Quantum computing for neutrino-nucleus scattering

Neutrino-nucleus cross section uncertainties are expected to be a dominant systematic in future accelerator neutrino experiments. The cross sections are determined by the linear response of the nucleus to the weak interactions of the neutrino, and are dominated by energy and distance scales of the order of the separation between nucleons in the nucleus. These response functions are potentially an important early physics application of quantum computers. Here we present an analysis of the resources required and their expected scaling for scattering cross section calculations. The current estimates of Trotter steps needed to achieve an energy resolution of 10 MeV and the number of CNOT gates for analyzing ${}^{40}\mathrm{Ar}$ highlights the need for significant improvements in algorithms. We also examine simple small-scale neutrino-nucleus models on modern quantum hardware. In this paper, we use variational methods to obtain the ground state of a three nucleon system (the triton) and then implement the relevant time evolution. To tame the errors in present-day NISQ devices, we explore the use of different error-mitigation techniques to increase the fidelity of the calculations.

Figure 1

Estimated number of Trotter steps for both splittings of the Hamiltonian and target resolutions $\delta \omega =10\mathrm{MeV}$ for Trotter-Suzuki formulas of different orders. The left panel shows the linear formulas Eqs. (13) and (14), and the right panel shows results for both a second order formula (solid lines) and a fourth order one (dotted lines).

Figure 2

Comparison of the analytical (solid lines) vs commutator bounds (dashed lines) $r_{2:A}$ and $r_{2:C}$ for the quadratic Trotter-Suzuki breakup with both splitting schemes ($\alpha$ for the left panel and $\beta$ for the right one) at fixed target resolution $\mathrm{\Delta }\omega =100\mathrm{MeV}$. The dashed lines for the $\beta$ splitting correspond to the choice ($V+K$) while the dotted lines are for the complementary choice ($K+V$) (see text for details).

Figure 3

Estimated gate count, in the $\mathrm{CNOT}+R_z$ basis, of the phase estimation kernel of the linear response algorithm of [17] as a function of the nucleon number. Shown are results for the $\beta$ splitting and both first and second order Trotter decompositions. The left panel is for an energy resolution $\mathrm{\Delta }\omega =10\mathrm{MeV}$ and the right is for $\mathrm{\Delta }\omega =100\mathrm{MeV}$. The solid lines correspond to the serial execution while parallelism is exploited for the dashed ones.

Figure 4

Estimated number of ancilla qubit needed for a fixed target precision of $\mathrm{\Delta }\omega =10\mathrm{MeV}$ (left panel) and 100 MeV (right panel) using either time evolution with both split methods described in the main text and qubitization. The inset shows the needed speedup the qubiterate needs to show with respect to the time evolution circuit for the base time $\tau =2\pi /\mathrm{\Delta }H$ in order to be competitive.

Figure 5

Estimated total gate count in $\mathrm{CNOT}+R_Z$ basis for the $\beta$ splitting and the variant based on the qubiterate.

Figure 6

Qubit mapping for a single fermion.

Figure 7

Probability of finding the three nucleons on the same site as a function of time using both a real QPU (black circles) and a simulated VM (red squares). See text for a description of the left panel.

Figure 8

Extrapolation procedures used to mitigate errors in the results for the three nucleon contact density $C_3(t)$ shown in Fig. 7. See main text for a description of the different panels.

Figure 9

Extrapolation procedures used to mitigate errors in the results for the two-body contact $P_{\mathrm{dyn}}$ defined in Eq. (40) of the main text. The rightmost vertical panel and the bottom horizontal panel are the same as in Fig. 8 and reported here for reference.

Figure 10

Extrapolation procedures used to mitigate errors in the results for the two-body contact $P_{\mathrm{dyn}}$ defined in Eq. (41) of the main text. The rightmost vertical panel and the bottom horizontal panel are the same as in Fig. 8 and reported here for reference.

Figure 11

The top panel shows, as a function of time, different measures of entanglement obtained analytically from the expected results. The black dashed line shows the single qubit entanglement entropy, the red (blue) solid line corresponds to the entanglement entropy of the pair of qubits 01 (03), and the solid green curve is the concurrence for the pair of qubits (03). For the pair 01 the concurrence is identically zero at all times. The bottom panel shows the fraction of runs where decoherence has been detected (same as in Fig. 8).

Figure 12

Estimated number of Trotter steps for both splittings of the Hamiltonian and two different target resolutions: $\delta \omega =100\mathrm{MeV}$ (solid lines) and $\delta \omega =10\mathrm{MeV}$ (dashed lines). The inset shows results for the best performing splitting (the $\beta$ one) in the region of the mass number of interest for neutrinos.

Figure 13

Number of Trotter steps required for a target precision ${\epsilon }_{\tau }=10\mathrm{MeV}$ (100 MeV) using $\beta$ splitting shown as solid (dashed) lines for the base time $\tau =2\pi /\mathrm{\Delta }H$.

Figure 14

Total number of Trotter steps required for a target precision ${\epsilon }_{\tau }=10\mathrm{MeV}$ (100 MeV) shown as solid (dotted) lines. In contrast to Fig. 13 these results are obtained for the full propagation time $T=2^{W-1}\tau$. See text for details.

Figure 15

Comparison of the analytical vs commutator bounds $r_{1:A}$ and $r_{1:C}$ for the linear Trotter-Suzuki breakup with both splitting schemes. The left panel corresponds to a target resolution $\mathrm{\Delta }\omega =10\mathrm{MeV}$ while the right panel corresponds to $\mathrm{\Delta }\omega =100\mathrm{MeV}$.

Figure 16

Empirical gate counts from a simulation of the naive implementation of $U_3/U_4$ for system sizes $M\in (8,3375)$. Also shown are the gate cost estimates for the fSWAP algorithm and the implementation of $U_2$ using Givens rotations (cf. Table 5).