Symmetry-adapted subspace-iteration CTMRG for two-dimensional tensor networks

The subspace-iteration corner transfer matrix renormalization group (SI-CTMRG) of Zhang and Corboz (arXiv:2607.15158, 2026) accelerates the contraction of infinite projected entangled-pair states (iPEPS) by replacing the truncated singular value decomposition (SVD) of the CTMRG renormalization step with randomized subspace iteration that directly builds the corner-isometry projectors, reported to be up to two orders of magnitude faster than standard CTMRG on GPUs. However, it is formulated only for dense tensors without global symmetries, and the paper's explicitly stated future-work item is precisely the extension of its contraction scheme to symmetric tensor networks. The open problem is to construct a quantum-number-preserving (block-sparse $U(1)$, ideally $SU(2)$) version of SI-CTMRG --- symmetry-resolved rangefinder initialization, power updates, small SVDs, projector reassembly, gauge fixing, and isometry recycling --- validated against symmetric SVD-based CTMRG on symmetry-conserving benchmark models.

Background

Infinite projected entangled-pair states (iPEPS) are variational tensor-network ansätze for ground states of two-dimensional quantum lattice models on the infinite lattice, characterized by a bond dimension $D$ per tensor (the dimension of the virtual indices linking neighboring tensors); evaluating any observable requires contracting an infinite two-dimensional network, which is done approximately by the corner transfer matrix renormalization group (CTMRG) method, introduced by Nishino and Okunishi and adapted to quantum systems by Orús and Vidal. CTMRG absorbs the environment into a finite boundary dimension $\chi$: it repeatedly doubles four corner matrices and half-infinite transfer matrices (together forming the double layer of the physical bra and ket tensors) and truncates them back to dimension $\chi$ by keeping the dominant singular subspace of an SVD of the grown corner. This SVD of a $\chi D^2 \times \chi D^2$ matrix dominates the cost, $O(\chi^3 D^6)$, and its replacement is the bottleneck of modern large-$D$ iPEPS calculations. Zhang and Corboz (arXiv:2607.15158, 2026) proposed SI-CTMRG: instead of computing the full SVD, one builds the projector onto the dominant corner subspace by randomized subspace iteration. Writing $M$ for the grown corner matrix, the isometries $X_n$ and $Y_n$ --- matrices whose columns span the estimated dominant $(\chi+p)$-dimensional left and right subspaces of $M$ --- are initialized as random Gaussian blocks (the rangefinder step of randomized linear algebra, in which a random sketch of a matrix is used to estimate its dominant subspace) and refined by power-iteration updates $X_n R_x = M^{T} M X_{n-1}$, and analogously for $Y_n$, so that at each step only the small $(\chi+p) \times (\chi+p)$ matrix $\rho = Y_n M X_n$ must be decomposed by SVD. Here $p$ is the oversampling parameter, the small number of dimensions retained beyond $\chi$ in the subspace estimate, which controls the accuracy of the randomized estimate (in the spirit of Halko, Martinsson, and Tropp's analysis of randomized matrix decompositions, which however applies to fixed matrices rather than the sequence of slowly varying corner matrices arising in a CTMRG fixed-point iteration). SI-CTMRG recycles the isometries at two distinct levels: within the CTMRG fixed-point loop, $X_n$ and $Y_n$ are recycled across CTMRG iterations once the dominant $\chi$-dimensional subspaces of successive steps are sufficiently aligned, measured by the mean squared sine $\overline{\sin^{2}\theta} \equiv \frac{1}{\chi}\sum_i \sin^{2}\theta_i < 10^{-3}$ of the principal angles $\theta_i$ between the current and previous estimated subspaces (the canonical angles between two subspaces, which vanish when the subspaces coincide); and, separately, the converged CTMRG environment together with the isometries is reused between iPEPS optimization iterations, except during line-search steps. The method was reported to run up to two orders of magnitude faster than standard CTMRG. The limitation that defines the present problem is that the method is formulated for dense tensors and does not exploit global symmetries. This matters because symmetric tensor networks --- tensors with conserved quantum numbers such as $U(1)$ particle number or $SU(2)$ total spin --- are stored block-sparsely, one dense block per allowed combination of quantum numbers, so the effective matrix dimensions and hence costs are dramatically reduced for charge- and spin-conserving models (e.g. Hubbard-type systems); block-sparse symmetric CTMRG based on per-block SVDs is standard since Singh, Pfeifer, and Vidal and Bauer, Corboz, Orús, and Troyer (these are exactly the two references the SI-CTMRG paper cites for symmetric tensor networks), but no published work combines quantum-number bookkeeping with randomized subspace iteration. As background for the algorithm to be reformulated, the source's remaining control constants are fixed heuristically: the oversampling parameter $p$ (5 during optimization, 15% of $\chi$ in the post-optimization $\chi$-ramp) and the recycling threshold $\overline{\sin^{2}\theta} < 10^{-3}$; the paper notes, citing the randomized linear-algebra literature, that the accuracy of such methods is mainly controlled by $p$ and the number of subspace iterations, but it does not pose a systematic study of these constants as an open question. Companion work by the same group replaces the SVD by QR decompositions (QR-CTMRG, Phys. Rev. B 113, L201106 (2026)) and extends the QR-based scheme to the honeycomb lattice with $C_{3v}$ point-group symmetry, the three-fold rotation and reflection symmetry group of the honeycomb lattice (Phys. Rev. B 113, 085109 (2026)), but these are dense-tensor, lattice-point-group methods and do not address internal quantum-number symmetry or the randomized subspace-iteration scheme of SI-CTMRG.

