Collective neutrino oscillations on a quantum computer

Abstract

We calculate the energy levels of a system of neutrinos undergoing collective oscillations as functions of an effective coupling strength and radial distance from the neutrino source using the quantum Lanczos (QLanczos) algorithm implemented on IBM Q quantum computer hardware. Our calculations are based on the many-body neutrino interaction Hamiltonian introduced in Patwardhan et al. (Phys Rev D 99, https://doi.org/10.1103/PhysRevD.99.123013, 2019). We show that the system Hamiltonian can be separated into smaller blocks, which can be represented using fewer qubits than those needed to represent the entire system as one unit, thus reducing the noise in the implementation on quantum hardware. We also calculate transition probabilities of collective neutrino oscillations using a Trotterization method which is simplified before subsequent implementation on hardware. These calculations demonstrate that energy eigenvalues of a collective neutrino system and collective neutrino oscillations can both be computed on quantum hardware with certain simplification to within good agreement with exact results.

Appendices

Appendix A: QITE algorithm

In this appendix, we briefly discuss the quantum imaginary-time evolution (QITE) algorithm which provides the basis for the quantum Lanczos (QLanczos) algorithm that is used to calculate the energy spectrum of the collective neutrino system that we studied.

In the QITE algorithm, the non-unitary imaginary-time evolution operator ${\mathfrak {U}} =e^{- \beta H_\text {block}}$ for imaginary-time evolution of an initial state can be expressed as

$$\begin{aligned} |\Psi (\beta )\rangle =c_n (e^{-\Delta \tau H_\text {block}})^n|\Psi _0\rangle \end{aligned}$$

(A1)

after dividing the imaginary-time evolution into n small steps with $\Delta \tau =\frac{\beta }{n}$. We included a normalization constant $c_n=\frac{1}{\sqrt{\langle \Psi _0|{\mathfrak {U}}^2|\Psi _0\rangle }}$. The sth step of this imaginary-time evolution is approximated by real-time evolution in terms of a unitary operator as

$$\begin{aligned} |\Psi _s\rangle =\frac{c_s}{c_{s-1}}e^{-\Delta \tau H_\text {block}}|\Psi _{s-1}\rangle \approx e^{-i\Delta \tau A[s]}|\Psi _{s-1}\rangle \end{aligned}$$

(A2)

where $c_s$ is a normalization constant that can be calculated recursively starting with $c_0=1$. The unitary update operators can be written as

$$\begin{aligned} A[s]=\sum _{i_0,\dots ,i_{n_q-1}}a_I[s] \sigma _{I} ~. \end{aligned}$$

(A3)

where $I=\{i_0,\dots , i_{n_q-1}\}$, $n_q$ is the number of qubits for the reduced block Hamiltonian, and $\sigma \in \{X,Y,Z\}$ are Pauli operators. The coefficients $a_I[s]$ are calculated by solving the linear system of equations $({{\varvec{{\mathcal {S}}}}}+{{\varvec{{\mathcal {S}}}}}^T)\cdot {{\varvec{a}}}={{\varvec{b}}}$ up to order ${\mathcal {O}}(\Delta \tau ^2)$, where

$$\begin{aligned} {\mathcal {S}}_{I}=\langle \sigma _{I} \sigma _{I'} \rangle \ , \ \ b_{I}=-i\sqrt{\frac{c_{s-1}}{c_s}}\langle \sigma _{I} H_\text {block} \rangle \end{aligned}$$

(A4)

These expectation values are calculated with respect to the state in the previous QITE step, $|\Psi _{s-1}\rangle$. The exponentially growing size of the matrices ${{\varvec{b}}}$ and ${{\varvec{{\mathcal {S}}}}}$ challenges the scalability of the algorithm. In our case, certain properties of the Hamiltonian, such as real matrix elements, help us reduce the size of these matrices. Due to the large number of measurements required for the calculation of matrix elements on quantum hardware accessible through cloud services, for a timely completion of calculations some of our measurements were performed using a noisy simulator instead. This resulted in a reduction of overall error but did not affect results significantly, as was ascertained by a comparison of results between sample quantum hardware and simulated runs.

We point out that circuit bundling, which is sometimes used to minimize the size of jobs submitted for quantum hardware runs, is not an option in our case, because the outcome of each step is used in the following step. Moreover, the QITE algorithm requires single- and two-qubit gates at each step resulting in a quantum circuit depth exceeding the decoherence time in NISQ devices. We took steps to shorten the quantum circuits by adapting methods we introduced in our earlier work [17, 19].

In detail, we started by determining the number of QITE steps in the implementation of our algorithm. We selected the number of steps that yielded convergence of energy expectation values within $\epsilon =0.001$ of the ground state energy. We used this criterion to estimate the required number of steps numerically. Then, we ran experiments on a noisy simulator of the quantum device that produced the matrix elements for ${\varvec{S}}$ and ${\varvec{b}}$ at each step, and used them to solve classically the system of equations $({{\varvec{{\mathcal {S}}}}}+{{\varvec{{\mathcal {S}}}}}^T)\cdot {{\varvec{a}}}={{\varvec{b}}}$ obeyed by the coefficients $a_I[s]$ introduced in Eq. (A3). Next, to shorten the depth of the quantum circuit so it is manageable by NISQ devices, we employed the isometry function from the IBM Qiskit library which uses the algorithm developed in [33] where a given isometry is decomposed into single-qubit and Controlled-NOT (CNOT) gates with the aim of having the least number of CNOT gates. We used this isometry function to simplify the circuits of the state at the sth step,

$$\begin{aligned} |\Psi _s\rangle =e^{-i\Delta \tau A[s]}e^{-i\Delta \tau A[s-1]}\cdot \cdot \cdot e^{-i\Delta \tau A[1]}|\Psi _0\rangle ~. \end{aligned}$$

(A5)

It turns out that the resulting quantum circuit at each QITE step has the same gate components but with different rotation angles making the implementation of the QITE algorithm possible on NISQ devices. We ran these quantum circuits on IBM Q quantum hardware to obtain experimental energy expectation values.

