Critical behavior of the Ising model by preparing the thermal state on a quantum computer

We simulate the critical behavior of the Ising model utilizing a thermal state prepared using quantum computing techniques. The preparation of the thermal state is based on the variational quantum imaginary time evolution (QITE) algorithm. The initial state of QITE is prepared as a classical product state, and we propose a systematic method to design the variational ansatz for QITE. We calculate the specific heat and susceptibility of the long-range interacting Ising model and observe indications of the Ising criticality on a small lattice size. We find the results derived by the quantum algorithm are well consistent with the ones from exact diagonalization, in both the neighborhood of the critical temperature and the low-temperature region.

Figure 1

An illustration of the measurement process. The specific heat $C_v$ and susceptibility $\chi$ of the Ising model can be calculated by the measurement in computational basis on quantum computers.

Figure 2

Example of circuits for the imaginary time evolution of Ising systems. The basic building blocks of the circuits are defined as $U_{ZY}\left(\theta \right)\equiv e^{-i\theta ZY},U_{YZ}\left(\theta \right)\equiv e^{-i\theta YZ}$. (a) The quantum circuit in the QITE-measure algorithm to carry out the imaginary time evolution $e^{\tau ZZ}|++\rangle$. $\mathrm{\Delta }\tau$ is the length of one time slice. (b) The variational ansatz converted from the QITE-measure circuit. $L$ is the number of circuit layers; ${\theta }_i,{\theta }_i^{\prime },i\in [1,L]$ are free variational parameters. (c) The variational ansatz for the nearest-neighbor one-dimensional Ising chain under the periodic boundary condition. Each layer consists of one layer of $ZY$-Pauli exponentials and one layer of $YZ$-Pauli exponentials, as shown in the dashed box. The figure shows the case of one layer.

Figure 3

Specific heat (left column) and susceptibility (right column) as a function of $K$ in 2D (upper row) and 3D (lower row) nearest-neighbor Ising models $(\alpha =\infty$). ED represents results from exact diagonalization. We see that the results from the noiseless quantum simulation are close to exact diagonalization, especially in the region far from the critical point. The black dashed lines in the four panels are the exact critical temperatures $K_c$ of the corresponding dimension in the infinite volume limit. The solid gray line in the upper left panel shows the peak movement as the system size is enlarged, which is fitted inspired by finite-size scaling (FSS). As the lattice size increases, the peaks of the specific heat and the leaps of the susceptibility are more obvious, and the transition points approach the exact critical point.

Figure 4

Specific heat as a function of $K$ in the 2D long-range interacting Ising model with interaction range $\alpha =1,2,3,\infty$, where smaller $\alpha$ indicates larger interaction range. The system size is $\left|\mathrm{\Lambda }\right|=3\times 3$. ED represents results from exact diagonalization. We see that for various $\alpha$ and $K$, the QITE results and the ED results are consistent. The black dashed line denotes the exact critical point of the 2D NNIM in the infinite volume limit. As $\alpha$ decreases, the peak of the specific heat curve shifts left, indicating that the effective dimension is raised for a larger interaction range.

Figure 5

The average absolute error of specific heat as a function of variational ansatz layers. We utilize the 2D nearest-neighbor Ising model with two volumes $\left|\mathrm{\Lambda }\right|=N_d\times N_d=3\times 3,4\times 4$. The limitation of the variational ansatz can be well controlled by increasing the layers. As the number of layers is larger than some transition layers $L^{*}$, the error has no obvious change. Theoretically, we can predict $L^{*}\in [1.5,3]$ for $N_d=3$ and $L^{*}\in [2,4]$ for $N_d=4$.

Figure 6

An illustration to explain the transition layer in Fig. 5. We plot the 2D nearest-neighbor Ising model with two volumes $\left|\mathrm{\Lambda }\right|=3\times 3,4\times 4$. Circles with solid edges are real spins. Circles with dashed edges are virtual spins from the periodic boundary condition. The spin at the origin (lower left corner) is denoted by the orange circle, which has three corresponding dashed-edge circles at three other corners. The yellow arrows denote the shortest path connecting the most remote spin to the spin at the origin. One layer of the variational ansatz generates the correlation between two spins at least 1 unit length apart. With at most $D{N}_d/2$ layers, the variational ansatz can connect all the spins in the graph.

Figure 7

The quantum circuit equivalent to the QITE-measure circuit in Fig. 2. Here $U_{ZY}\left(\theta \right)\equiv e^{-i\theta ZY},U_{YZ}\left(\theta \right)\equiv e^{-i\theta YZ}$. This circuit can be expressed perfectly with only one layer of the variational ansatz in Fig. 2.