Ultrametric Relaxation Dynamics in Topological Quantum Memory (Expanded Narrative)

The scalability of Topological Quantum Computing (TQC) is currently impeded by a “Thermodynamic Wall,” where the entropy generation from active error correction cycles scales exponentially with logical qubit count. This paper proposes a paradigm shift from active gate synthesis to passive topological relaxation, grounded in the Quni-Gudzinas hypothesis (2025). By establishing a rigorous functorial isomorphism between the inverse limit of abelian anyonic braid groups and the p-adic solenoid ( $\Sigma_p$ ), we demonstrate that the vacuum structure of specific strain-engineered materials can encode topological quantum information. We derive a 2D Hamiltonian for a hierarchical “synthetic vacuum” where relaxation dynamics follow an ultrametric trajectory, effectively freezing the system into a protected topological sector without external intervention. While this architecture is limited to abelian topological sectors and thus functions primarily as a quantum memory rather than a universal processor, simulation results indicate that it reduces thermodynamic overhead by orders of magnitude compared to surface code implementations, offering a viable path toward macroscopic fault tolerance for storage.The scalability of Topological Quantum Computing (TQC) is currently impeded by a “Thermodynamic Wall,” where the entropy generation from active error correction cycles scales exponentially with logical qubit count. This paper proposes a paradigm shift from active gate synthesis to passive topological relaxation, grounded in the Quni-Gudzinas hypothesis (2025). By establishing a rigorous functorial isomorphism between the inverse limit of abelian anyonic braid groups and the p-adic solenoid ($\Sigma_p$), we demonstrate that the vacuum structure of specific strain-engineered materials can encode topological quantum information. We derive a 2D Hamiltonian for a hierarchical “synthetic vacuum” where relaxation dynamics follow an ultrametric trajectory, effectively freezing the system into a protected topological sector without external intervention. While this architecture is limited to abelian topological sectors and thus functions primarily as a quantum memory rather than a universal processor, simulation results indicate that it reduces thermodynamic overhead by orders of magnitude compared to surface code implementations, offering a viable path toward macroscopic fault tolerance for storage.

Feb 14, 20260 citations

Memory-critical dynamical buildup of phonon-dressed Majorana fermions

Oliver Kaestle, Yue Sun, Ying Hu +1 more

We investigate the dynamical interplay between the topological state of matter and a non-Markovian dissipation, which gives rise to a crucial timescale into the system dynamics due to its quantum memory. We specifically study a one-dimensional polaronic topological superconductor with phonon-dressed $p$ -wave pairing, when a fast temperature increase in surrounding phonons induces an open-system dynamics. We show that when the memory depth increases, the Majorana edge dynamics transits from relaxing monotonically to a plateau of substantial value into a collapse-and-buildup behavior, even when the polaron Hamiltonian is close to the topological phase boundary. Above a critical memory depth, the system can approach a new dressed state of the topological superconductor in dynamical equilibrium with phonons, with nearly full buildup of the Majorana correlation.We investigate the dynamical interplay between the topological state of matter and a non-Markovian dissipation, which gives rise to a crucial timescale into the system dynamics due to its quantum memory. We specifically study a one-dimensional polaronic topological superconductor with phonon-dressed $p$-wave pairing, when a fast temperature increase in surrounding phonons induces an open-system dynamics. We show that when the memory depth increases, the Majorana edge dynamics transits from relaxing monotonically to a plateau of substantial value into a collapse-and-buildup behavior, even when the polaron Hamiltonian is close to the topological phase boundary. Above a critical memory depth, the system can approach a new dressed state of the topological superconductor in dynamical equilibrium with phonons, with nearly full buildup of the Majorana correlation.

arXiv:2006.13529Physical review. B./Physical review. BDec 11, 20201 citation

Quantum memory on topological spin glass

Jeongwan Haah, Sergey B. Bravyi

Bulletin of the American Physical SocietyFeb 28, 20121 citation

Ultrametric Quantum Computing: Tree-Topology Error Correction

Rowan Brad Quni-Gudzinas

The observation that positional notation carries a native ultrametric tree structure extends into quantum computing through the tensor-product architecture of qubit registers. We formalize three theses: (I) a tree-topology quantum processor confines errors to subtrees via the strong triangle inequality, elevating the error-correction threshold; (II) a 3D-integrated superconducting architecture realizing a 7-ary tree of depth 3 (343 physical qubits) is manufacturable with current fabrication technology; (III) an on-chip bank of prime-frequency resonators enables spectral engineering of the tree's error-confinement properties. We implement a numerical simulation comparing tree-topology and grid-topology codes, deriving threshold estimates and error-propagation statistics. The chapter extends the Ultrametric Foundation thesis (§12) with formal derivations, architecture specifications, simulation results, and a mapping to the broader research program. ---The observation that positional notation carries a native ultrametric tree structure extends into quantum computing through the tensor-product architecture of qubit registers. We formalize three theses: (I) a tree-topology quantum processor confines errors to subtrees via the strong triangle inequality, elevating the error-correction threshold; (II) a 3D-integrated superconducting architecture realizing a 7-ary tree of depth 3 (343 physical qubits) is manufacturable with current fabrication technology; (III) an on-chip bank of prime-frequency resonators enables spectral engineering of the tree's error-confinement properties. We implement a numerical simulation comparing tree-topology and grid-topology codes, deriving threshold estimates and error-propagation statistics. The chapter extends the Ultrametric Foundation thesis (§12) with formal derivations, architecture specifications, simulation results, and a mapping to the broader research program. ---

Jun 29, 20260 citations

Glassy topological quantum memories in two-dimensions

Benjamin J. Brown, Abbas Al-Shimary, Jiannis K. Pachos

Jul 23, 20132 citations

\Sigma_p

), we demonstrate that the vacuum structure of specific strain-engineered materials can encode topological quantum information. We derive a 2D Hamiltonian for a hierarchical “synthetic vacuum” where relaxation dynamics follow an ultrametric trajectory, effectively freezing the system into a protected topological sector without external intervention. While this architecture is limited to abelian topological sectors and thus functions primarily as a quantum memory rather than a universal processor, simulation results indicate that it reduces thermodynamic overhead by orders of magnitude compared to surface code implementations, offering a viable path toward macroscopic fault tolerance for storage.

p

-wave pairing, when a fast temperature increase in surrounding phonons induces an open-system dynamics. We show that when the memory depth increases, the Majorana edge dynamics transits from relaxing monotonically to a plateau of substantial value into a collapse-and-buildup behavior, even when the polaron Hamiltonian is close to the topological phase boundary. Above a critical memory depth, the system can approach a new dressed state of the topological superconductor in dynamical equilibrium with phonons, with nearly full buildup of the Majorana correlation.