Rovibrational energy levels of H$_2$O by quantum computing

Abstract

We calculate rovibrational energy levels of H$_2$O using a trapped-ion quantum computer. We first derive the qubit form of Watson's Hamiltonian, including the rovibrational coupling terms. In a second step, we employ a variant of the quantum-selected configuration-interaction method to calculate rovibrational energy levels. A truncated form of the qubit Hamiltonian is used to generate correlated rovibrational wave functions on the quantum computer by time evolution, and a basis set is selected by sampling from the measured probability distribution. The rovibrational energy levels are obtained by constructing a Hamiltonian matrix using the selected basis set, and diagonalizing the matrix using a classical computer. We show that an accuracy of a few cm$^{-1}$ can be achieved for low-lying rovibrational energy levels.

Figures (8)

• Figure 1: (a) Number of terms $L_{\text{q}}$ in the qubit Hamiltonian for $v_{\rm max} =1$ ($\bullet$) and $v_{\rm max}=3$ ($\blacktriangle$). The number of qubits $N_{\rm q}^{\rm rot}$ used to represent the rotational part of the wave function is indicated above the plot. The total number of qubits is $N_{\rm q}=N_{\rm q}^{\rm rot}+N_{\rm q}^{\rm vib}=N_{\rm q}^{\rm rot}+3$ for $v_{\rm max}=1$ and $N_{\rm q} = N_{\rm q}^{\rm rot}+6$ for $v_{\rm max}=3$. The solid and dotted lines represent power-law fits of the form $L_{\text{q}}=c_{v_{\rm max}}J^{\kappa_{v_{\rm max}}}$, with $(c_1,\kappa_1)=(167.7,1.08)$ and $(c_3,\kappa_3)=(474,1.07)$. (b) Absolute values $|h_{\bm{k}}|$ of the expansion coefficients in the qubit Hamiltonian [see Eq. \ref{['Eq:qubitHamiltonian']}], for $J=1$. The coefficients $h_{\bm{k}}$ are sorted according to size and the number of qubits the Pauli operator $P_{\bm{k}}$ operates on.
• Figure 2: (a) Rovibrational energy levels $\Delta E$ relative to the rovibrational ground state for $0\le J \le 6$ obtained using the full rovibrational Hamiltonian defined in Eq. \ref{['Eq:Hdef']}. The maximum vibrational quantum number is $v_{\rm max}=3$. Energy levels of proton singlet states are drawn by solid lines and triplet states are drawn by dotted lines. (b) Energy spectrum in the range $3700\text{ cm$^{-1}$}\le\Delta E\le 4000\text{ cm$^{-1}$}$ for $J=0$. We compare the energy levels obtained using the Hamiltonians $H_{\rm RR} + H_{\rm HO}$ (first column), $H_{\rm RR} + H_{\rm HO} + V_{\rm anharm}$ (second column), $H_{\rm RR} + H_{\rm HO} + V_{\rm anharm} + H_{\rm vibCor}$ (third column), and $H$ (fourth column). See Eqs. \ref{['Eq:Hdef']}--\ref{['Eq:Hrovibdef']} for definitions of the different Hamiltonians. (c) Energy spectrum in the range $3800\text{ cm$^{-1}$}\le\Delta E\le 4300\text{ cm$^{-1}$}$ for $J=3$.
• Figure 3: Distributions of basis states $|\langle v_1v_2v_3|\psi(\tau,N_{\rm ST})\rangle|^2$ for (a) $N_{\rm ST}=3$, $\tau=0.24$ fs, and (b) $N_{\rm ST}=1$, $\tau=1.5$ fs. The exact, noise-free distribution is shown with red, filled circles, and the distribution estimated using Reimei ($N_{\text{shot}}=10^3$) is shown with blue squares. The states having an incorrect value of the parity $(-1)^{v_3}$ are displayed by open squares. The populations $|c_{v_1v_2v_3}^0|^2$ in the vibrational ground state are shown by black stars as a reference.
• Figure 4: Energy levels estimates $\delta E_{v_1v_2v_3}$ for $v_1v_2v_3$ equal to (a) 000, (b) 010, (c) 020, (d) 100, and (e) 001, as a function of the number of basis functions in the basis set. $J=0$, $v_{\rm max}=3$ (corresponding to a number of qubits $N_{\rm q} =6$). The values of $\delta E$ obtained by sampling of the Suzuki-Trotter distribution \ref{['Eq:pmitDef']} are shown with filled red circles, those obtained by sampling the mitigated distribution \ref{['Eq:pmitDef']} obtained using Reimei are shown with blue, filled squares. The time step size is fixed to $\tau=0.24$ fs [same as in Fig. \ref{['Fig3']}(a)], and the number of Suzuki-Trotter steps varies from $N_{\rm ST} =0$ to $N_{\rm ST} =7$. The value of $N_{\rm ST}$ is indicated inside the curve symbols. The energies at $N_{\rm ST} =0$ (meaning that the basis set contains only the initial state $|v_1v_2v_3\rangle$) are much higher than the largest energies shown in the panels. We have $\delta E_{000}= 22.5$ cm$^{-1}$, $\delta E_{010}= 1680.2$ cm$^{-1}$, $\delta E_{020}= 3241.2$ cm$^{-1}$, $\delta E_{100}= 4188.6$ cm$^{-1}$, and $\delta E_{001}= 4210.6$ cm$^{-1}$ for $N_{\rm ST} =0$ ($|\Omega|=1$; same values for the "No noise", "Reimei" and "Optimal" curves). The exact eigenenergies obtained by diagonalization of $\bm{H}$ are shown with horizontal, dotted lines, the diamond symbol indicates $\delta E$ obtained by sampling from the perturbation-theory distribution \ref{['Eq:pPTDef']}, and the $\delta E$'s obtained by the optimal method are shown with purple triangles.
• Figure 5: Energy levels estimates $\delta E_{v_1v_2v_3}$ for $v_1v_2v_3$ equal to (a) 000, (b) 010, (c) 020, (d) 100, and (e) 001, as a function of the number of basis functions, for $J=0$, $v_{\rm max}=7$ ($N_{\rm q} =9$). The same curve styles as in Fig. \ref{['Fig4']} are employed. The number of Suzuki-Trotter steps is fixed to $N_{\rm ST} =1$ and the time step $\tau=n\tau_0$ with $\tau_0=0.48$ fs and $n=0, 1, \ldots, 5$ (indicated inside the curve symbols). The energies at $n =0$ (only one basis functions) are not shown, and take the values $\delta E_{000}(n=0)= 23.3$ cm$^{-1}$, $\delta E_{010}(n=0)= 1680.0$ cm$^{-1}$, $\delta E_{020}(n=0)=3241.9$ cm$^{-1}$, $\delta E_{100}(n=0)=4189.4$ cm$^{-1}$, and $\delta E_{001}(n=0)=4211.4$ cm$^{-1}$ (same values for the "No noise", "Reimei" and "Optimal" curves).