Nuclear two point correlation functions on a quantum computer

The calculation of dynamic response functions is expected to be an early application benefiting from rapidly developing quantum hardware resources. The ability to calculate real-time quantities of strongly correlated quantum systems is one of the most exciting applications that can easily reach beyond the capabilities of traditional classical hardware. Response functions of fermionic systems at moderate momenta and energies corresponding roughly to the Fermi energy of the system are a potential early application because the relevant operators are nearly local, and the energies can be resolved in moderately short real time, reducing the spatial resolution and gate depth required. This is particularly the case in quasielastic electron and neutrino scattering from nuclei, a topic of great interest in the nuclear and particle physics communities and directly related to experiments designed to probe neutrino properties. In this work we use current quantum hardware and error mitigation protocols to calculate response functions for a highly simplified nuclear model through calculations of a 2-point real time correlation function for a modified Fermi-Hubbard model in two dimensions with three distinguishable nucleons on four lattice sites.

Figure 1

Real and imaginary parts of the response function at different lattice sites using time orderings $A1$, $A2$, $B1$ and $B2$ compared against the numerically exact calculation, for the problem described in the text. The top panels are for $\mathbf{q}=(0,1)$ while the bottom panels are for $\mathbf{q}=(1,1)$.

Figure 2

Circuits for the calculation of the correlator $⟨Z_1(t)Z_3⟩$ using the time evolutions $B1$ (left) and $B2$ (right).

Figure 3

Real and imaginary parts of the response function wave vector $\mathbf{q}=(0,1)$ using time ordering $A1$, $A2$ (left panel) and $B1$, $B2$ (right panel) obtained from the five qubit machine Ourense [40]. For the Trotterized time evolution, black lines, green and red dots denote the numerically exact result, the bare estimate from the QPU and the error mitigated results, respectively. The dashed grey line represents the exact dynamics with no time step errors.

Figure 4

Real and imaginary parts of the response function wave vector ${\mathbf{q}}_1=(1,1)$ using time ordering $A1$, $A2$ (left panels) and $B1$, $B2$ (right panels) ran on the five qubit machine Ourense. Black lines, green and red dots denote respectively the expected result, the bare estimate from the QPU and the error mitigated results. The dashed grey line represents the exact dynamics with no time step errors.

Figure 5

Circuit decomposition for the three-body propagator for $e^{-i{H}_M^{(3bd)}t}$. The angle in the single qubit rotations is $\theta =\tau U/2$.

Figure 6

Circuit identities used in the synthesis of the full circuit for pair correlators as described in the text.