Appendix B: transition probabilities calculated on different 7-qubit hardware

In this section, we present the collective neutrino oscillations transition probability results that we obtained from currently available 7-qubit IBM Quantum hardware. Fig. 6a, b compares the 3-neutrino (4-neutrino) oscillations transition probability values from $|\Psi _{\text {initial}}\rangle =|\nu _e \nu _e \nu _e\rangle$ ($|\Psi _{\text {initial}}\rangle =|\nu _e \nu _e \nu _e \nu _e\rangle$) to $|\Psi _{\text {final}}\rangle =|\nu _e \nu _e \nu _e\rangle$ ($|\Psi _{\text {final}}\rangle =|\nu _e \nu _e \nu _e \nu _e\rangle$). In each panel, the transition probabilities obtained from exact Trotterization, noiseless simulator, noisy simulator of the particular quantum hardware, and 7-qubit IBM Quantum hardware are plotted. We ran the experiments on IBM Quantum Casablanca, Jakarta, and Lagos (from top to bottom) devices. These devices have processor type of Falcon r5.11H and their gate maps can be seen in Fig. 5. In each of these runs we used $N_{\text {shots}}=8192$ and each experiment were run $N_{\text {runs}}=2$ times. For the experiments run in Casablanca and Jakarta we used qubits [2,1,3] ([2,1,3,5]) for 3-qubit (4-qubit) runs and for the experiments run in Lagos we used qubits [3,5,6] ([1,3,5,6]) for 3-qubit (4-qubit) runs. The experiments conducted on IBM Quantum Casablanca and Jakarta were run on 9/23/2021 and the experiments run on Lagos were run on 9/16/2021.

Fig. 5

The gate map of the qubits of IBM Quantum 7-qubit devices Casablanca, Jakarta, and Lagos

Table 1 Calibration data for three different 7-qubit IBM Quantum systems. IBM-reported calibration data for experiments conducted on systems Casablanca and Jakarta on 9/23/2021 and for Lagos on 9/16/2021. For Casablanca and Jakarta, we used qubit layouts [2,1,3] and [2,1,3,5], and for Lagos, we used [3,5,6] and [1,3,5,6] for 3- and 4-qubit runs, respectively. The values reported are $T_1$, $T_2$, readout assignment error ($e_{RO}$), single-qubit $\sqrt{X}$ gate error ($e_{SX}$), and 2-qubit CNOT gate error for qubit pairs used ($e_{2Q}$)

Table 2 Comparison of the mean values for three different 7-qubit IBM Quantum systems. The mean values are listed for $T_1$, $T_2$, $e_{RO}$, $e_{SX}$ and $e_{2Q}$ for 3- and 4-qubit values, respectively. The last two columns represent the error values calculated from the plots of quantum hardware collected data as seen in Fig. 6 from the experiments run on 7-qubit devices Casablanca, Jakarta, and Lagos

Fig. 6

The transition probability amplitude comparison for transition from (a) $|\text {initial}\rangle =|\nu _e \nu _e\nu _e\rangle$ to $|\text {final}\rangle =|\nu _e\nu _e\nu _e\rangle$, (b) $|\text {initial}\rangle =|\nu _e \nu _e\nu _e\nu _e\rangle$ to $|\text {final}\rangle =|\nu _e\nu _e\nu _e\nu _e\rangle$ as a function of time between exact time evolution calculations, noiseless simulated Trotterization and values obtained using Trotterization implemented on the three different 7-qubit IBM Q hardware (Casablanca, Jakarta, and Lagos, from top to bottom). The error mitigation method applied is readout error mitigation (ROEM). The parameters are chosen such that $\sin ^2\left( 2\theta \right) =0.10$, $\omega _p=p\omega _0$ and $\mu (r)/\omega _0=1.0$. The experiments were run $N_{\text {run}}=3$ times on 9/23/2021 for systems Casablanca and Jakarta and on 9/16/2021 for Lagos. The error bars represent $\pm \sigma$

In Table 1, we present the IBM-reported $T_1$, $T_2$, readout error ($e_{RO}$), SX gate error ($e_{SX}$), and two-qubit CNOT gate error ($e_{2Q}$) values for each qubit used on each device. These qubits were chosen such that the reported CNOT error rates for the chosen qubit pairs are minimum throughout the device. In Table 2, we report the mean values for the IBM-reported calibration values demonstrated in Table 1 and 3- and 4-qubit errors, $e_{\text {3-qubit}}$ and $e_{\text {4-qubit}}$ obtained from the plots. The errors from the plots were calculated by finding the standard deviation of the hardware data from the exact Trotterization data and then adding it all up for each data point. As a result, as seen in Fig. 6 there is not a significant difference between hardware data obtained from each device. 3-qubit results are better than the 4-qubit results since 3-qubit quantum circuit is much shorter than 4-qubit one. Even though, 4-qubit data demonstrates a significant hardware noise the collective neutrino oscillations are visible in both cases. For 3-qubit experiments run on different 7-qubit devices, some of the data points lie in error bar limits of the exact Trotterization data. Another interesting observation from these data is that noisy simulator does not capture all quantum hardware error since it does not include errors such as cross-talk between qubits.