Recommender engine for continuous-time quantum Monte Carlo methods

Recommender systems play an essential role in the modern business world. They recommend favorable items such as books, movies, and search queries to users based on their past preferences. Applying similar ideas and techniques to Monte Carlo simulations of physical systems boosts their efficiency without sacrificing accuracy. Exploiting the quantum to classical mapping inherent in the continuous-time quantum Monte Carlo methods, we construct a classical molecular gas model to reproduce the quantum distributions. We then utilize powerful molecular simulation techniques to propose efficient quantum Monte Carlo updates. The recommender engine approach provides a general way to speed up the quantum impurity solvers.

Figure 1

(a) The quantum impurity problem consists of an impurity with local interaction embedded in a bath of non-interacting fermions. (b) The CT-QMC method maps the quantum impurity model to a one-dimension classical molecular gas model. Each red dot represents an interaction vertex in the interaction expansion Eq. (2). The length of periodic imaginary time is the inverse temperature $\beta$. Configurational bias Monte Carlo simulation of the molecular gas recommends efficient updates to the CT-QMC simulation.

Figure 2

(a) The Legendre coefficients ${\mathcal{V}}_{\ell }^{(2,3)}$ of the two- and three-body interactions in Eq. (3). (b) The log weight of the CT-QMC, Eq. (2), and of the molecular gas model, Eq. (3). Each red dot represents a test sample which is not used for training. The physical parameters of the quantum impurity model (1) are $\beta =100,U=-2,\varepsilon =0.2,$ and $\lambda =1.0$.

Figure 3

(a) The two-body and (b) the three-body interaction potentials for various hybridization strengths $\lambda$. Positive (negative) value means attractive (repulsive) interactions due to the minus sign in Eq. (3). The physical parameters are identical to Fig. 2.

Figure 4

The weight ratio of adding a vertex at the imaginary time $\tau$ to the configuration $\mathcal{C}$. The red solid line is $w(\mathcal{C}\cup \{\tau \left\}\right)/w\left(\mathcal{C}\right)$ and the blue dashed line is $e^{-E(\mathcal{C}\cup \{\tau \left\}\right)+E\left(\mathcal{C}\right)}$. The red stars indicate the location of the existing vertices in $\mathcal{C}$. For visibility we offset their vertical positions in the graph. Inset shows the same plot in the linear scale which highlights the three dominant peaks. The physical parameters $\beta ,U,\varepsilon$ are identical to Fig. 2.

Figure 5

The autocorrelation times of the total expansion order. The improvement of the ordinary CT-QMC results (blue dots) [9, 11] is more significant in the difficult parameter region where the hybridization strength $\lambda$ or the inverse temperature $\beta$ is large. Increasing the trial steps $N$ in the CBMC simulation of the molecular gas model (3) further reduces the autocorrelation time. In (a) $\beta =100$ and in (b) $\lambda =1.5$, the other physical parameters $U,\varepsilon$ are identical to Fig. 2.