Quantum computing of the ${}^6\mathrm{Li}$ nucleus via ordered unitary coupled clusters

The variational quantum eigensolver (VQE) is an algorithm to compute ground and excited state energy of quantum many-body systems. A key component of the algorithm and an active research area is the construction of a parametrized trial wave function—a so-called variational ansatz. The wave function parametrization should be expressive enough, i.e., represent the true eigenstate of a quantum system for some choice of parameter values. On the other hand, it should be trainable, i.e., the number of parameters should not grow exponentially with the size of the system. Here, we apply VQE to the problem of finding ground and excited state energies of the odd-odd nucleus ${}^6\mathrm{Li}$. We study the effects of ordering fermionic excitation operators in the unitary coupled clusters ansatz on the VQE algorithm convergence by using only operators preserving the $J_z$ quantum number. The accuracy is improved by two orders of magnitude in the case of descending order. We first compute optimal ansatz parameter values using a classical state-vector simulator with arbitrary measurement accuracy and then use those values to evaluate energy eigenstates of ${}^6\mathrm{Li}$ on a superconducting quantum chip from IBM. We post-process the results by using error mitigation techniques and are able to reproduce the exact energy with an error of $3.8\%$ and $0.1\%$ for the ground state and for the first excited state of ${}^6\mathrm{Li}$, respectively.

Figure 1

Training curve in a semilog scale for fermionic UCC ansatz with different ordering. The best shuffle curve is taken among 20 independent runs. The grey area corresponds to the 1% margin, which is acceptable in most applications.

Figure 2

Training curve in a semilog scale for all different ansätze with different training strategies. The grey area corresponds to the 1% margin, which is acceptable in most applications.

Figure 3

$\mathrm{U}(\theta ,\varphi )$ gate decomposition. Decomposition of $U(\theta ,\varphi$) in terms of elementary gates where $R(\theta ,\varphi )=R_z(\varphi +\pi )R_y(\theta +\pi /2)$ and $R_z\left(\theta \right)=e^{-i\theta Z/2}, R_y\left(\theta \right)=e^{-i\theta Y/2}$.

Figure 4

Excitation-preserving ansatz for one single register (e.g., for the protons). The circuit starts from the initial state $|001000\rangle$ and is built with the excitation preserving gate $U(\theta ,\varphi )$ in a pyramidal and efficient way.