Robust contraction of infinite two-dimensional tensor networks with non-Hermitian transfer matrices

The 2018 faster-contraction methods for infinite 2D tensor networks — transfer-matrix VUMPS and the fixed point corner method (FPCM) — matured environment computation for Hermitian, symmetric networks, but their asymmetric versions remain unreliable: asymmetric FPCM tends to get trapped in suboptimal fixed points, no algorithm solves the top and bottom boundary-MPS fixed points of a non-Hermitian transfer operator in a single optimization, and biorthogonalization-based projector construction is numerically fragile. The problem is to give non-Hermitian infinite 2D tensor networks a principled and stable contraction scheme: a fixed-point algorithm that reliably computes the opposing boundary fixed points of the non-Hermitian transfer operator, with a well-defined truncation criterion, robust projector construction, and convergence to the most accurate boundary environment attainable at a given bond dimension, validated on standard asymmetric benchmarks.

Background

Contracting an infinite two-dimensional (2D) tensor network means computing its partition function per site, or the norm and local expectation values of the quantum state it represents. This contraction is the computational bottleneck of two widely used tools in many-body physics: the infinite projected entangled pair state (iPEPS) ansatz — a variational wavefunction for 2D quantum lattices built from a repeating local tensor, whose bond dimension $D$ bounds the entanglement the state can carry — and tensor-network representations of 2D classical partition functions. Exact contraction of generic 2D networks is computationally intractable, so the network is approximated by a boundary matrix product state (MPS): a one-dimensional chain of tensors with bond dimension $\chi$ that encodes the environment of each lattice site.

Two algorithm families build this boundary. The corner transfer matrix renormalization group (CTMRG), introduced by Nishino and Okunishi, grows corner and edge environment tensors by absorbing rows and columns of the network and truncates them via projectors; iteration-wise it behaves like a power method. Fishman, Vanderstraeten, Zauner-Stauber, Haegeman and Verstraete accelerated this family in two ways: transfer-matrix VUMPS, an adaptation of the variational uniform matrix product state (VUMPS) algorithm of Zauner-Stauber et al. that diagonalizes the row-to-row transfer operator — a matrix product operator (MPO), i.e. an operator on one lattice row with the same tensor-chain structure as an MPS — using Krylov-type eigensolvers — iterative eigenvalue algorithms that need only matrix–vector products with the operator — instead of power-method iterations; and the fixed point corner method (FPCM), which keeps the CTMRG corner ansatz but solves for the environment tensors directly from fixed-point equations. Both bring large speedups over CTMRG, especially near criticality, where the transfer-operator spectral gap closes and power methods stagnate.

These constructions lean on Hermitian or reflection/rotation-symmetric structure. For example, in the norm of a PEPS the doubled row-to-row transfer operator is Hermitian, a variational principle bounds the truncation error of the boundary MPS, and CTMRG projectors can be obtained from Hermitian eigendecompositions. Many networks of interest lack this structure: classical partition functions with non-Hermitian transfer-operator unit cells (frustrated magnets, lattice gauge theories), chiral or otherwise asymmetric PEPS, and the effective non-Hermitian networks arising in open-system (Lindblad) simulations, i.e. dissipative quantum dynamics governed by a stochastic master equation. For such networks the leading up and down fixed points of the row-to-row transfer operator — and, likewise, the leading left and right fixed points of the column-to-column transfer operator, together "the four boundary MPSs" — are distinct and must be treated together. The standard technique is biorthogonalization, introduced by Huang in transfer-matrix DMRG (TMRG, the application of DMRG — the density matrix renormalization group, the standard variational method for one-dimensional quantum lattice systems — to transfer operators) and applied to CTMRG by Corboz, Rice and Troyer: the two fixed-point MPSs are gauged so that their overlap becomes the identity, and the renormalization projectors are built from this biorthogonal pair, requiring pseudoinversion of the overlapping singular-value spectrum. This route is numerically fragile — the inverses can be ill-conditioned, the results are sensitive to which singular value decomposition routine is used, and accuracy depends on the gauge of the tensors — and, as benchmarked in the source paper, asymmetric FPCM alone tends to converge to suboptimal fixed points ('getting stuck') unless interleaved with CTMRG steps.