Problem Statement

Construct and validate a symmetry-adapted subspace-iteration CTMRG (SI-CTMRG) for infinite two-dimensional tensor networks with conserved quantum numbers. Concretely, reformulate the SI-CTMRG renormalization step of arXiv:2607.15158 for block-sparse tensors carrying global $U(1)$ charges (and, ideally, non-Abelian $SU(2)$ quantum numbers), including symmetry-resolved initialization of the randomized rangefinder on the quantum-number blocks of the grown corner matrix, sector-wise power updates, the small SVD of the $\rho = Y_n M X_n$ form performed block by block, reassembly and gauge fixing (absorbing the unitary factors of the small SVD that fix the basis inside the estimated subspace) of the resulting isometries so that they respect the block structure, and the recycling of isometries --- both across CTMRG iterations, where the source gates recycling on the mean squared sine of the principal angles between successive estimated subspaces, and between iPEPS optimization steps, where the source reuses the converged environment and isometries except during line-search steps. The algorithm must reproduce, on symmetry-conserving benchmark models (e.g. the spin-1/2 Heisenberg antiferromagnet and a charge-conserving model such as the Hubbard or $t$-$J$ model), the observables and energies of the established symmetric SVD-based CTMRG to the same accuracy at equal boundary dimension $\chi$, while exhibiting the symmetry-induced reduction of effective matrix dimensions (equivalently, of cost at fixed accuracy). The overarching objective is to make the speed of SI-CTMRG available for the symmetry-conserving models where it currently cannot be used at all, at the accuracy that symmetric SVD-based CTMRG already delivers there.

References

1. Yining Zhang and Philippe Corboz, "Fast two-dimensional tensor-network contraction via subspace iteration", arXiv:2607.15158 (2026), https://arxiv.org/abs/2607.15158
2. Yining Zhang, Qi Yang, and Philippe Corboz, "Accelerating two-dimensional tensor network contractions using QR decompositions", Phys. Rev. B 113, L201106 (2026), DOI 10.1103/ptls-kr9z, arXiv:2505.00494, https://arxiv.org/abs/2505.00494
3. Qi Yang and Philippe Corboz, "Efficient iPEPS simulation on the honeycomb lattice via QR-based corner transfer matrix renormalization group", Phys. Rev. B 113, 085109 (2026), DOI 10.1103/9gmp-byx8, arXiv:2509.05090, https://arxiv.org/abs/2509.05090
4. Sukhwinder Singh, Robert N. C. Pfeifer, and Guifré Vidal, "Tensor network states and algorithms in the presence of a global U(1) symmetry", Phys. Rev. B 83, 115125 (2011), DOI 10.1103/PhysRevB.83.115125, arXiv:1008.4774, https://arxiv.org/abs/1008.4774
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6. Nathan Halko, Per-Gunnar Martinsson, and Joel A. Tropp, "Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions", SIAM Review 53, 217 (2011), DOI 10.1137/090771806, arXiv:0909.4061, https://arxiv.org/abs/0909.4061
7. Tomotoshi Nishino and Kouichi Okunishi, "Corner Transfer Matrix Renormalization Group Method", J. Phys. Soc. Jpn. 65, 891 (1996), DOI 10.1143/JPSJ.65.891, https://doi.org/10.1143/JPSJ.65.891
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9. Juraj Hasik and Philippe Corboz, "Incommensurate Order with Translationally Invariant Projected Entangled-Pair States: Spiral States and Quantum Spin Liquid on the Anisotropic Triangular Lattice", Phys. Rev. Lett. 133, 176502 (2024), DOI 10.1103/PhysRevLett.133.176502, https://doi.org/10.1103/PhysRevLett.133.176502