Speaker
Description
Quantum computation has shown great potential for studying many-body systems, where computational challenges become substantial due to the exponential growth of Hilbert spaces with the number of particles. Such approaches have also been explored in nuclear physics. In these frameworks, wavefunctions obtained from quantum simulation methods can be used to compute expectation values of one- and two-body operators.
In this work, we propose applying the Hadamard test to evaluate expectation values of a non-unitary operator within the nuclear shell model framework. Specifically, we calculate the quadrupole moment of ${}^{6}$He for the first excited $2^+$ state, where the wavefunction is obtained using the variational quantum eigensolver (VQE).
Since the wavefunction is expressed in the M-scheme basis, the quadrupole operator must also be represented in the same scheme. This is achieved by transforming the one-body transition density (OBTD) operator from the J-scheme to the M-scheme, which introduces an additional coupling term. The operator can then be written in second quantization form as
\begin{equation}
\hat{O} = \sum_{i,j} \langle \phi_i \mid O \mid \phi_j \rangle a_i^\dagger a_j,
\end{equation}
where $\mid \phi_j \rangle$ are single-particle states. Using the Jordan–Wigner transformation, the creation and annihilation operators are mapped to Pauli strings, yielding
\begin{equation}
\hat{Q} = 0.6062Z_0 - 0.6062Z_1 - 0.8573X_1X_2 - 0.8573Y_1Y_2,
\end{equation}
indicating that the system is encoded on three qubits.
The Hadamard test is then employed to evaluate the expectation value by decomposing the operator into its unitary components and measuring each term separately. The measurement is performed on an ancilla qubit, where the probabilities of obtaining the $0$ and $1$ outcomes, $p(0)$ and $p(1)$, are used to compute the expectation value as their difference. Finally, the reduced matrix element of the quadrupole operator is evaluated and scaled appropriately to obtain the quadrupole moment of ${}^{6}$He. This approach will be extended to other lighter nuclei as well.