Quantum computational method of finding the ground-state energy and expectation values

We propose a quantum computational way of obtaining a ground-state energy and expectation values of observables of interacting Hamiltonians. It is based on the combination of the adiabatic quantum evolution to project a ground state of a noninteracting Hamiltonian onto a ground state of an interacting Hamiltonian and the phase estimation algorithm to retrieve the ground-state energy. The expectation value of an observable for the ground state is obtained with the help of the Hellmann-Feynman theorem. As an illustration of our method, we consider a displaced harmonic oscillator, a quartic anharmonic oscillator, and a potential scattering model. The results obtained by this method are in good agreement with the known results.

Figure 1

(Color online) $\text{Re}\left\{a_0\left(t\right)\right\}$ as a function of $t/T_0$ for (a) $\lambda =0$ and $E_c=0$, (b) $\lambda =0.9$ and $E_c=0$, (c) $\lambda =0.9$ and $E_c=0.25$, (d) $\lambda =\sqrt{2}$ and $E_c=0$, and (e) $\lambda =\sqrt{2}$ and $E_c=1.0$. Here $N=4$ and $T_0=2\pi \hslash /W_0=4\pi$ with $W_0=\hslash \omega /2$.

Figure 2

(Color online) (a) Probability $p_n\left(t\right)={\left|a_n\left(t\right)\right|}^2$ of qubits in the computational basis $|n⟩$, and (b) the instantaneous ground-state energy $E_0\left(t\right)$ in the unit of $\hslash \omega$ as a function of the dimensionless time $t/T_R$ for $\lambda =\sqrt{6}$. In (b), $f\left(t\right)$ is an adiabatically switching-on function. (c) At $t=T_R$, $p_n$ obeys the Poisson distribution of a coherent state.

Figure 3

(Color online) Ground state energy $E_0\left(\alpha \right)$ in the unit of $\hslash \omega$ as a function of $\alpha$ for (a) $\lambda =0$ and (b) $\lambda =1$. The points are numerical results. The line in (a) is the plot of $f\left(\alpha \right)=\frac{1}{2}\left(\alpha +1\right)$. In (b), the line is the plot of $g\left(\alpha \right)=\frac{3}{2}\alpha$.

Figure 4

(Color online) For a quartic anharmonic oscillator, (a) $E_0\left(\lambda \right)$ in the unit of $\hslash \omega$ and (b) $\Delta x^2$ as a function of $\lambda$. Here $N=6$ and $T_R=15T_0$.

Figure 5

(Color online) (a) Energy levels $E_n$ (in arbitrary units) and fidelity $F_n$ as a function of $g$. Here $N=6$ and $\Delta =10/64$.