Rakovszky and Khemani (2023) recast classical low-density parity-check (LDPC) codes as quantum spin Hamiltonians on non-Euclidean graphs - the geometries on which good quantum LDPC codes are built - and generalized the Kramers-Wannier (KW) duality to map a code to its transpose code. A code whose Tanner graph is self-dual can sit at a self-dual point where the KW map acts as an internal, non-invertible symmetry, and the authors posed the follow-up questions of computing the fusion relations of these symmetries and using them to constrain possible critical points, as part of understanding phase transitions in non-Euclidean geometries. Subsequent work has computed the fusion algebra of the duality operators on arbitrary bipartite graphs and constructed gapped, frustration-free self-dual points at which the duality symmetry is spontaneously broken. What remains open is the central physical question: whether continuous (critical) self-dual phase transitions can exist on expander-like geometries at all, and what the non-invertible duality symmetry and its anomaly forbid or allow for the phase transitions of transverse-field LDPC code models there.

Background

A classical low-density parity-check (LDPC) code is a binary linear code whose parity checks each act on only a bounded number of bits; its check matrix σ\sigma is equivalently encoded in the Tanner graph, the bipartite graph whose two vertex classes are the nn bits and the mm checks, with an edge whenever a check acts on a bit. The physics of such a code is captured by a transverse-field Hamiltonian H=−J∑v^Bv^−g∑vXvH=-J\sum_{\hat v} B_{\hat v}-g\sum_v X_v, where Bv^B_{\hat v} multiplies the Pauli ZZ operators on the bits of check v^\hat v and XvX_v is a transverse field. Its two fixed-point limits are a symmetry-broken code phase (the frozen codewords, symmetry-broken by the logical operators ∏vXv\prod_v X_v corresponding to codewords) and a trivial paramagnet. On expander graphs - sparse graphs in which every sufficiently small vertex set has a boundary comparable to its own size, so that no perimeter-versus-area distinction survives - Tanner-code constructions yield good classical codes and, via chain-complex constructions (a chain complex is a sequence of binary vector spaces with 'boundary' maps whose composition vanishes - the algebraic skeleton underlying quantum CSS codes, i.e., stabilizer quantum codes whose parity checks are products of Pauli XX or of Pauli ZZ operators only), the recently discovered good quantum LDPC codes that combine a finite rate with a code distance growing linearly in blocklength. This is the motivation for studying phases of matter beyond finite-dimensional Euclidean lattices.

Rakovszky and Khemani generalized the Kramers-Wannier duality of the one-dimensional Ising model to arbitrary classical codes: there is a duality map relating the transverse-field model of a code C\mathcal{C} to that of its transpose code CT\mathcal{C}^T (the code whose check matrix is σT\sigma^T), exchanging check violations with spins and symmetries with redundancies. In the original Ising chain this map squares to a lattice translation times the sum of the identity and the global spin-flip symmetry; when restricted to a single Hilbert space it is not invertible, and at the self-dual coupling J=gJ=g it becomes an internal non-invertible symmetry - a symmetry implemented by an operator that is not invertible and closes under a fusion algebra (composition rules with sums of operators on the right-hand side) rather than under a group, with the critical Ising point as the canonical example of a self-dual critical point. For a code whose Tanner graph admits a reversing automorphism - a graph symmetry that exchanges the bit and check vertices; equivalently, a code isomorphic to its transpose - the transverse-field code model is likewise self-dual under this generalized KW duality, and the same possibility of self-dual critical points arises, now on graphs that may be far from any Euclidean lattice.

Two developments define the current frontier. First, the fusion algebra of these duality operators is now known in the general graph setting: Gorantla, Shao and Tantivasadakarn constructed the KW and Wegner (the electric-magnetic self-duality of lattice Z2\mathbb{Z}_2 gauge theories) duality operators of generalized transverse-field Ising models on arbitrary bipartite graphs - explicitly including the Tanner graphs of parity-check codes, i.e., precisely the models above - as ZX-calculus (a diagrammatic calculus for linear maps on qubits) and tensor-network operators, and computed DρDρ′=Tρ∘ρ′C\mathsf{D}_\rho\mathsf{D}_{\rho'}=T_{\rho\circ\rho'}\mathsf{C}, where TπT_\pi implements graph automorphisms, ρ,ρ′\rho,\rho' are reversing automorphisms, and the condensation operator C\mathsf{C} is proportional to the sum over the invertible Z2\mathbb{Z}_2 symmetries in ker⁡σ\ker\sigma (the code's logical operators); they also classified when the duality mixes with such graph symmetries. Second, Gorantla and Huang constructed duality-preserving deformations of LDPC and CSS codes in transverse field whose self-dual, frustration-free points are provably gapped in the thermodynamic limit with the duality symmetry spontaneously broken, showing that self-dual points need not be critical but can realize first-order coexistence of the trivial and code phases. Beyond such solvable points, essentially nothing is known about the transitions on expander geometries: the standard diagnostics of criticality (diverging correlation length, perimeter- versus area-law scaling of order and disorder parameters (symmetry-breaking expectation values and their dual, symmetry-twisting operators)) degenerate on graphs where the boundary of a set scales like its volume, and classical Ising models on expanders are understood mainly for specific locally tree-like families. The question is what replaces this picture, and what role the non-invertible duality symmetry plays, for code models on genuinely non-Euclidean graphs.

Problem Statement

For transverse-field Hamiltonians built from classical LDPC codes on non-Euclidean (e.g., expander) geometries, characterize the phase transitions between the symmetry-broken code phase and the trivial paramagnet, and determine how the Kramers-Wannier non-invertible duality symmetry constrains them. Concretely, for codes that are self-dual under the generalized KW duality: (i) establish whether the self-dual point can host a genuinely critical (continuous) phase transition on expander-like graphs, or whether the transition is forced to be first-order or gapped, and how the answer depends on the graph family and its automorphisms (including translation-like symmetries with which the duality can mix); and (ii) derive the constraints that the non-invertible duality symmetry, its fusion algebra, and its anomaly (the obstruction, following from the fusion algebra, to simultaneously preserving the duality symmetry and having a unique gapped ground state) impose on the possible critical points and phase boundaries - for example on allowed ground-state degeneracies, on spontaneous breaking of the duality, on symmetry-enforced gaplessness, and on which critical behaviors, if any, are compatible with the non-Euclidean geometry. This is the surviving core of the question posed by Rakovszky and Khemani; the fusion-rule computation they requested as one deliverable has since been carried out on arbitrary bipartite graphs, so answers may build on the known fusion algebra, and the open objective is the nature and constraint structure of the transitions themselves.

References

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