Quantum simulation of water-molecule bond angles using an NMR quantum computer

Determining the properties of molecules, such as bond angles and bond lengths, is an important part of materials science. Quantum simulators are expected to efficiently determine them by calculating their energy spectrum. Compared with the previous single-resonant quantum eigensolver adopted in simulating water molecule [Phys. Rev. Lett. 122, 090504 (2019)], we propose a multiresonant quantum eigensolver that can search for the energy spectrum of molecules in a parallel manner. Our approach can exponentially save the cost of finding the energy spectrum by a factor of $2^r$, where $r$ is the number of ancillary qubits. As an interesting demonstration, we have designed and implemented an experiment to determine the bond angle of water molecules on a four-qubit nuclear spin quantum processor. We experimentally estimate the ground and first excitation energies and their corresponding states for effective water Hamiltonians with different H-O bond angles. The experimental results clearly show that the molecule structure with $\angle \text{H-O}\sim {100}^{\circ }$ bond angle is most stable. We believe that our approach sheds light on further applications in solving quantum chemical problems.

Figure 1

Schematic diagram of the method based on the quantum multiresonant transitions. It contains a probe qubit, $r$ ancillary qubits, and $n$ work qubits. The probe qubit is used to observe the resonance phenomenon, and ancillary qubits can help us locate the specific energy level range where the resonance occurs. By measuring the probe qubit and the ancillary qubit, we can know if it is resonant and at which level it is resonant. In addition, the evolution is governed by the Hamiltonian in Eq. (1), which can drive the initial state ${|0\rangle |0\rangle }^{\otimes r}{|0\rangle }^{\otimes n}$ to $|1\rangle |k\rangle |v_j\rangle$, where $|v_j\rangle$ is the eigenstate of $A$ and $|k\rangle$ is the computational basis of ancillary register.

Figure 2

Schematics of the dynamical evolution of Hamiltonian design (1). (a) Block structure of Hamiltonian ${\mathcal{H}}_{\text{1}}$ for finding the eigenvalues ${\lambda }_j$ of $A$. White empties are full-zeros matrices, and $I$ and $E=E_2^{\otimes n}$ are, respectively, identity and full-ones matrices in the same dimension as the work system. $I_0{=1/2I+\left(\right|0\rangle \langle 0|)}^{\otimes n}$ and $A_j^{\prime }=A+(h_j-1/2)I_2^{\otimes n}=A+[3/2-({\omega }_{j-1}+{\omega }_x)]I_2^{\otimes n}$. ${\omega }_x$ is the parameter of sweep frequency. (b) Virtual Bloch sphere in two levels ${\left\{\right|0\rangle |0\rangle }^{\otimes r}{|0\rangle }^{\otimes n},|1\rangle |k\rangle |v_j\rangle \}(k=0,...,R-1;j=1,...,N)$. $H_{00}=3/2$, $H_{01}=H_{10}^{*}=c\langle v_j|E_2^{\otimes n}{|0\rangle }^{\otimes n}$, and $H_{11}={\lambda }_j+3/2-({\omega }_k+{\omega }_x)$. The orange arrow means $f_j=c|\langle v_j|E_2^{\otimes n}{|0\rangle }^{\otimes n}|$ and the green arrow represents the detuning $H_{11}-H_{00}={\omega }_k+{\omega }_x-{\lambda }_j$. When ${\omega }_x$ is tuned to the resonance point $H_{00}=H_{11}$, ${|0\rangle |0\rangle }^{\otimes r}{|0\rangle }^{\otimes n}$ will be rotated towards the south. Otherwise, no changes happen. (c) Energy structure of the coupled probe-work system in the subspace ${\left\{\right|0\rangle |0\rangle }^{\otimes r}{|0\rangle }^{\otimes n},|1\rangle |k\rangle |v_j\rangle ,j=1,...,N\}$ for each $|k\rangle$. The coupling $f_j$ will cause the transition from ${|0\rangle |0\rangle }^{\otimes r}{|0\rangle }^{\otimes n}$ to $|1\rangle |k\rangle |v_j\rangle$ when the resonance happens for each pair of $|k\rangle$ and $|v_j\rangle$. Measuring $P_{|1\rangle |k\rangle }$ will give the resonant peak in the range $[{\omega }_k,{\omega }_{k+1}]$.

Figure 3

NMR quantum circuit to realize quantum chemistry simulation. For mQRT, two qubits (${\mathrm{Q}}_1$ and ${\mathrm{Q}}_2$) are used as probe and register qubits and the other two (${\mathrm{Q}}_3$ and ${\mathrm{Q}}_4$) are work qubits. For iQRT, ${\mathrm{Q}}_1$ is used as a probe qubit and the rest are work qubits. It starts from ${|0\rangle }^{\otimes 4}$, which is prepared via the initialization method. $U_1$ is applied to realize the dynamics of $\mathcal{H}$. Last, we measure the state of the work qubits in the ancillary subspace $|10\rangle$ for mQRT, and in the ancillary subspace $|1\rangle$ for iQRT.

Figure 4

Molecular properties and the Hamiltonian parameters of the ${}^{13}\mathrm{C}$-labeled crotonic acid (${\mathrm{C}}_4{\mathrm{H}}_6{\mathrm{O}}_2$). Left: the molecular structure of crotonic acid and the coupling structure of the four-qubit quantum processor. Right: the encoding, controllability, readability, and coherent time $T_2$ of C1–C4. The diagonal and off-diagonal elements give the chemical shifts ${\nu }_i$ and the scalar coupling strengths $J_{ij}$ (in units of Hz), respectively.

Figure 5

The pulse sequence for the initialization step involves single-qubit rotations in the four-qubit NMR system. Each small block represents a single-qubit rotation, with the rotation angle and axis indicated in the circuit and legend. Here, $R\left(\theta \right)$ denotes the single-qubit rotation gate with the angle of $\theta$. $J$ is the coupling value between different spins with the subscripts omitted. $U(1/2J)=e^{-i\frac{\pi }{4}{\sigma }_z^j{\sigma }_z^k}$ is the free evolution between different spins.

Figure 6

Several examples of the final states in the experiments are shown. (a) and (c) respectively depict the density matrix of the final states obtained by quantum state tomography at the first and last scanning points, while (b) corresponds to the resonant point, where the final state exhibits a distinct characteristic compared to the initial state. Here, Hartree is the energy unit. The state number means the index of the computational basis from $|0000\rangle$ to $|1111\rangle$.

Figure 7

Experimental results of simulating ${\mathrm{H}}_2\mathrm{O}$ molecule with different bond angles. (a) The measured probabilities $P_{|1\rangle }$ as a function of the sweep frequency via mQRT. $P_{|1\rangle |0\rangle }$ and $P_{|1\rangle |1\rangle }$ are responsible for the first half and the back half of the frequency range, respectively. (b) The measured probabilities $P_{|1\rangle }$ around the resonance points via iQRT for the molecules with five bond angles $\angle =\{{90}^{\circ },{96}^{\circ },{100}^{\circ },{108}^{\circ },{114}^{\circ }\}$. The right-hand side of each figure shows the first-excited state spectrum for each bond angle, while the left-hand side displays the ground-state spectrum. The results presented here focus specifically on the resonance points. (c) The found ground-state and first-excited-state energies as a function of the bond angle in the ${\mathrm{H}}_2\mathrm{O}$ molecule. The curve is a theoretical numerical simulation, and the points are derived from the experiments. It is observed that the most stable bond angle is around ${100}^{\circ }$.