Computation of relativistic and many-body effects in atomic systems using quantum annealing

We report results for the computation of relativistic effects in quantum many-body systems using quantum annealers. An average accuracy of 98.9% in the fine-structure splitting of boron-like ions with respect to experiments is achieved using the Quantum Annealer Eigensolver (QAE) algorithm on the D-Wave Advantage 5000-qubit hardware, which is substantially higher than that attained on a gate-based quantum device to date. We obtain these results in the framework of the many-electron Dirac theory. We implement QAE using a hybrid quantum annealing method that includes an alternative qubit encoding scheme and decomposing the problem into smaller ones based on perturbation theory.

Figure 1

Illustration of our QAE workflow. The Lagrange multiplier ${\lambda }^{\left\{0\right\}}$ is initialized to the CSF energy associated with the most dominant configuration, $\langle {\mathrm{\Phi }}_0\left|H\right|{\mathrm{\Phi }}_0\rangle$. $\left\{a_{\alpha }\right\}$ denotes the list of expansion coefficients from the relativistic CI expansion of the wave function. $E_f$ and $|{\mathrm{\Psi }}_f\rangle$ refer to the final ground-state energy and its corresponding wave function obtained using the presented workflow, respectively. Floating qubit encoding is abbreviated as FQE in the figure.

Figure 2

Variation of energy $E$ with $\lambda$, while minimizing the energy functional $\epsilon$.

Figure 3

Error with respect to relCI in QAE with Simulation that takes into account connectivity and with the D-Wave QPU for ${}^{2}P_{1/2}$ and ${}^{2}P_{3/2}$ states for all the considered ions.

Figure 4

Error in energy with respect to relCI versus number of Repeats in our workflow for (a) ${}^{2}P_{1/2}$ and (b) ${}^{2}P_{3/2}$ states of B-like Kr using the D-Wave Advantage QPU with steepest descent. The dashed lines present the error at every repeat, while circles present the error of the global sample. Different colors denote different repetition of the experiment.

Figure 5

Error in energy with respect to relCI versus number of Repeats in our workflow for (a), (c), (d), and (f) ${}^{2}P_{1/2}$ and (b), (d), (f), and (h) ${}^{2}P_{3/2}$ states of all the considered boron-like atoms using the simulation, taking hardware architecture into consideration. The dashed lines present the error at every repeat, while circles denote the error in the global sample. Different colors denote different repetitions of the experiment. Panels (a,b), (c,d), (e,f), and (g,h) correspond to boron-like Ca, Fe, Kr, and Mo, respectively.

Figure 6

Error in energy with respect to relCI versus number of Repeats in our workflow for (a), (c), (d), and (f) ${}^{2}P_{1/2}$ and (b), (d), (f), and (h) ${}^{2}P_{3/2}$ states of all the considered boron-like atoms using the D-Wave hardware. The dashed lines present the error at every repeat, while circles denote the error in the global sample. Different colors denote different repetitions of the experiment. Panels (a,b), (c,d), (e,f), and (g,h) correspond to boron-like Ca, Fe, Kr, and Mo, respectively.