Coarse-grained belief propagation for simulation of interacting quantum systems at all temperatures

We continue our numerical study of quantum belief propagation initiated in [Phys. Rev. A 77, 052318 (2008)]. We demonstrate how the method can be expressed in terms of an effective thermal potential that materializes when the system presents quantum correlations, but is insensitive to classical correlations. The thermal potential provides an efficient means to assess the precision of belief propagation on graphs with no loops. We illustrate these concepts using the one-dimensional quantum Ising model and compare our results with exact solutions. We also use the method to study the transverse field quantum Ising spin glass for which we obtain a phase diagram that is largely in agreement with the one obtained in [Phys. Rev. B 78, 134424 (2008).] using a different approach. Finally, we introduce the coarse-grained belief propagation (CGBP) algorithm to improve belief propagation at low temperatures. This method combines the reliability of belief propagation at high temperatures with the ability of entanglement renormalization to efficiently describe low-energy subspaces of quantum systems with local interactions. With CGBP, thermodynamic properties of quantum systems can be calculated with a high degree of accuracy at all temperatures.

Figure 1

(Color online) Upper bound on $\Vert V_{\text{eff}}^j\Vert$ (left) and correlations (right) of the Ising chain with critical transverse field. These exact values are obtained from a Jordan-Wigner transform. Note that $\Vert V_{\infty }^j\Vert$ decays much faster than the correlations (notice the different length scales). The values for $\beta =10$ are in a different color as an aid to the eye.

Figure 2

Calculating $m_{4\to 5}$ from $m_{3\to 4}$ in an iteration of BP algorithm with $l=4$. In the first step $e^{h_{4,5}}$ is added to $m_{3\to 4}$ using the $\odot$ product. Then, the first spin is traced out yielding $m_{4\to 5}$.

Figure 3

(Color online) Error on the energy density as a function of temperature for the critical Ising chain. For QBP with window size $l=10$, we show the error estimate of Eq. (13) and the true error obtained by comparison with analytical solution. Also shown is the finite-size error for exact diagonalization of a 11-site chain and error caused by iTEDB (Ref. 10) with parameters $\delta T=0.001$ and $\chi =150$, both of which require equivalent computational resources.

Figure 4

Schematic illustration of the procedure to calculate the message from a node to its parent. First, two messages from its children are merged using the $\odot$ product. The bare hamiltonian term relating the current node to its parent is also added. Tracing out the leaves of this eight-spin message gives the message to the node’s parent.

Figure 5

(Color online) The Edwards-Anderson order parameter for transverse field quantum Ising spin glass on a degree-3 Cayley tree with random parallel boundary fields. The total depth of the tree is 12, and the plots show the average of 100 instances of random boundary fields. Values of $q_{\text{EA}}$ range from 0 to 1 and contours are equally spaced. The error on the order parameter is estimated using the procedure outlined at the end of Sec. 2B.

Figure 6

(Color online) Schematics of entanglement renormalization for a ternary MERA of a one-dimensional lattice.

Figure 7

(Color online) Energy vs temperature for the one-dimensional critical quantum Ising model. Presented are the plain BP results for $l=10$ (dark blue), CGBP results for various levels of coarse graining with $\chi =4$ and $l=5$ (various colors), the plain ER result $\chi =4$ (light green) along with the exact result obtained from Jordan-Wigner transformations (black). The labels on the temperature axis correspond to the switching temperatures between various levels of coarse graining in the CGBP algorithm. For example, plain BP is very accurate down to $T_1\approx 0.22$, where BP combined with one level of coarse graining (red) becomes more accurate. The inset shows the absolute error in the combined CGBP result with respect to the exact solution (blue), and the error estimate calculated using the procedure given in Sec. 4B. While plain BP and plain ER are both very inaccurate for temperature range $0.005\lesssim T\lesssim 0.22$, combining the two using CGBP yields very accurate results.

Figure 8

(Color online) The exact error in the energies calculated at the first and second levels of coarse graining, $\left|E_1-E_{exact}\right|$ and $\left|E_2-E_{exact}\right|$ are plotted along with $\left|E_1-E_2\right|$. Note that for both $\left|E_1-E_{exact}\right|$ and $\left|E_2-E_{exact}\right|$, MERA error dominates at high temperatures, and BP error dominates at low temperatures. The minimum of their difference $\left|E_1-E_2\right|$ occurs when the BP error at level 1 of coarse graining is equal to the MERA error at level 2 of coarse graining. Furthermore, the high temperature portion of $\left|E_1-E_2\right|$ is dominated by the MERA error at level 2 and the low-temperature portion of $\left|E_1-E_2\right|$ is dominated by the BP error at level 1.