Probabilistic low-rank factorization accelerates tensor network simulations of critical quantum many-body ground states

We provide evidence that randomized low-rank factorization is a powerful tool for the determination of the ground-state properties of low-dimensional lattice Hamiltonians through tensor network techniques. In particular, we show that randomized matrix factorization outperforms truncated singular value decomposition based on state-of-the-art deterministic routines in time-evolving block decimation (TEBD)– and density matrix renormalization group (DMRG)–style simulations, even when the system under study gets close to a phase transition: We report linear speedups in the bond or local dimension of up to 24 times in quasi-two-dimensional cylindrical systems.

Figure 1

Compression steps in the benchmarked tensor networks of bond dimension $\chi$ and local dimension $d$. The truncated SVDs (highlighted in red) retain at most $\chi$ largest singular values (red bar) and produce isometric tensors (shaded). (a) Diagrammatic representation of the MPS used in TEBD. (b) Absorbtion of the nearest-neighbor evolution exponential in form of a four-link tensor “block” (top to bottom). (c) Absorbtion of a sum over $K$ Kronecker products results in a different compression problem. (d) Binary tree TN ansatz for variational energy minimization. (e) Two-site update: The energy of an effective Hamiltonian (left) is minimized by optimizing adjacent tensors in form of a four-link matrix which is then rank-$\chi$ factorized (right).

Figure 2

Speedup in compression step due to RSVD in TEBD simulations of increasing local dimensions. (a) 2D Ising model ($L=30$) as a function of the width $W$. (b) 1D Ising model ($L=100$) as a function of local spin $S$. Dark and light blue points represent data at bond dimensions $\chi =100$ and $\chi =150$, respectively, both from block update and fixed convergence criteria (1D: $\delta E={10}^{-13}$, 2D: $\delta E={10}^{-8}$ except $W=8$ was stopped before convergence). Error bars indicate a $10\%$ error estimate in speedups. Orange crosses show speedup in higher precision target $\delta E={10}^{-14}$, $\chi =100$. Dashed lines are linear fits for $d>10$ with slopes $\approx 0.10,0.13$ (2D) and 0.18,0.21 (1D) for $\chi =100,150$ respectively.

Figure 3

Dependency of RSVD speedup on the bond dimension in TEBD simulations. (a) 2D Ising model, speedup at width $W=6$, $L=30$, and $\delta E={10}^{-8}$. Blue and gray: block (B) and product (P) updates, respectively. (b) 1D Ising model, speedup at spin $S=5$, $L=100$ and $\delta E={10}^{-13}$. Both panels share the same $x$ axis.

Figure 4

Relative errors in 1D TEBD simulations with TSVD and RSVD. (a) Error in final-state energy $\mathrm{\Delta }E$. (b) Error in magnetization $\mathrm{\Delta }M$. Each group of three bars shows the error with increasing bond dimensions $\chi =50,75,100$ (left to right) at a given transverse field (both panels share the same $x$ axis). Black (orange) bars correspond to TSVD (RSVD) results at convergence thresholds $\delta E={10}^{-11}$ (light shaded) and ${10}^{-13}$. All errors have been obtained from extrapolated ground state values for $S=5,L=100$ in the ferro- ($h=1.0$) and paramagnetic ($h=2.0$) phases as well as close to the critical point.

Figure 5

Correlation properties and singular values in TEBD simulated ground states in various transverse fields $h$, for the 1D Ising model at $S=5,L=100$ [left panels (a), (c)] and the 2D Ising model at $W=6,L=30$ [right panels (b), (d)]. Panels (a) and (b): Magnetization $M$ (black) and estimates for the correlation length $\overline{\xi }/L$ (gray dashed). Errors are smaller than point sizes. Insets show von Neumann entropies $S_N\left(j\right)$ from singular values on MPS bonds. Inset (a): $h=1.5$ (green), 1.762 (purple with light-purple fit $S_N^C\left(j\right)$, see text), 1.774 (red), 2.0 (orange). Inset (b): $h=2.0$ (green), 3.04 (red), 3.5 (orange). Panels (c) and (d): Decay exponents $\gamma$ of singular values. Upward (downward) pointing triangles indicate even (odd) sectors, respectively. Shaded area encloses fit errors. Insets show singular values (black) at a central bond in the invariant sector for near-critical fields and a polynomial fit ${\sigma }_k\approx {(C_{1}k+C_2)}^{-\gamma }$ (orange, $C_2=0$ in 2D). Inset (c): $h=1.7735$. Inset (d): $h=3.04$. Panels share $x$ ($y$) axes.

Figure 6

Singular values ${\lambda }_k$ monitored over algorithm runtime in 1D Ising models at, or close to, the critical field. Both panels show values in the invariant sector at a central lattice bipartition with $\chi =100$. The dashed lines indicate the truncation at ${\chi }_0=50$. (a) Imaginary TEBD ($S=5$, $L=100$, $h=1.7735$, $\delta E={10}^{-13}$, block update B). The time step $dt$ was subsequently reduced to $d{t}_i=0.4\times 0.7^i$ for $i=0,\cdots ,11$ (red to black). (b) TTN ground-state search ($S=1/2$, $L=128$, $h=1.0$), after $i=0,\cdots ,4$ network updates (red to black).