Quantum Computing the Deuteron#

Author: Jingyi Li

In this introduction we will briefly discuss the 2018 paper, Cloud Quantum Computing of an Atomic Nucleus.

From the paper we can read that the authors employed the projection of hamiltonian operators onto quantum qubits (X,Y,Z) to construct a quantum computatable hamiltonian by the Pionless EFT, a systematically improvable and model-independent approach to nuclear interactions in a regime where the momentum scale \(Q\) of the interesting physics. The important part of this paper is how the deuteron creation and annihilation operator is mapped onto qubits using Jordan-Wigner transformation. Qubits can be used by quantum computers for operations based on Pauli matrices (denoted as \(X_{q}, Y_{q}\), and \(Z_{q}\) on qubit \(q\) ).

Defining the Hamiltonian#

The deuteron Hamiltonian is

(1)#\[\begin{equation} H_{N}=\sum_{n, n^{\prime}=0}^{N-1}\left\langle n^{\prime}|(T+V)| n\right\rangle a_{n^{\prime}}^{\dagger} a_{n} . \end{equation}\]

The operators \(a_{n}^{\dagger}\) and \(a_{n}\) here are the creation and annihilation operators. A deuteron is created or annihilated in the harmonic-oscillator \(s\)-wave state \(|n\rangle\). The matrix elements of the kinetic and potential energy are represented by:

(2)#\[\begin{equation} \begin{aligned} \left\langle n^{\prime}|T| n\right\rangle=& \frac{\hbar \omega}{2}\left[(2 n+3 / 2) \delta_{n}^{n^{\prime}}-\sqrt{n(n+1 / 2)} \delta_{n}^{n^{\prime}+1}\right.\\ &\left.-\sqrt{(n+1)(n+3 / 2)} \delta_{n}^{n^{\prime}-1}\right] \\ \left\langle n^{\prime}|V| n\right\rangle=& V_{0} \delta_{n}^{0} \delta_{n}^{n^{\prime}} \end{aligned} \end{equation}\]

Here, \(V_{0}=-5.68658111 \mathrm{MeV}\), and \(n, n^{\prime}=0,1, \ldots N-1\), for a basis of dimension \(N\). We also set \(\hbar \omega=7 \mathrm{MeV}\).

Mapping the deuteron onto qubits#

Quantum computers manipulate qubits by operations based on Pauli matrices (denoted as \(X_{q}, Y_{q}\), and \(Z_{q}\) on qubit \(q\) ). The deuteron creation and annihilation operators can be mapped onto Pauli matrices via the Jordan-Wigner transformation, defined as,

(3)#\[\begin{equation} \begin{aligned} &a_{n}^{\dagger} \rightarrow \frac{1}{2}\left[\prod_{j=0}^{n-1}-Z_{j}\right]\left(X_{n}-i Y_{n}\right) \\ &a_{n} \rightarrow \frac{1}{2}\left[\prod_{j=0}^{n-1}-Z_{j}\right]\left(X_{n}+i Y_{n}\right) . \end{aligned} \end{equation}\]

Note that other mappings can be used, with the various mappings having benefits and drawbacks that may apply to your problem of interest.

A spin up \(|\uparrow\rangle\) (down \(|\downarrow\rangle\) ) on qubit \(n\) corresponds to zero (one) deuteron in the state \(|n\rangle\). For \(N=2,3\) we have the components of the Hamiltonian (all numbers are in units of \(\mathrm{MeV}\))

(4)#\[\begin{equation} \begin{aligned} H_{2}=& 5.906709 I+0.218291 Z_{0}-6.125 Z_{1} \\ &-2.143304\left(X_{0} X_{1}+Y_{0} Y_{1}\right) \end{aligned} \end{equation}\]
(5)#\[\begin{equation} H_{3}=H_{2}+9.625\left(I-Z_{2}\right)-3.913119\left(X_{1} X_{2}+Y_{1} Y_{2}\right) \end{equation}\]