Methods

This section describes the theoretical background of the algorithms implemented in Ace-TN. We follow the notation established in [Richards2025]. For a comprehensive review of tensor network methods see [Orus2014], [Cirac2021].

iPEPS Ansatz

The infinite projected entangled-pair state (iPEPS) ansatz approximates an infinite two-dimensional quantum many-body state through a periodic tensor network [Verstraete2004], [Jordan2008]. Each site \(i\) of the square lattice is associated with a rank-5 site tensor \(A^{[s_i]}_{l_i u_i r_i d_i}\) carrying a physical index \(s_i\) of dimension \(d\) and four bond indices \(l_i, u_i, r_i, d_i\) each of dimension \(D\):

The bond dimension \(D\) controls the amount of entanglement captured by the ansatz, with \(D = 1\) corresponding to a product state. Increasing \(D\) systematically improves the approximation.

Double-Layer Tensor

Expectation values require contracting the tensor-network state with its conjugate, forming a double-layer tensor network. The double-layer tensor at site \(i\) is obtained by contracting \(A_i\) with \(A^\dagger_i\) over the physical index and fusing pairs of bond indices:

\[a_{(l_i l'_i)(u_i u'_i)(r_i r'_i)(d_i d'_i)} = \sum_{s_i} A^{[s_i]}_{l_i u_i r_i d_i} \, A^{[s_i]\dagger}_{l'_i u'_i r'_i d'_i}\]

Diagrammatically, the ket and bra layers are contracted over the physical index \(s_i\) and each pair of bond indices is fused into a single leg of dimension \(D^2\):

Boundary Tensor Approximation

The infinite double-layer network cannot be contracted exactly. It is approximated by boundary tensors surrounding each site: corner transfer matrices \(C^k_i\) (rank-2) and edge tensors \(E^k_i\) (rank-3) for \(k = 1,2,3,4\) [Nishino1996], [Orus2009]. The environment of a site is then approximated by the finite arrangement:

Here \(C^k\) are rank-2 corner tensors and \(E^k\) are rank-3 edge tensors. Each corner carries two boundary bonds of dimension \(\chi\), and each edge carries two boundary bonds and one double-layer bond. The boundary tensors encode the effect of the infinite lattice surrounding each site. Their computation is the task of the CTMRG algorithm.

• CTMRG
  • Directional Moves
  • Projector Calculation
  • Convergence
  • References
• Full Update
  • Imaginary-Time Evolution
  • QR Decomposition
  • Gate Application
  • Norm Tensor Construction
  • ALS Solution
  • Reassembly
  • Fast Full Update
  • References
• Simple Update
  • Gamma–Lambda Decomposition
  • Bond Update Procedure
  • Measurements
  • References

References

[Verstraete2004]

6. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, arXiv:cond-mat/0407066 (2004).

[Jordan2008]

10. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. I. Cirac, Classical simulation of infinite-size quantum lattice systems in two spatial dimensions, Phys. Rev. Lett. 101, 250602 (2008).

[Nishino1996]

20. Nishino and K. Okunishi, Corner transfer matrix renormalization group method, J. Phys. Soc. Jpn. 65, 891 (1996).

[Orus2009]

18. Orús and G. Vidal, Simulation of two-dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction, Phys. Rev. B 80, 094403 (2009).

[Orus2014]

18. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of Physics 349, 117 (2014).

[Cirac2021]

10. 1. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).

[Richards2025]

1. 4. 19. Richards and E. S. Sørensen, Ace-TN: GPU-Accelerated Corner-Transfer-Matrix Renormalization of Infinite Projected Entangled-Pair States, arXiv:2503.13900 (2025).