Toward scalable quantum computations of atomic nuclei

Abstract

We solve the nuclear two-body and three-body bound states via quantum simulations of pionless effective field theory on a lattice in position space. While the employed lattice remains small, the usage of local Hamiltonians including two- and three-body forces ensures that the number of Pauli terms scales linearly with increasing numbers of lattice sites. We use an adaptive ansatz grown from unitary coupled cluster theory to parametrize the ground states of the deuteron and $^3$He, compute their corresponding energies, and analyze the scaling of the required computational resources. Our quantum simulations reproduce exact benchmarks for $^2$H and $^3$He within 100 keV, requiring at most 30 layers in the ansatz and thus resulting in modest circuit depths. Additionally, we find the number of shots required to reach a given precision scales linearly in the lattice size and more mildly in the system size. Based on the agreement with exact benchmarks and mild scaling, we conclude that this can be an efficient, scalable approach for quantum computations of nuclear ground states, particularly to prepare initial states for quantum phase estimation or other filtering algorithms.

Figures (5)

• Figure 1: Scaling of the number of samples $N_{\rm shots}$ required to reach a given precision in the lattice extent $L$ and the system size $A$. Going from $\varepsilon = 1~\mathrm{MeV}$ (filled circles) to 0.1 MeV (open circles) requires two orders of magnitude more samples. For $A=2,3$ we are able to make simplifications such that ADAPT-VQE computations only require $n_q = 2 L^3, 3L^3$ qubits, respectively, rather than the general $n_q = 4L^3$.
• Figure 2: Ground state energies and state fidelities obtained using ADAPT-VQE to solve the deuteron. Results are shown as a function of the number of optimization steps $N_{\rm opt}$. We optimize the parameters $\theta_{\alpha\beta}$ for a selection of 10 operators $\hat{A}_{\alpha\beta}$ in each epoch $\alpha$, where the start of an epoch and the selection of a new set of operators is indicated by the open points. In each epoch, we use at most 100 optimization iterations to identify an optimal set of parameters. The exact ground state energy and the threefold degenerate first excited-state energy for our Hamiltonian, computed via exact diagonalization, are indicated as dashed black and dotted gray lines, respectively.
• Figure 3: Same as Fig. \ref{['fig:h2_no_noise']}, but for $^{3}\mathrm{He}$. State fidelities were only computed at the end of each epoch, so the dashed lines in the right panel are only intended to guide the eye.
• Figure 4: Differences of ground state energies of the deuteron with the exact ground state energy $E_\mathrm{FCI}$ of the deuteron as computed in ADAPT-VQE simulations including measurement noise for $N_{\rm shots} = 1000$ (blue), 10000 (red) and without measurement noise (black). Results are shown as a function of the number of optimization steps $N_{\rm opt}$. The open points with error bars are results at the end of each optimization epoch, with uncertainties estimated as $E_\mathrm{corr} / \sqrt{N_{\rm shots}}$ based on the correlation energy and the number of shots. For clarity, we only show 10% of the evaluated energies obtained during the optimization as small, filled-in points. The estimated uncertainty on the energy during the optimization is indicated by the band.
• Figure 5: Number of $T$ (top) and sequential CNOT (bottom) gates in the circuit for our ADAPT-VQE ansatz as the number of exponentials $N_{\rm e}$ is increased for the deuteron and $^3$He. As the number of exponentials is increased, the ansatz grows more complicated and the circuit grows due to the added operators. At the same time, the error to the exact ground state energy ($E-E_\mathrm{FCI}$ plotted on the $x$ axis) is systematically decreased. We only show points with $N_{\rm e} \geq 1$, where the number of $T$ and CNOT gates is larger than zero.