Efficient Tensor Network Ansatz for High-Dimensional Quantum Many-Body Problems

We introduce a novel tensor network structure augmenting the well-established tree tensor network representation of a quantum many-body wave function. The new structure satisfies the area law in high dimensions remaining efficiently manipulatable and scalable. We benchmark this novel approach against paradigmatic two-dimensional spin models demonstrating unprecedented precision and system sizes. Finally, we compute the ground state phase diagram of two-dimensional lattice Rydberg atoms in optical tweezers observing nontrivial phases and quantum phase transitions, providing realistic benchmarks for current and future two-dimensional quantum simulations.

Figure 1

(a) An aTTN for a $8\times 8$ 2D system: The disentanglers in $\mathcal{D}(u)$ are applied to the TTN state $|{\psi }_{\mathrm{TTN}}⟩$ across the boundaries ${\partial }_{\nu }$ of each link $\nu$, in order to fulfill the area law depicted (b) for a sublattice $\mathcal{A}$ (shaded region) and its boundary $\partial \mathcal{A}$ (purple dots). (c),(d) Relative error of the Ising model ground state energy computed with the aTTNs and the TTNs. While for $L=8$ the precision achieved with the two methods is the same, a clear improvement emerges for $L=64$.

Figure 2

Relative error $\epsilon$ of the 2D Heisenberg ground-state energy as a function of the system linear size $L$ for the TTN, aTTN, NNS [68], NAQS [69], EPS [70], PEPS [58], 2D-DMRG [64] (circles, squares and triangles indicate open BC, periodic BC and cylindrical BC, respectively) each compared with the best available estimates obtained by MC with the same BC (for pbc [22], for obc [58] obtained via ALPS library [75, 76, 77]).

Figure 3

Up: Phase diagram as a function of the detuning $\mathrm{\Delta }$ and the nearest-neighbors interaction energy $V_{nn}$. The disordered phase is characterized by a substantially uniform distribution of the excitations, while in the phases ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_4$ the excitations are distributed as shown in the upper (${\mathbb{Z}}_4$) and lower (${\mathbb{Z}}_2$) insets. Down: Renormalized structure factor $S^{\prime }(\mathbf{k})=S(\mathbf{k})/S(\mathbf{0})$ for $V_{nn}=46\mathrm{MHz}$ and (a) $\mathrm{\Delta }=28\mathrm{MHz}$ (${\mathbb{Z}}_2$ phase) and (b) $\mathrm{\Delta }=12\mathrm{MHz}$ (${\mathbb{Z}}_4$ phase). Other parameters: $\mathrm{\Omega }=4\mathrm{MHz}$.