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Approximate Realization of Four-Dimensional Topological QuantumCodes on Two-Dimensional Architectures via Machine Learning andEntanglement-Assisted Floquet Engineering

Parham Ghayour

Published May 8, 2026• 0 citations
Publisher

Abstract

Abstract Four-dimensional (4D) topological quantum codes occupy a privileged position in quantum information theory: they are among the only known models exhibiting passive self-correction via macroscopic energy barriers and thermodynamic stability. However, physical quantum hardware is inherently limited to two or three spatial dimensions, making literal implementation of such codes impossible. In this work, we ask a different question: whether the logical and operational features of 4D topology can be approximated on two-dimensional (2D) hardware, even if the underlying phase of matter cannot. We introduce a framework for the approximate realization of 4D topological quantum codes on 2D architectures by combining three resources: synthetic dimensions induced by time-periodic control, entanglementassisted nonlocal connectivity, and machine-learning-based optimization. A mapping is constructed from the 4D toric code Hamiltonian to a periodically driven 2D system whose effective Floquet Hamiltonian reproduces the desired stabilizer structure up to controlled error. Virtual dimensions are realized using ancilla registers and time-indexed dynamics, while Bell-pair teleportation and graph-state primitives flatten geometric constraints and enable constant-depth realization of high-dimensional adjacency relations. Machine learning is deployed at three levels: (i) error-aware compilation and Floquet scheduling, (ii) learned stabilization and control pulse design, and (iii) neural decoding robust to mapping-induced distortions. We define precise operational metrics for approximation accuracy using channel distance, logical error rates, and effective barrier proxies, and provide rigorous error bounds from Trotter–Suzuki expansions and Floquet–Magnus theory. Numerical experiments demonstrate that entanglement-assisted routing suppresses geometric overhead, actionable neural decoding significantly reduces logical failure rates at finite size, and Floquet control yields tunable convergence toward ideal 4D dynamics. Although true thermodynamic self-correction remains forbidden in two dimensions, the resulting system exhibits an emergent regime of effective dimensional behavior in which key logical features of four-dimensional topology are operationally reproduced. Our results establish that dimensionality is not solely a geometric constraint but also a programmable resource mediated by entanglement, control, and inference. This work opens a pathway toward exploiting higher-dimensional code structure on near-term planar hardware and introduces a general paradigm for dimensionally enhanced quantum architectures.

DOI
10.21203/rs.3.rs-8331217/v1

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Floquet codes are a novel class of quantum error-correcting codes with dynamically generated logical qubits arising from a periodic schedule of noncommuting measurements. We utilize the interpretation of measurements in terms of condensation of topological excitations and the rewinding of measurement sequences to engineer new examples of Floquet codes. In particular, rewinding is advantageous for obtaining a desired set of instantaneous stabilizer groups on both toric and planar layouts. Our first example is a Floquet code with instantaneous stabilizer codes that have the same topological order as the three-dimensional (3D) toric code(s). This Floquet code also exhibits a splitting of the topological order of the 3D toric code under the associated sequence of measurements, i.e., an instantaneous stabilizer group of a single copy of the 3D toric code in one round transforms into an instantaneous stabilizer group of two copies of the 3D toric code up to nonlocal stabilizers in the following round. We further construct boundaries for this 3D code and argue that stacking it with two copies of the 3D subsystem toric code allows for a transversal implementation of the logical non-Clifford controlled-controlled- Z gate. We also show that the coupled-layer construction of the X-cube Floquet code can be modified by a rewinding schedule such that each of the instantaneous stabilizer codes is finite depth equivalent to the X-cube model up to toric codes; the X-cube Floquet code exhibits a splitting of the X-cube model into a copy of the X-cube model and toric codes under the measurement sequence. Our final 3D example is a generalization of the 2D Floquet toric code on the honeycomb lattice to three dimensions, which has instantaneous stabilizer codes with the same topological order as the 3D fermionic toric code. Published by the American Physical Society 2024
Quantum error-correcting codes on graph geometries beyond regular euclidean lattices have emerged as a promising route toward robust quantum information processing. A major challenge in quantum error correction is the trade-off between favorable properties of codes constructed in high dimensionality and the need for locality in practical implementations on near-term hardware. Sparse intermediate geometries may combine the advantages of both, but how their structure governs quantum information dynamics and code performance remains unclear. Here we show that the spectral dimension dsd_sds​, defined from the low-energy spectrum of the graph Laplacian, is the universal parameter connecting these properties. Using Floquet random Clifford circuits on two distinct sparse graph architectures with tunable spectral dimension, we find that dsd_sds​ governs both quantum information spreading and the scaling of the resulting quantum error-correcting code distance. We identify a geometry-induced entanglement phase transition from localized area-law to ergodic volume-law, whose universality is determined by dsd_sds​ and exhibits long-range percolation criticality. In the ergodic phase, the circuit acts as a CSS code encoder whose code distance is set by the competition between two logical sectors --- the cost of severing the graph and the length of its shortest winding cycle --- which connectivity pushes in opposite directions. The spectral dimension organizes this trade-off, highlighting a maximum at intermediate dsd_sds​ retaining sparse connectivity and physical realizability.
Quantum error correction (QEC) is crucial for realizing scalable quantum technologies, and topological quantum error correction (TQEC) has emerged as the most experimentally advanced paradigm of QEC. Existing homological and topological code constructions, however, are largely confined to orientable two-manifolds with simple boundary conditions. In this work, we develop a unified algebraic-topological framework for TQEC based on homology, cohomology, and intersection theory, which characterizes exactly when an arbitrary-dimensional manifold (with or without boundary) can serve as a quantum memory, thereby extending the standard 2D homological-code picture to arbitrary dimension and to manifolds with boundary via Poincaré-Lefschetz duality. Building on this classification, we introduce concrete code families that exploit nontrivial topology beyond the planar and toric settings. These include ``3-torus code'' and higher-dimensional ``volume codes'' on compact manifolds with mixed XXX- and ZZZ-type boundaries. We further give a topological construction of qudit TQEC codes on general two-dimensional cell complexes using group presentation complexes, which unifies and extends several known quantum LDPC and homological-product-like constructions within a single geometric language. Finally, we combine the theoretical framework with numerical simulations to demonstrate that changing only the global topology can yield improved logical performance at fixed entanglement resources. Taken together, our results provide a systematic set of topological design principles for constructing and analyzing TQEC codes across dimensions and boundaries, and they open new avenues for topology-aware fault-tolerant quantum architectures.