2qubits

About

Implementation of a Python simulation of a quantum system of a few qubits. A variational estimate of the ground state energy of the deuteron is simulated on a Qiskit quantum circuit. This is done in two different ways, one being the representation of each quantum harmonic oscillator state the nucleus can occupy as a qubit, the other being the relative mixing between different orbital angular momentum states of the system mapped into a quantum circuit. Secondly, some read-out error mitigation algorithms are implemented.

The present work is done in collaboration with TIFPA (Trento Institute for Fundamental Physics and Applications) research centre and the University of Trento as part of a quantum computing internship.

Deuteron: the deuterium nucleus

The deuteron is the bound state of a proton and a neutron. In order to estimate the binding energy of the system, there are two main approaches one could follow involving a simulation on a quantum computer.

EFT Hamiltonian

The first being the one mapping the creation and destruction operators of the deuteron into combinations of Pauli gates. The ladder operators $a_n^\dagger$ and $a_n$ create and annihilate a deuteron in the harmonic oscillator state $|n \rangle$. The Hamiltonian (from pionless effective field theory) is of the form

$$ H_N = \sum_{n,n'=0}^{N-1} \langle n' | T+V | n \rangle \,a_{n'}^\dagger a_n. $$

Thanks to the Jordan-Wigner transformation, the Hamiltonian is expressed in terms of Pauli matrices, allowing us to map it into a quantum circuit with $N$ qubits. By calculating the variational ground state energies for $H_1$, $H_2$ and $H_3$, one can extrapolate the infinite-basis binding energy using the harmonic oscillator variant of Lüscher formula.

Relative mixing between $^3 S_1$ and $^3 D_1$ partial waves

The second being the study of the relative mixing between the only two allowed states for the ground state deuteron: a isospin singlet, spin triplet state with $L=0$ ($^3 S_1$) and $L=2$ ($^3 D_1$). One can consider the two-level system built from the $S$ wave $\ket{\phi_S}$ and $D$ wave $\ket{\phi_D}$ such that the Hamiltonian is of the form:

$$ H = \left(\begin{matrix} \braket{\phi_S|H|\phi_S} & \braket{\phi_S|H|\phi_D} \\ \braket{\phi_D|H|\phi_S} & \braket{\phi_D|H|\phi_D}\end{matrix}\right). $$

By introducing a relative mixing angle $\xi$, one can calculate the optimal mixing between the two partial waves in order to obtain the lowest energy:

$$ \ket{\psi(\xi)} = \sin\left(\frac{\xi}{2}\right)\ket{\phi_S} + \cos\left(\frac{\xi}{2}\right) \ket{\phi_D}. $$

How to run

The code is a simple Jupyter notebook running Python. All you need to do is to make sure you installed the Python and Qiskit libraries contained in the first block of code of the file deuteron.ipynb. Make sure you run the blocks in order, since in some cases comparisons between previous and new results are performed.

Results

Here you can find an example of some results obtained with the code in deuteron.ipynb:

                               

References

The EFT approach is mostly taken from the work presented here. The relative mixing Hamiltonian can be found here instead. Read-out error correction procedures and rigorous confidence intervals estimates are explained in the appendices here and here.

Acknowledgements

• A. Roggero (professor and internship tutor)
• Diego Scantamburlo

License

The code here presented is released under version 3 of the GNU General Public License.