Simulations of subatomic many-body physics on a quantum frequency processor

Simulating complex many-body quantum phenomena is a major scientific impetus behind the development of quantum computing, and a range of technologies are being explored to address such systems. We present the results of the largest photonics-based simulation to date, applied in the context of subatomic physics. Using an all-optical quantum frequency processor, the ground-state energies of light nuclei including the triton (${}^3\mathrm{H}$), ${}^3\mathrm{He}$, and the alpha particle (${}^4\mathrm{He}$) are computed. Complementing these calculations and utilizing a 68-dimensional Hilbert space, our photonic simulator is used to perform subnucleon calculations of the two- and three-body forces between heavy mesons in the Schwinger model. This work is a first step in simulating subatomic many-body physics on quantum frequency processors—augmenting classical computations that bridge scales from quarks to nuclei.

Figure 1

Quantum simulation for subatomic physics. Ideally, quantum simulation applied to both QCD (left side) and EFT (right side) will enable high-precision predictions of static and dynamic properties of nuclei and nuclear matter. EFT parameters may be determined from experiment or by a complementary program of classical and quantum simulation.

Figure 2

Experimental setup including our all-optical quantum frequency processor.

Figure 3

Results of the nuclear ground-state energies for ${}^3\mathrm{H}$, ${}^3\mathrm{He}$, and ${}^4\mathrm{He}$ nuclei computed with the QFP (blue data points) for Hilbert spaces with effective spatial extent $L$ with estimated systematic uncertainties. Also shown are the leading-order extrapolations [see Eq. (C2)] to infinite model spaces with propagated uncertainties (gray bands), the resulting extrapolated energies $E_{\infty }$ (red point at right), the tuned NLO EFT predictions (dark blue solid line), and the known high-precision values of the binding energies (dashed gray line).

Figure 4

The $e^{+}{e}^{-}$ densities ($\langle \rho \rangle$) and energy density in the electric field ($\langle E^2\rangle$) for $Q=-1$ static charges separated by one spatial lattice site. The left panel shows the raw distributions, the center panel has the vacuum removed, and the right panel also has the contributions from the individual charges removed. The horizontal black dashed lines are the analytic values of the local densities, while the error bands (not seen on this scale) represent fluctuations over the last ten iterations of the VQE. The values shown in each panel are presented in Table 6 in Appendix pp6.

Figure 5

The left panel shows the potential between two like-sign static-charged sources (upper curves) and between opposite-sign charges (lower curves) as a function of separation (in lattice units). The symmetric gray curves represent the extracted lattice potential, including the presence of image charges. The blue and green bands represent the infinite-volume potentials using correlated extracted values of the couplings and masses (see Appendix pp5 for details). The center panel shows the three-body potential for three like-sign static charges. The right panel shows the three-body potential for two like-sign and one opposite-sign static charges.

Figure 6

Nucleon-nucleon phase shifts in the singlet (left) and triplet (right) $S$-wave channels (in degrees) as a function of relative momentum. The solid (black) lines correspond to high-precision results obtained with the CD Bonn potential. The dashed (red) lines correspond to results obtained from the NLO pionless EFT, while the shaded (red) region includes estimates from higher orders in the pionless EFT expansion that are not included in the interactions employed on the QFP.

Figure 7

Monte Carlo samples and 68% confidence ellipses of the couplings and masses of the $QQ$ and $Q\overline{Q}$ two-body interactions. The ellipse associated with the $QQ$ system (left panel) is described by eigenvectors $(0.176,0.984)$ and $(-0.984,0.176)$ with associated semiaxis radii 0.177 and 0.984. The ellipse associated with the $Q\overline{Q}$ system (right panel) is described by eigenvectors $(-0.977,0.211)$ and $(-0.211,-0.977)$ with associated semiaxis radii 0.332 and 0.0236.

Figure 8

The potential between three static charges represented by Jacobi coordinates in one dimension, $R_1=|r_2-r_1|$ and $R_2=|r_3-\frac{R_1}{2}|$, with $r_{1,2,3}$ the $QQQ$ or $QQ\overline{Q}$ distances from the origin. The physical configurations of static charges on the lattice associated with the blue and green paths through the grid of three-body potential values are depicted by the schematic diagrams at the corners.

Figure 9

Local properties of the vacuum with three static charges, with $Q$ and $\overline{Q}$ located at lattice sites (0, 2) and 1, respectively. The uncertainties, which are too small to be visible, represent the stability of these local properties over wave functions extracted over the last ten VQE iterations. The horizontal dashed lines are the values calculated through exact diagonalization. The values shown in each panel are presented in Table 7.

Figure 10

Local properties of the vacuum and a single static charge located at $r=0$ that creates a heavy meson. The uncertainties, which are too small to be visible, represent the stability of these local properties over wave functions extracted over the last ten VQE iterations. The horizontal dashed lines are the values calculated through exact diagonalization. The values shown in each panel are presented in Tables 8 and 9.