Quantum Simulation of Resonant Transitions for Solving the Eigenproblem of an Effective Water Hamiltonian

It is difficult to calculate the energy levels and eigenstates of a large physical system on a classical computer because of the exponentially growing size of the Hilbert space. In this work, we experimentally demonstrate a quantum algorithm which could solve this problem via simulated resonant transitions. Using a four-qubit quantum simulator in which two qubits are used as ancillas for control and measurement, we obtain the energy spectrum of a 2-qubit low-energy effective Hamiltonian of the water molecule. The simulated transitions allow the state of the quantum simulator to transform and access large regions of the Hilbert space, including states that have no overlap with the initial state. Furthermore, we make use of this algorithm to efficiently prepare specific eigenstates on the simulator according to the measured eigenenergies.

Figure 1

Schematic diagram of the quantum algorithm. The bottom two lines denote register $R$ consisting of one ancillary qubit and an $n$-qubit system. The evolution governed by the Hamiltonian $\mathcal{H}$ in Eq. (1) can drive the register $R$ to resonate with the probe qubit.

Figure 2

(a) Molecular properties of ${}^{13}\mathrm{C}$-iodotrifluoroethylene (${\mathrm{C}}_2{\mathrm{F}}_3\mathrm{I}$). The chemical shifts ${\nu }_j$ and the scalar coupling constants $J_{jk}$ (in units of Hz) are given on and above the diagonal of the table, respectively. The spin-lattice relaxation time $T_1$ and spin-spin relaxation time $T_2$ are in the last two columns. (b) Quantum circuit to implement the algorithm. The logic gate sequence between parentheses performs the operator $e^{-i\mathcal{H}\tau /M}$, where $M$ denotes the number of time steps and $\mathcal{H}$ is the Hamiltonian of the simulator.

Figure 3

(a) Energy spectrum of the water molecule obtained using the quantum algorithm. The blue lines and red dots represent the theoretical and experimental probability $P$ of the probe qubit in the state $|0⟩$ as a function of frequency $\omega$. Here, ${\omega }_{a,b,c,d}=0.22$, 1.80, 1.82, 1.84. (b) Experimental NMR signal spectrum of ${}^{13}\mathrm{C}$ in the final state at ${\omega }_a=0.22$ and ${\omega }_b=1.80$ [both indicated in (a)].

Figure 4

Experimental state tomography of the quantum simulator: (a)–(d) respectively show the real parts of experimental reconstructed density matrices at frequencies ${\omega }_{a,b,c,d}$ indicated in Fig. 3. The rows and columns labeled 1–4 represent computational basis states from $|0100⟩$ to $|0111⟩$, and those labeled 5–8 represent basis states from $|1000⟩$ to $|1011⟩$. The subspaces ${|00⟩}_{12}$ and ${|11⟩}_{12}$ appear with negligible probabilities and are not shown. All imaginary parts of the density matrices are also negligible.