrid quantum-classical work flow to compute thermodynamic properties. This work flow contains many options that can significantly affect the efficiency and the accuracy of the results, including classical optimizer attributes, number of ansatz repetitions, and how the vibrational Schrödinger equation is solved to determine vibrational modes. We also analyze the effects of these options by employing robust statistics along with simulations and experiments on actual quantum hardware. We show that, through careful selection of work flow options, nearly order-of-magnitude increases in accuracy are possible at equivalent computing time.

Figure 1

Schematic diagram of the parametrized circuit (ansatz) using two repetitions of $R_Y$ gates and four qubits which are linearly entangled with controlled-$X$ gates. The application of the last two $X$ gates allows us to construct the HF state in a meaningful way for a specific initialization of all the circuit parameters.

Figure 2

(a) The MAE of the PES as a function of ansatz repetitions and measurement shots. (b) The computed $C_p$ for each PES. (c) The total number of shots used in each calculation, which includes shots used in the resampling procedure. (d–g) Representative PESs corresponding to selected numbers of measurement shots and ansatz repeats.

Figure 3

Comparison of four implementations of the VQE classical optimizer algorithm. (a) Stochastic gradient descent without bootstrapping or resampling (SGD). (b) With bootstrapping and resampling $(\mathrm{SGD}+\mathrm{BR})$. (c) With bootstrapping only $(\mathrm{SGD}+\mathrm{B}$). (d) With resampling only $(\mathrm{SGD}+\mathrm{R}$). The simulations include shot noise only and include four ansatz repetitions and 512 measurement shots

Figure 4

Comparison of harmonic, Morse, and spline fits of a classically computed FCI/cc-pVDZ PES. The top panel shows the fits of the PES. The middle panel shows vibrational energy levels as horizontal lines, which have been shifted to zero by their zero-point energies. The bottom panel shows predicted heat capacity and experimental data as green plus signs.

Figure 5

Schematic diagram of the topology of $\mathrm{ibmq}\_\mathrm{rome}$ (top) and the quantum circuit used for VQE (bottom).

Figure 6

Results from ibmq_rome for LiH. The PES, vibrational energy levels, and $C_p$ are shown in the top, middle, and bottom panels, respectively. Green “plus signs” are experimental data. The spline fits of the ibmq_rome data were smoothed to enable a better fit to the sparse data.