NMR Implementation of a Molecular Hydrogen Quantum Simulation with Adiabatic State Preparation

It is difficult to simulate quantum systems on classical computers, while quantum computers have been proved to be able to efficiently perform such kinds of simulations. We report an NMR implementation simulating the hydrogen molecule (${\mathrm{H}}_2$) in a minimal basis to obtain its ground-state energy. Using an iterative NMR interferometer to measure the phase shift, we achieve a 45-bit estimation of the energy value. The efficiency of the adiabatic state preparation is also experimentally tested with various configurations of the same molecule.

Figure 1

(a) General schematic circuit for calculating molecular energies. (b) Molecular structure of quantum register (${\mathrm{CHCl}}_3$). (c) Pulse sequence for the adiabatic process to prepare the ${}^{13}\mathrm{C}$ nucleus in the ground state $|\Psi ⟩$ of the molecular Hamiltonian $H$. (d) Pulse sequence to implement the controlled-$U_k$ operation, where $\theta =\arctan (\frac{2H(1,2)}{H(1,1)-H(2,2)})=0.226$, $\gamma =\frac{\pi }{2}-\theta =1.3458$, $\beta ={\beta }_k^0-{\varphi }_{k-1}^{\prime }$ (${\varphi }_{k-1}^{\prime }$ is calculated from the result of last iteration and ${\beta }_k^0=-\frac{8^{k-1}\tau }{2}\mathbf{(}H(1,1)+H(2,2)\mathbf{)}$), $\alpha =\frac{8^{k-1}\tau }{2}\sqrt{4H(1,2{)}^2+\mathbf{(}H(1,1)-H(2,2){\mathbf{)}}^2}$ and $d=\frac{\alpha }{\pi J_{wa}}$ for the $k$th iteration. Totally 15 iterations are performed in our experiment. The $U$ sequences are optimized as we chose a suitable $\tau =1.94112172562605$.

Figure 2

Fidelity of ASP as a function of evolution time $T$ for different nucleus distances $r$ in a single run. The solid lines represent numerical simulations while the squares and the triangle are data points. The triangle in (a) denotes the configuration of ASP for our molecular simulation. The fidelity approaches 1 as the evolution time grows according to the adiabatic theorem. The adiabatic evolution is divided into 100 steps in numerical simulations and 8 steps in the experiments.

Figure 3

(a) Measured energy values for 15 iterations. The solid blue line denotes the spline fit of experimental values (the circles) and the dash red line denotes the theoretical expectation. The experimental value approaches theoretical expectation exponentially. (b) Experimental ${}^1\mathrm{H}$ spectra. The reference spectrum ($A$) from initial state $|{\psi }_{\mathrm{in}}⟩$ and ($B$)–($D$) the spectra observed after iterations $k=0$, 1, and 14, which provide the relative phase information. The phase measured in each iteration is used in its next iteration.