Trapped-ion quantum simulation of collective neutrino oscillations

It is well known that the neutrino flavor in extreme astrophysical environments changes under the effect of three contributions: the vacuum oscillation, the interaction with the surrounding matter, and the collective oscillations due to interactions between different neutrinos. The latter adds a nonlinear contribution to the equations of motion, making the description of their dynamics complex. In this work we study various strategies to simulate the coherent collective oscillations of a system of $N$ neutrinos in the two-flavor approximation using quantum computation. This was achieved by using a pair-neutrino decomposition designed to account for the fact that the flavor Hamiltonian, in the presence of the neutrino-neutrino term, presents an all-to-all interaction that makes the implementation of the evolution dependent on the qubit topology. We analyze the Trotter error caused by the decomposition demonstrating that the complexity of the implementation of time evolution scales polynomially with the number of neutrinos and that the noise from near-term quantum device simulation can be reduced by optimizing the quantum circuit decomposition and exploiting a full-qubit connectivity. We find that the gate complexity using second order Trotter-Suzuki formulas scales better with system size than with other decomposition methods such as quantum signal processing. We finally present the application and the results of our algorithm on a real quantum device based on trapped-ion qubits.

Figure 1

Sketch of the environment in a core-collapse supernova assuming spherical symmetry. Near the proto-neutron star the large local neutrino density causes an important effect from pairwise neutrinos scattering, while at large radii the interaction with electrons prevails instead. Moreover, we assume a narrow cone of forward peaked emitted neutrinos.

Figure 2

Exact evolution of a system of $N=4$ neutrinos with initial state $|{\mathrm{\Psi }}_0⟩=|0011⟩$. The panel (a) shows the evolution of the expectation value $⟨Z_i⟩$ while the panel (b) shows the evolution of the flavor inversion probability.

Figure 3

SWAP network for $N=4$ qubits that implements the two-body propagator $U_2(t)$ using a chain of linearly connected qubits. Vertical lines represent the qubit interaction and crosses the SWAP operation. The pair ordering influences the total error; here, we use the map $(q_0,q_1,q_2,q_3)=({\nu }_0,{\nu }_2,{\nu }_1,{\nu }_3)$.

Figure 4

Implementation of the two-body propagator $U_2(t)$ on a qubit system with all-to-all connectivity. This scheme can implement the optimal ordering of pairs to minimize the decomposition error. For our system this corresponds to the qubit to neutrino map $(q_0,q_1,q_2,q_3)=({\nu }_0,{\nu }_1,{\nu }_2,{\nu }_3)$.

Figure 5

Top panels: inversion probability after a single Trotter step for different values of the time steps $dt$. Panel (a) shows the evolution for neutrinos ${\nu }_0$ and ${\nu }_3$, and panel (b) for ${\nu }_1$ and ${\nu }_2$. The dotted black curve is the exact evolution, and the solid blue line is the one obtained by applying the implementation proposed in this work [Eqs. (16) and (17)] together with the scheme in Fig. 4. The dashed orange and red lines are the evolution using the implementation proposed in Ref. [37] and described by Eq. (15), and the scheme from Fig. 3. Panel (c) shows the error in spectral norm between the exact propagator and the two Trotter decompositions.

Figure 6

Time evolution of the inversion probability for neutrino ${\nu }_1$ using a time step $dt=16{\mu }^{-1}$. The dotted black curve is the exact evolution, the orange dashed curve is the Trotter decomposition with the best possible SN, and the solid blue curve uses the OO achievable using the full connectivity.

Figure 7

Decomposition of the two-body propagator $u_{ij}(dt)$ in Eq. (23) using native gates and setting $\alpha =-dt{J}_{ij}$.

Figure 8

Scheme for implementing a single Trotter step. We first apply the single-qubit gates corresponding to the one-body propagator $U_1(dt)$ and then the pair propagator using the optimal ordering implementing $U_2(dt)$.

Figure 9

Single Trotter step evolution for the inversion probability starting with the initial state $|{\mathrm{\Psi }}_0^{(4)}⟩=|0011⟩$. Panel (a) is for neutrinos ${\nu }_0$ and ${\nu }_3$, and panel (b) is for ${\nu }_1$ and ${\nu }_2$. The dotted line represents the ideal results using the exact propagator, while the solid black line indicates the ideal result obtained using one Trotter step. The results obtained from experiments on QSM H1-2 are represented by data points and error bars with and without caps corresponding to 68% and 90% confidence intervals, respectively.

Figure 10

Single Trotter step evolution for the inversion probability starting with the initial state $|{\mathrm{\Psi }}_0^{(8)}⟩=|00001111⟩$. Panel (a) shows result for the pair $({\nu }_0,{\nu }_7)$, panel (b) for $({\nu }_1,{\nu }_6)$, panel (c) for $({\nu }_2,{\nu }_5)$, and panel (d) for $({\nu }_3,{\nu }_4)$. Curves and data points follow the same convention used in Fig. 9.

Figure 11

Real-time evolution of a system with $N=4$ neutrinos for time step $dt=4{\mu }^{-1}$ and for a total time of $T=kdt$ with $k\in [1,10]$ using the QSM H1-2 trapped-ion device. The top panel shows neutrinos ${\nu }_0$ and ${\nu }_3$ and the bottom one ${\nu }_1$ and ${\nu }_2$. Dashed black lines are the ideal evolution, which, using this small time step, is almost the same as the Trotter approximated propagation.

Figure 12

Number of two-qubit operations $\mathcal{C}$, as a function of $N$, needed to evolve the system of a total time $T=40{\mu }^{-1}$ with an error ${ϵ}_1(T)\le \epsilon =0.15$. The solid blue line corresponds to the first order bound and the green band to the real scaling achievable using different pair ordering. The solid orange line refers to the second order bound instead, and the yellow band is the corresponding real achievable complexity. Using the optimal decomposition, for the system of $N=4$ neutrinos, the number of two-qubit universal operations needed is equal to $10N(N-1)/2=60$ and corresponds to 180 CNOT or $ZZ$ gates, which we actually use to obtain the last point in Fig. 11.