Variational quantum eigensolver for frustrated quantum systems

Hybrid quantum-classical algorithms have been proposed as a potentially viable application of quantum computers. A particular example—the variational quantum eigensolver, or VQE—is designed to determine a global minimum in an energy landscape specified by a quantum Hamiltonian, which makes it appealing for the needs of quantum chemistry. Experimental realizations have been reported in recent years and theoretical estimates of its efficiency are a subject of intense effort. Here we consider the performance of the VQE technique for a Hubbard-like model describing a one-dimensional chain of fermions with competing nearest- and next-nearest-neighbor interactions. We find that recovering the VQE solution allows one to obtain the correlation function of the ground state consistent with the exact result. We also study the barren plateau phenomenon for the Hamiltonian in question and find that the severity of this effect depends on the encoding of fermions to qubits. Our results are consistent with the current knowledge about the barren plateaus in quantum optimization.

Figure 1

Convergence of the VQE energy to the true ground state energy for the model of Eq. (3) versus the number of layers, on condition $V_1=2t,V_2=t$. Cases of $n\le 4$ qubits are not shown as they converged to exact solution within two layers.

Figure 2

Density-density correlation function between spatially separated lattice sites. Solid line denotes the exact solution as obtained by virtue of exact diagonalization of the Hamiltonian (3); dashed lines stand for the VQE solutions with different ansatz depths. Here $V_1=2t,V_2=t$.

Figure 3

Relative error of the correlation function as obtained with the VQE method, $C_{VQE}$, with respect to the exact solution, $C_{\mathrm{exact}}$, as a function of the number of layers, $L$.

Figure 4

Barren plateau effect for the Hamiltonian of Eq. (3) with $V_1=t$ and $V_2=0$ (dashed lines), as well as $V_1=2t$ and $V_2=t$ (solid lines) versus the number of qubits as realized by virtue of Jordan-Wigner mapping. Diamonds: four qubits; circles: six qubits; triangles: eight qubits; squares: 10 qubits. Markers in Figs. 5 and 7 follow the same convention.

Figure 5

Barren plateau effect for the Hamiltonian of Eq. (3) with $V_1=2t$ and $V_2=t$ depending on the number of qubits as performed by means of Bravyi-Kitaev transformation.

Figure 6

A two-qubit entangler gate consisting of a two-qubit rotation parametrized by a two $\widetilde{{\theta }_3}$, sandwiched between one-qubit rotations, parametrized by $\widetilde{{\theta }_1}$, $\widetilde{{\theta }_2}$, $\widetilde{{\theta }_4}$, and $\widetilde{{\theta }_5}$.

Figure 7

Barren plateau effect for the transverse field Ising model of Eq. (4) away from criticality with $h=0.1$ (dashed lines) and at the critical point $h=1$ (solid lines).