Experimental Determination of Electronic States via Digitized Shortcut to Adiabaticity and Sequential Digitized Adiabaticity

A combination of digitized shortcut to adiabaticity (STA) and sequential digitized adiabaticity is implemented in a superconducting quantum device to determine the electronic states in two example systems, the ${\mathrm{H}}_2$ molecule and the topological Bernevig-Hughes-Zhang (BHZ) model. For ${\mathrm{H}}_2$, a short internuclear distance is chosen as a starting point, at which the ground and excited states are obtained via digitized STA. From this starting point, a sequence of internuclear distances is built. The eigenstates at each distance are sequentially determined from those at the previous distance via the digitized adiabaticity, leading to the potential energy landscapes of ${\mathrm{H}}_2$. The same approach is applied to the BHZ model and the valence and conduction bands are obtained with a high degree of accuracy along the $X$-$\mathrm{\Gamma }$-$X$ line cut of the first Brillouin zone. Furthermore, a numerical simulation of this method is performed to successfully extract the ground states of hydrogen chains with lengths of between three and six atoms.

Figure 1

A schematic diagram of the (a) continuous and (b) digitized adiabatic or STA processes.

Figure 2

The determination of the ground state of ${\mathrm{H}}_2$ via a digitized STA eigensolver for the internuclear distance at $R=0.05\text{Å}$. (a) A quantum circuit diagram of the $M$-step digitized STA process. (b) The evolution of the state fidelities (in comparison to the exact ground state) in four digitized STA experiments ($1\le M\le 4$). (c) The evolution of the state energies in four digitized STA experiments. The square, circle, triangle, and diamond denote the total number $M=1,2,3,4$ of digitized steps, respectively.

Figure 3

The fidelity of the experimental density matrix ${\rho }_{\exp }(t)$ in comparison to the theoretical simulation $|{\mathrm{\Psi }}_{\mathrm{theo}}(t)⟩⟨{\mathrm{\Psi }}_{\mathrm{theo}}(t)|$ for the experiment shown in Fig. 2. The open symbols represent the results without state purification. The solid symbols represent the results by selecting the pure state $|{\psi }_i(t)⟩$ with the largest population after the diagonalization of ${\rho }_{\exp }(t)$. The square, circle, triangle, and diamond denote the total number $M=1,2,3,4$ of digitized steps, respectively.

Figure 4

The determination of the first excited state of ${\mathrm{H}}_2$ via a digitized STA eigensolver for the internuclear distance at $R=0.05\text{Å}$. (a) A quantum circuit diagram of the $M$-step digitized STA process. (b) The evolution of the state fidelities in four digitized STA experiments ($1\le M\le 4$) in comparison with the exact result. (c) The evolution of the state energies in four digitized STA experiments. The square, circle, triangle, and diamond denote the total number $M=1,2,3,4$ of digitized steps, respectively.

Figure 5

The experimental determination of the potential energy landscapes for both the ground and excited states of ${\mathrm{H}}_2$ via the digitized STA and sequential digitized adiabatic eigensolvers. (a) The two eigenergies (${\epsilon }_{n=0,1}$) as functions of the internuclear distance $R$. The quantum circuit diagram of a sequential digitized adiabaticity is shown in the inset. In the large-distance range ($1.05\text{Å}\le R\le 1.65\text{Å}$), the experimental results of the state fidelities and eigenenergies are enlarged in (b) and (c). The symbols (circles and diamonds) represent the experimental results, while the solid lines represent the exact results from the theoretical calculation.

Figure 6

The BHZ-model experiment. (a),(b) The quantum circuit diagrams of digitized STA to determine the ground and excited eigenstates at the starting wave vectors, $k_{x_0}=\pm 0.1a^{-1},\pm 3.1a^{-1}$. (c) The quantum circuit diagram of sequential digitized adiabaticity for other wave vectors, $k_x\ne k_{x_0}$. (d) The experimental determination of the valence and conduction bands along the $X$-$\mathrm{\Gamma }$-$X$ line cut. The inset is an enlarged view of (d) around the Dirac point. The symbols (circles and diamonds) represent the experimental results, while the solid lines represent the exact results from the theoretical calculation.

Figure 7

The ground-state energy landscapes of the (a) three-, (b) four-, (c) five-, and (d) six-atom hydrogen chains, where $R$ is the internuclear distance between two adjacent atoms. The red circles are obtained from the numerical simulation of digitized STA and sequential digitized adiabaticity, while the black solid lines are the exact results.