Parametrized quantum circuit for weight-adjustable quantum loop gas

Motivated by the recent success of realizing the topologically ordered ground state of the exactly solvable toric code model by a quantum circuit on the real quantum device [Satzinger et al., Science 374, 1237 (2021)], here we propose a parametrized quantum circuit (PQC) with the same real-device-performable optimal structure to represent quantum loop gas states with adjustably weighted loop configurations. Combining such a PQC with the variational quantum eigensolver, we obtain the accurate quantum circuit representation for the toric code model in an external magnetic field with any field strength, where the system is not exactly solvable. The topological quantum phase transition in this system is further observed in the optimized circuits by measuring the magnetization and topological entanglement entropy.

Figure 1

(a) Schematic figure of the toric code model in an external magnetic field $B$ applied along the $z$ direction on the $6\times 5$ lattice under OBCs, where qubits (indicated by circles) are located on each bond of the lattice. Here, the blue cross connecting four blue circles indicates an $A_s$ term and the green square connecting four green circles denotes a $B_p$ term. Note that $A_s$ terms are also present at the corners and edges of the lattice under OBCs, thus involving only two and three qubits, respectively. The loops (indicated by red) are formed by local qubit states ${|1\rangle }_i$, while qubits not along any loops are ${|0\rangle }_i$. (b) Schematic figure of the Ansatz state $\left|\mathrm{\Psi }\right(\mathit{\theta })\rangle$ generated by a parametrized loop gas circuit. In these two figures, the shade intensity of red for loops represents a weight of the corresponding loops.

Figure 2

Ground state energy $E_0$ obtained by the VQE calculations as a function of $x$ for $H_{\text{TCM}}\left(x\right)$ on different clusters, $L_x\times L_y=3\times 3$, $4\times 3$, and $4\times 4$ (indicated by blue circles, green squares, and red triangles, respectively), which correspond to $N\left(N_p\right)=12\left(4\right)$, $17\left(6\right)$, and $24\left(9\right)$, respectively (also see the insets of Fig. 4). For comparison, the ground-state energy $E_{\mathrm{ED}}$ obtained by the ED method are also plotted by solid lines with the same colors. The inset shows the corresponding energy difference per qubit, $\mathrm{\Delta }{e}_0\equiv (E_0-E_{\mathrm{ED}})/N$, indicated by the same symbols. The dashed line displays the extrapolated values of $\mathrm{\Delta }{e}_0$ to $N\to \infty$ estimated by fitting $\mathrm{\Delta }{e}_0$ for the three different clusters with a second-order polynomial of $1/N$.

Figure 3

Same as Fig. 2 but for the ground-state magnetization $\langle m_z\rangle$.

Figure 4

Same as Fig. 2 but for the ground-state TEE $S_{\mathrm{topo}}$. The insets show the subsystems with the tripartition to extract $S_{\mathrm{topo}}$ for the corresponding systems.