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Holographic Emergence and Spectral Stability: Bridging the HaPPY Code and a Spectral Operator Framework

Mohamed Nour kayad

Published Aug 24, 2026• 0 citations
Publisher

Abstract

Recent advances in quantum error correction and holographic tensor networksculminating in the HaPPY code and its experimental realizations on trapped-ion processorsprovide concrete evidence that spacetime geometry can emerge from the quantum information encoded on a boundary. In this paper we bridge the Harlow-Pastawski-Preskill-Yoshida (HaPPY) tensor network with a spectral operator framework inspired by critical-line states with Re(s) = 1/2. We argue that spacetime curvature, gravitational binding, and particle connement can be understood as macroscopic manifestations of an underlying quantum error-correcting code. The central object of our proposal is a spectral isomorphism conjecture linking holographic boundary states to eigenstates of a self-adjoint operator whose spectrum is supported on the criticalline. We complement this conjecture with structural results on Stiefel-manifold retractions anddiscuss the status of the framework as a program of conjectures to be tested numerically and experimentally.

DOI
10.5281/zenodo.22074285

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Recent advances in quantum error correction and holographic tensor networksculminating in the HaPPY code and its experimental realizations on trapped-ion processorsprovide concrete evidence that spacetime geometry can emerge from the quantum information encoded on a boundary. In this paper we bridge the Harlow-Pastawski-Preskill-Yoshida (HaPPY) tensor network with a spectral operator framework inspired by critical-line states with Re(s) = 1/2. We argue that spacetime curvature, gravitational binding, and particle connement can be understood as macroscopic manifestations of an underlying quantum error-correcting code. The central object of our proposal is a spectral isomorphism conjecture linking holographic boundary states to eigenstates of a self-adjoint operator whose spectrum is supported on the criticalline. We complement this conjecture with structural results on Stiefel-manifold retractions anddiscuss the status of the framework as a program of conjectures to be tested numerically and experimentally.
The unification of quantum mechanics and general relativity remains one of the major open problems of theoretical physics. The Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence provides a valuable theoretical framework for this effort via a holographic duality between a theory of quantum gravity in asymptotically AdS spacetime and a conformal quantum field theory on the lower-dimensional boundary. Here, we implement a toy model of this duality called the HaPPY code, a quantum error-correcting code in the form of a tensor network with hyperbolic entanglement patterns, on a trapped-ion quantum computer. We present the first experimental confirmation of the Faulkner-Lewkowycz-Maldacena formula in this model - a key test of the holographic correspondence. We then enrich it with non-stabilizerness, or magic, and observe entropic precursors expected of emergent gravity. Finally, we present and measure a code construction whose entropic behavior is reminiscent of a highly quantum wormhole. Our experiments illustrate how quantum computers can serve as testbeds for modeling the emergence of spacetime.
This paper unifies holographic quantum error-correcting codes, the internaldynamics of black holes, and the Einstein–Rosen bridge within a mathematicalframework centered on the category of constraint networks and spectral flow. Byencoding tensor networks as weighted hypergraphs and introducing a faithful functor, a strict duality between bulk spacetime geometry and boundary quantuminformation is established. In this framework, the code distance is determined bythe Cheeger isoperimetric lower bound and is shown to be equivalent to the invariant features of the Laplacian spectral flow under coarse-graining; black holeevaporation is described as a percolation phase transition caused by the failure ofinternal entanglement structures, where the failure of the condensation operatorautomatically triggers the splitting operator, leading to branching of quantum extremal surfaces and quantitative constraints on information release governed by theinformation conservation axiom; the formation of the ER bridge corresponds to atopological mutation induced by the rewiring functor, whose throat area is rigidlydetermined by the information-potential difference across the boundary and thenetwork compression ratio. The two axioms of information conservation and computability run throughout the entire treatment, reducing all geometric connectivitydecision problems to polynomial-time solvable graph algorithms. We further givespecific numerical predictions for the code-distance enhancement of non-Abeliansymmetric tensor networks, the acceleration of wormhole connectivity computability, and the critical behavior of the condensation–splitting phase transition, anddesign a simulation verification scheme independent of external datasets.
We generalize the Pastawski-Yoshida-Harlow-Preskill (HaPPY) holographic quantum error-correcting code to provide a toy model for bulk gauge fields or linearized gravitons. The key new elements are the introduction of degrees of freedom on the links (edges) of the associated tensor network and their connection to further copies of the HaPPY code by an appropriate isometry. The result is a model in which boundary regions allow the reconstruction of bulk algebras with central elements living on the interior edges of the (greedy) entanglement wedge, and where these central elements can also be reconstructed from complementary bounda ry regions. In addition, the entropy of boundary regions receives both Ryu-Takayanagi-like contributions and further corrections that model the δArea4GN \frac{\delta \mathrm{Area}}{4{G}_N} 4GN​δArea​ term of Faulkner, Lewkowycz, and Maldacena. Comparison with Yang-Mills theory then suggests that this δArea4GN \frac{\delta \mathrm{Area}}{4{G}_N} 4GN​δArea​ term can be reinterpreted as a part of the bulk entropy of gravitons under an appropriate extension of the physical bulk Hilbert space.
This study extends the framework of non-equilibrium Hamiltonian field theory and algebraic coset partitioning established in preceding architectures to holographic quantum information manifold sectors, providing an explicit formulation for the macroscopic emergence of anti-de Sitter (AdS) spacetimes driven by the variational minimization of codimension-two entangling surfaces inside boundary conformal field theories (CFT). By embedding the Helfrich elastic bending grammar into the generalized Ryu-Takayanagi entropy functional, the spontaneous emergence of localized bulk horizons coupled to quantum error-correcting stabilizer codes is formalized. Partitioning the total configuration space under tensor network constraints demonstrates that non-local entanglement geometries undergo algebraic degeneration into exactly three universal holographic cosets.
The AdS/CFT correspondence conjectures a holographic duality between gravity in a bulk space and a critical quantum field theory on its boundary. Tensor networks have come to provide toy models to understand these bulk-boundary correspondences, shedding light on connections between geometry and entanglement. We introduce a versatile and efficient framework for studying tensor networks, extending previous tools for Gaussian matchgate tensors in 1 + 1 dimensions. Using regular bulk tilings, we show that the critical Ising theory can be realized on the boundary of both flat and hyperbolic bulk lattices, obtaining highly accurate critical data. Within our framework, we also produce translation-invariant critical states by an efficiently contractible tensor network with the geometry of the multiscale entanglement renormalization ansatz. Furthermore, we establish a link between holographic quantum error-correcting codes and tensor networks. This work is expected to stimulate a more comprehensive study of tensor network models capturing bulk-boundary correspondences.
This report offers a comprehensive academic synthesis of the information-theoretic paradigm in fundamental physics, in which spacetime geometry, vacuum structure, and gravitational dynamics arise from the entanglement and error-correcting properties of quantum many-body systems. Beginning with the Toric Code, Color Codes, and string-net condensates, the review explains how gauge fields, fermionic statistics, and virtual particle structure emerge from stabilizer constraints and topological order. It then explores Fracton phases and subsystem symmetries as geometric substrates capable of reproducing the constraints and potentials of linearized gravity. The analysis extends into holographic quantum error correction—including MERA, the HaPPY code, and random tensor networks—to show how bulk locality, black hole entropy, and AdS geometry emerge from entanglement structure. The Hyperbolic Fracton Model is examined as a synthesis that unifies fractonic mobility constraints with holographic scaling and area-law entropy. The report supports a unified “code-theoretic” framework in which physical laws and spacetime arise from the structure of an underlying quantum code.