Quantum implementation of the unitary coupled cluster for simulating molecular electronic structure

In classical computational chemistry, the coupled-cluster ansatz is one of the most commonly used ab initio methods, which is critically limited by its nonunitary nature. The unitary modification as an ideal solution to the problem is, however, extremely inefficient in classical conventional computation. Here, we provide experimental evidence that indeed the unitary version of the coupled-cluster ansatz can be reliably performed in a physical quantum system, a trapped-ion system. We perform a simulation on the electronic structure of a molecular ion (${\mathrm{HeH}}^{+}$), where the ground-state energy surface curve is probed, the energies of the excited states are studied, and bond dissociation is simulated nonperturbatively. Our simulation takes advantages from quantum computation to overcome the intrinsic limitations in classical computation, and our experimental results indicate that the method is promising for preparing molecular ground states for quantum simulations.

Figure 1

(a) The conceptual procedure and (b)–(f) the experimental realization of the UCC simulation to find the ground state of the target molecule. (b) The Hartree-Fock basis states of the target molecule, ${\mathrm{HeH}}^{+}$. (c) The mapping of the basis states on the energy levels of ${}^{171}\mathrm{Yb}^{+}$ including cluster amplitudes $t_{11}, t_{12}$, and $t_2$, which are controlled by the duration of microwave pulses. (d) The microwave pulse sequence for the preparation of the UCC ansatz. The effective time evolution operator $e^{T-T^{†}}$ is expanded by the Suzuki-Trotter scheme (see SM, Sec. C: Quantum implementation of the UCC ansatz [31]). (e) The measurement of energy $〈H〉$ of the UCC ansatz given cluster amplitudes. (f) A classical minimum search algorithm (Nelder-Mead) is applied to determine the cluster amplitudes for the next UCC ansatz and the final ground state.

Figure 2

The search process for the minimum energy at $R=1.7\text{a.u.}$ assisted by the classical Nelder-Mead algorithm with the UCC ansatz. The measured energy $〈H〉$ (dots) and the fidelity of the prepared state (bars) to the ideal ground state depending on the number of iterations (a) with the full six parameters and (b) with two parameters. For both cases, the algorithm converges to the ground-state energy obtained by exact diagonalization with decent fidelity of the state. Red dots show the successful steps that contribute to the convergence. (c) The side view and (d) the bottom view of the searching process with two parameters for the successful steps. Atomic units (a.u.) are used for the energy through all the figures.

Figure 3

The ground-state energy of ${\mathrm{HeH}}^{+}$ depending on the internucleus distance $R$. The error bars of the experimental data mainly come from the quantum projection noise of 1000 repetitions for each term of the Hamiltonian (2).

Figure 4

Applications of the UCC simulation. (a) The ground-state energy of ${\mathrm{HeH}}^{+}$ subject to a static electric field along the nuclei axis for different strengths. (b) A comparison between the UCC quantum simulation and the perturbation theory at a given internucleus distance $R=1.7\text{a.u.}$ (c) The search process for energies of the excited states of $H$ by finding the ground-state energy of the Hamiltonian ${(H-\lambda )}^2$ by scanning the values of $\lambda$. When $\lambda$ is the energy of an excited state, the experimental minimum value of $〈{(H-\lambda )}^2〉$ tends to be zero. We can also calculate the excited-state energy from other nonzero values of $〈{(H-\lambda )}^2〉$. If $\lambda$ is on the left (right) side of the excited energy, the positive (negative) solution of $E_{\mathrm{meas}}=\langle {(H-\lambda )}^2\rangle$ provides the excited energy.