The source paper flags, as separate optional directions, what could bring the asymmetric case to parity with the symmetric one: a modification of asymmetric FPCM that avoids suboptimal fixed points (its empirical remedy of interleaving FPCM with CTMRG steps is effective and robust, but its mechanism is unexplained); a generalization of transfer-matrix VUMPS that solves two opposing fixed points — for example both the top and bottom fixed points of the row-to-row transfer operator — in the same optimization, in analogy with two-sided non-Hermitian TMRG; and more robust techniques for the projectors, possibly along the lines of the subspace-expansion stabilization used in non-Hermitian TMRG, which systematically enlarges the truncation subspace to keep the recursion well-conditioned. Later work clarified that the failures have a structural origin: Tang, Verstraete and Haegeman showed in 2025 that gauge transformations on the virtual indices of a 2D network can change the entanglement structure of the transfer operator's eigenstates, so single-sided boundary-MPS optimization can fail entirely for non-normal transfer operators (operators that do not commute with their adjoint), and that no variational principle is available at the truncation step of the biorthogonalization formalism, leaving the optimal truncation scheme for non-Hermitian networks undefined.

Problem Statement

Derived from the outlook of Fishman et al. (Phys. Rev. B 98, 235148 (2018)), the problem is to give the approximate contraction of infinite 2D tensor networks with non-Hermitian (asymmetric) transfer structure the same principled and robust foundation that exists in the Hermitian case, so that the faster contraction methodology becomes reliable for generic networks rather than only symmetric ones.

Concretely, construct and validate a contraction scheme for a generic non-Hermitian infinite 2D tensor network in the setting considered by the source — translation-invariant networks on an infinite square lattice with a single-site unit cell, with enlarged unit cells and other lattices as natural extensions rather than requirements — applicable to classical partition functions with non-Hermitian transfer-operator unit cells, asymmetric or chiral PEPS norms and expectation values, and effective non-Hermitian (e.g. Lindblad-type) networks. The source presents the corresponding directions as separate optional suggestions; they are collected here as the linked facets of the single objective of a principled and robust non-Hermitian contraction scheme:

1. The opposing fixed points of the non-Hermitian transfer operator (the leading up and down boundary MPSs of the row-to-row transfer operator, and analogously the left and right pair of the column-to-column transfer operator) are computed reliably, replacing the current practice in which each of the four boundary MPSs is solved for in a separate single-sided optimization. No particular algorithmic architecture is prescribed. Since no variational principle currently governs the truncation step in the biorthogonalization formalism — a limitation made explicit by Tang, Verstraete and Haegeman (2025) rather than by the 2018 outlook — the scheme should also supply a principled criterion for truncating the boundary MPS at bond dimension $\chi$.
2. The renormalization projectors (or equivalent environment update) are numerically stable, avoiding the ill-conditioned pseudoinversions and singular-value-decomposition sensitivity of present biorthogonalization-based CTMRG/FPCM.
3. The iteration converges to the most accurate boundary environment attainable at the given $\chi$ for asymmetric networks, rather than to suboptimal fixed points. As in the source's own benchmarks, "most accurate ... attainable at the given $\chi$" is operationalized by errors in local observables or in the partition function/norm per site at fixed $\chi$, measured against cross-method and extrapolated references. Understanding the mechanism that prevents such trapping is desirable — the source itself only speculates about why its FPCM+CTMRG remedy helps — but is not a mandatory deliverable.

A related diagnostic question, raised by Tang, Verstraete and Haegeman (Phys. Rev. B 111, 035107 (2025)) rather than by the 2018 outlook, is contemporaneous context for the same objective rather than an additional mandatory deliverable: characterizing when biorthogonalization-based truncation remains reliable under (local and nonlocal) gauge transformations of the network, whether normalizing gauge transformations always exist, and how nonlocal ones could be found.

The source's directions are constructive: they ask for a working scheme, not for a proof that some class of schemes cannot exist. The field's standard asymmetric benchmarks (e.g. the gauge-transformed classical Ising network and the non-Hermitian (complex-symmetric) PEPS of the source paper, the triangular-lattice antiferromagnetic Ising partition function in its two inequivalent tensor representations, or Lindblad-type networks) provide the natural references for assessing such a scheme, through accuracy as a function of $\chi$, stability metrics (conditioning of inverted spectra, sensitivity to decomposition routines), and computational cost relative to asymmetric CTMRG. The fourth direction of the original outlook — integrating the faster contraction methods into variational PEPS optimization and assessing the overall speedup for ground-state searches — has been substantially realized for symmetric networks and is not a required deliverable of this problem; the non-Hermitian integration it would enable, where the contraction currently remains blocked precisely by the instabilities above, is a natural downstream application of the scheme sought here rather than an additional mandatory facet.

References

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