Quantum Quasi-Monte Carlo Technique for Many-Body Perturbative Expansions

High order perturbation theory has seen an unexpected recent revival for controlled calculations of quantum many-body systems, even at strong coupling. We adapt integration methods using low-discrepancy sequences to this problem. They greatly outperform state-of-the-art diagrammatic Monte Carlo simulations. In practical applications, we show speed-ups of several orders of magnitude with scaling as fast as $1/N$ in sample number $N$; parametrically faster than $1/\sqrt{N}$ in Monte Carlo simulations. We illustrate our technique with a solution of the Kondo ridge in quantum dots, where it allows large parameter sweeps.

Figure 1

Comparison of the convergence rates for QQMC and DiagQMC. Here $Q_n(N)$ is the expansion coefficient of the occupation number of the Anderson impurity model at order $n$ as a function of the number of integrand evaluations $N$. Each result is normalized to the exact analytic result $Q_n^{\text{Bethe}}$.

Figure 2

Expansion coefficients $Q_n$ for the Anderson impurity occupation number relative to the analytic result $Q_n^{\text{Bethe}}$. QQMC methods converge at rates close to $1/N$ with the number of integrand evaluations $N$. For visibility, the data have been smoothed (see Supplemental Material [26]). The black lines indicate exact $1/N$ (dotted) and $1/\sqrt{N}$ (dashed) convergence. Each run was performed with one Sobol’ sequence. Inset: Cartoon of quantum dot setup.

Figure 3

Current at finite-bias voltage through the Anderson impurity at $T=0$, sweeping through several interaction regimes. Each point is a different QQMC calculation up to order $n=10$, including series resummation [11]. The error bars are a combination of integration error and truncation error of the resummation; the latter dominates. By construction, the data are symmetric with respect to the particle-hole symmetric point ${ϵ}_d=-U/2$. Inset: Coulomb diamond in the Coulomb blockade picture (large $U$). Regions where current can flow are shaded gray. The dashed line indicates the scan shown in the main plot (varying ${ϵ}_d$ for fixed $V/U=1/7$).

Figure 4

Comparison of convergence of $Q_8$ for different methods of integration: evaluating unwarped integrand with a Sobol’ sequence (cyan), DiagQMC (red), warped integral sampled with Mersenne Twister pseudorandom numbers (green), or Sobol’ sequence (blue). For the warped cases, we used Eq. (6) with $h^{(i)}(v_i)=\exp (-v_i/\tau ),\tau =0.95$. After an initial warping with exponential functions $\tau =1.1$, we can apply an additional warping obtained by projection (orange) [26]. For visibility, the data (except Sobol’ and DiagQMC) have been smoothed in the same way as in Fig. 2.