Simulation of collective neutrino oscillations on a quantum computer

In astrophysical scenarios with large neutrino density, like supernovae and the early universe, the presence of neutrino-neutrino interactions can give rise to collective flavor oscillations in the out-of-equilibrium collective dynamics of a neutrino cloud. The role of quantum correlations in these phenomena is not yet well understood, in large part due to complications in solving for the real-time evolution of the strongly coupled many-body system. Future fault-tolerant quantum computers hold the promise to overcome much of these limitations and provide direct access to the correlated neutrino dynamic. In this work, we present the first simulation of a small system of interacting neutrinos using current generation quantum devices. We introduce a strategy to overcome limitations in the natural connectivity of the qubits and use it to track the evolution of entanglement in real-time. The results show the critical importance of error-mitigation techniques to extract meaningful results for entanglement measures using noisy, near term, quantum devices.

Figure 1

Pictorial representation of the swap network used in our simulation in the case of $N=4$ neutrinos.

Figure 2

Layout of the IBM Quantum Canary Processor Vigo [27]. Shown are the five qubits, labeled from 0 to 4, and their connectivity denoted as solid black lines.

Figure 3

Inversion probability $P_{\mathrm{inv}(t)}$ for neutrinos 1 and 4: the red circle and brown square correspond to the bare results, the blue triangle and the green diamond are obtained after error mitigation (see text). The left panel (VM) are virtual machine results while the right panel (QPU) are results obtained on the Vigo [27] quantum device.

Figure 4

Inversion probability $P_{\mathrm{inv}}(t)$ for neutrinos 2 and 3. The notation is the same as for Fig. 3.

Figure 5

Inversion probability $P_{\mathrm{inv}}$ at the initial time $t=0$ for the first neutrino. Black solid circles are results from the Vigo QPU [27] while the red squares correspond to results obtained using the VM with simulated noise. Also shown are extrapolations to the zero noise limit, for both the QPU (green line) and the VM (blue line), together with the extrapolated value (greed triangle up and blue triangle down respectively). The dashed orange line denotes the result for a maximally mixed state.

Figure 6

Single spin entanglement entropy for neutrino 2. Black square are bare results obtained from the QPU, red triangles are results obtained by amplifying the noise to $\epsilon /{\epsilon }_0=3$, the blue circles are obtained using Richardson extrapolation, the turquoise plus symbols indicate results obtained using the standard exponential extrapolation and the green diamonds correspond to the results obtained from a shifted exponential extrapolation using the maximum value of the entropy (indicated as a dashed orange line).

Figure 7

Pair entanglement entropy for the neutrino pair (1,2) starting as $|e⟩\otimes |e⟩$ (left panel) and pair (2,4) which starts as the flavor state $|e⟩\otimes |x⟩$ (right panel). Results obtained directly from the QPU are shown as black squares ($r=1$) and red triangles ($r=3$) while blue circles and green diamonds indicate mitigated results using Richardson and the shifted exponential extrapolations respectively. For the shifted exponential ansatz we use the maximum value of the entropy (indicated as a dashed orange line).The magenta triangle indicates a mitigated result with shifted exponential extrapolation below zero within error bars.

Figure 8

Extended concurrence $\tilde{C}$ for two pairs of neutrinos, (1,2) in the left and (2,4) in the right panel. The convention for the curves and date point used here is the same as in Fig. 7. The gray area indicates the region where the concurrence $C(\rho )$ is zero. The maximum value for the concurrence is shown as a dashed orange line.

Figure 9

Panel (a) shows the error in matrix 2-norm Eq. (a3) of the two approximations $U_1$ and $U_2$ described in the text. Panel (b) shows the state fidelity and the right panels show results for the inversion probability $P_{\mathrm{inv}}(t)$. Panel (c) is for neutrino 1 while panel (d) is for neutrino 2.

Figure 10

Single spin entanglement entropy for all four neutrinos. Black square are bare results obtained from the QPU, the blue circles are obtained using Richardson extrapolation and the green diamonds correspond to the results obtained from a shifted exponential extrapolation using the maximum value of the entropy (dashed orange line). The magenta triangle indicates a mitigated result with shifted exponential extrapolation below zero within errorbars.

Figure 11

Pair entanglement entropy for all pair of neutrinos. Black square are bare results obtained from the QPU, red triangles are results obtained by amplifying the noise to $\epsilon /{\epsilon }_0=3$, the blue circles are obtained using Richardson extrapolation and the green diamonds correspond to the results obtained from a shifted exponential extrapolation using the maximum value of the entropy (indicated as a dashed orange line). The magenta triangle points are mitigated results with shifted exponential extrapolation below zero within errorbars.

Figure 12

Entanglement concurrence for all the pairs of qubits. The maximum value for the concurrence is shown as a dashed orange line.