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  "id": "oai:papers.qnfo.org:computation-as-hamiltonian-isomorphism-a-mode-based-taxonomy-of-physical-computi",
  "slug": "computation-as-hamiltonian-isomorphism-a-mode-based-taxonomy-of-physical-computi",
  "title": "Computation as Hamiltonian Isomorphism: A Mode-Based Taxonomy of Physical Computing Substrates",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Special-purpose accelerators, neuromorphic chips, and optical processors are today evaluated by qubit or transistor counts, yet the machines that deliver the largest effective speedups are those whose physical dynamics directly embodies the target problem. This paper addresses a specific gap: no existing framework defines when a physical system's relaxation dynamics is equivalent to a problem's solution structure, and no hardware taxonomy classifies machines by the mechanism that achieves many candidate solutions simultaneously. We formalize \"Hamiltonian isomorphism\" as an equivalence relation between a target problem's cost structure and a physical system's energy function, and we propose classifying computing hardware by its physical simultaneity mechanism rather than by its information unit. We replace the qubit with the \"mode\" (a unit of symmetry) as the fundamental resource count. As a principal quantitative result, we derive that the thermodynamic floor for a gate-based logical operation under a stated error-correction overhead of 10² to 10³ physical operations per logical operation is 2.87× 10^-19 to 2.87× 10^-18 joules at 300 K, while a relaxation machine that reaches its ground state once pays only the single-bit resetting cost of 2.87× 10^-21 joules per output read — a separation of two to three orders of magnitude that motivates the taxonomy. We prove which structural conditions a problem class must satisfy to admit a Hamiltonian isomorphism, and we show the classification problem itself is at least as hard as known isomorphism problems for computable structures. The framework changes how accelerator architects state design goals: from maximizing gate fidelity to engineering better Hamiltonians.",
  "abstract_markdown": "Special-purpose accelerators, neuromorphic chips, and optical processors are today evaluated by qubit or transistor counts, yet the machines that deliver the largest effective speedups are those whose physical dynamics directly embodies the target problem. This paper addresses a specific gap: no existing framework defines when a physical system's relaxation dynamics is *equivalent* to a problem's solution structure, and no hardware taxonomy classifies machines by the mechanism that achieves many candidate solutions simultaneously. We formalize \"Hamiltonian isomorphism\" as an equivalence relation between a target problem's cost structure and a physical system's energy function, and we propose classifying computing hardware by its physical simultaneity mechanism rather than by its information unit. We replace the qubit with the \"mode\" (a unit of symmetry) as the fundamental resource count. As a principal quantitative result, we derive that the thermodynamic floor for a gate-based logical operation under a stated error-correction overhead of $10^{2}$ to $10^{3}$ physical operations per logical operation is $2.87\\times 10^{-19}$ to $2.87\\times 10^{-18}$ joules at 300 K, while a relaxation machine that reaches its ground state once pays only the single-bit resetting cost of $2.87\\times 10^{-21}$ joules per output read — a separation of two to three orders of magnitude that motivates the taxonomy. We prove which structural conditions a problem class must satisfy to admit a Hamiltonian isomorphism, and we show the classification problem itself is at least as hard as known isomorphism problems for computable structures. The framework changes how accelerator architects state design goals: from maximizing gate fidelity to engineering better Hamiltonians.",
  "date": "2026-10-15",
  "first_published": "2026-10-15",
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  "url": "https://papers.qnfo.org/papers/computation-as-hamiltonian-isomorphism-a-mode-based-taxonomy-of-physical-computi",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "Computation as Hamiltonian Isomorphism: A Mode-Based Taxonomy of Physical Computing Substrates"
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