Quantum von Neumann Machine and the Cost of Coherent Programmes

Hutan Ashrafian

Published Oct 31, 2026• 0 citations

Abstract

A quantum von Neumann machine that retains its stored programme in superposition must execute the selected operations without destroying coherence between programme branches, creating a resource requirement that cannot be assessed by examining each programme separately. I establish this distinction for programmes that select controlled-Z operations on the edges of a graph. Every classically selected programme uses only Clifford operations, yet coherent execution requires a number of injected non-Clifford T states proportional to the programme length for every fixed normalised diamond-distance tolerance below one half. For a graph with m edges and n active data vertices, the lower bound for exact execution is m + n states, whilst an explicit implementation uses at most 4m. The lower bound permits arbitrary programme-dependent phases, stabilizer workspace, measurements and adaptive control. Its extension to imperfect execution bounds both the probability of substantial resource consumption and the expected consumption, without imposing a separate limit on workspace or measurement count. The proof derives a uniform bound on Pauli expectations of the joint programme-data output and uses its stabilizer subspaces to constrain every low-resource trajectory. A supporting criterion identifies precisely when a Clifford instruction library admits an exact Clifford interpreter under the chosen binary programme encoding. These results separate the cost of interpreting coherent instructions from the cost of the operations selected by a classical programme, providing a quantitative constraint on compilation and resource estimation for stored-programme quantum computation.

DOI: 10.5281/zenodo.23841118

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Quantum von Neumann Machine and the Cost of Coherent Programmes

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Published Oct 31, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.23841118

Related papers

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Together, the TUS OS, periodically modulated decay and dual-Poisson results establish three distinct exact contract-relative regimes: strict finite separation: 4 → 64; unbounded finite separation: 2 → N; finite-capacity breakdown: 2 → ∅finite for every τ > 0. These are three instances of one general structural law, not three identical recovery mechanisms. TUS OS uses admitted experiment families that collectively separate its canonical carriers, whereas the stochastic-process results use direct causal-state decoding at a declared inspection boundary. The results establish that finite-capacity behavioural sufficiency does not imply positive uniform worst-case recoverability of native predictive identity, and that exact representation cost is not a context-free property of the native system alone. It depends on the complete typed obligation imposed on an admitted realisation. The paper does not claim physical quantum execution, quantum advantage, a universal scalar resource, equivalence of TUS OS recovery with causal-state transparency, or finite-capacity breakdown for prior-weighted dual-Poisson transparency. Supporting General Theory Programme This paper is Paper 6 in the TUS OS quantum-operational and Answerability research programme. Building on Papers 1–5, it develops the general theory of contract-relative quantum realisations of classical processes, showing that quantum resource requirements are not intrinsic to a process alone but depend on the governing observational, recovery, and transparency contract. Paper 7 extends this into operational observability and the classification of contract-induced resource transitions; Papers 8–9 develop generic Answerability theory and calculus; Paper 10 proves the protected and executable realisation of TUS OS within that calculus; and Paper 11 closes the programme with A General Theory of Answerability, establishing Structural Identity, Recognition, the canonical mandatory protected core, and lawful Realisation Freedom. Paper 1 — Canonical Tetrahedral Qubit-State Geometry in the TUS OS TriadXOR Completion, Receipt Verification and the Qubit-Geometric Semantic WitnessEstablishes the canonical tetrahedral geometry, unique XOR completion, exact qubit-SIC representation and centred-face trine geometry. Paper 2 — The TUS OS Operational CalculusClassical Construction of Exact Qubit Prepare–Measure Behaviour from a Receipt-Custodied Engine ArchitectureEstablishes exact full-carrier qubit prepare–measure behaviour from the deterministic classical engine under the declared Coordinate-Incidence Operational Harness. For each lawful foundation, the 64 six-bit carriers map to exactly 27 anchor-relative qubit preparations, while the separately declared face Harness gives the exact trine table. Paper 3 — From Finite Relational Structure to Qubit Operational FormAmbient Foundations, Canonical Subatlas, Exact Global Quantum Dimension, GUIDANCE Witness, Sequential Closure and Global Representation in TUS OSEstablishes the repeated-common-effect static Atlas, its exact minimum complex Hilbert dimension 4, exact carrier-reconstruction structure and deterministic predictive-state minimum 64 for the frozen complete principal future language. Paper 4 — Technical Volume I — Formal Foundations and the Complete-Future CorpusNative Contract, Statistical Discrimination and the Invariant Probability ModuleEstablishes the frozen complete principal terminal future-experiment language, complete statistical separation of distinct carrier pairs, exact parity-resolved discrimination results, complement reachability and the 33-dimensional rational invariant probability module. Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic SynthesisEstablishes universal complement-support orthogonality and its propagation under exact common-CPTP dynamics, the resulting common-CPTP process bounds, recovery-complete LANDING and GUIDANCE results, exact 64-dimensional selected-line instrument/readout minima at their declared contracts, an exact classical–quantum hybrid construction, and the native structural/predictive quotient correspondence. The state-only common-CPTP minimum remains bounded within 32–64 rather than being fixed at 64. Paper 5 — TUS OS as an Anchored Analog Quantum SystemNative Witnesses, Predictive Synthesis, and Access-Contract Resource SeparationDefines Analog Quantum Views and Anchored Analog Quantum Systems, proves that TUS OS is a proper Anchored Analog Quantum System on its 64-input abstract canonical corpus, elevates the inherited structural/predictive correspondence into the Predictive Synthesis Identity Theorem, and composes the recognition, predictive, resource and downstream-conservation results into the TUS OS Anchored Analog Quantum System Theorem. Paper 6 — Contract-Relative Quantum Realisations of Classical ProcessesPredictive Compression, Vanishing Worst-Case Causal-State Transparency, and Finite-Dimensional BreakdownGeneralises the access-dependent resource phenomenon beyond the TUS-specific discovery corpus by formalising the contract-relative achievable-resource object (S,C,𝔐,ε) ⟼ Ach(S,C;𝔐,ε), and establishes three exact contract-relative regimes: strict finite separation, unbounded finite separation and finite-capacity breakdown. These are instantiated respectively by the TUS OS 4→64 separation, the periodically modulated decay 2→N exact prediction-to-transparency separation, and dual-Poisson finite-capacity breakdown for every positive uniform worst-case transparency threshold. 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Aug 14, 20260 citations

n

copies of a

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-dimensional system, we construct a Clifford+T circuit that block-encodes the symmetric projector with operator-norm error at most

ε

using

\widetilde{O}(n)

T gates, where the tilde-

O

notation suppresses polylogarithmic factors in

n

d

, and

1/ε

. Conversely, using stabilizer nullity, we prove that any Clifford+T implementation of such a block encoding with error

ε<\frac{1}{2}\sqrt{1-1/n}

requires at least

2n-2

T gates, even allowing arbitrary positive normalization and clean stabilizer ancillas. Taken together, these results establish the nearly optimal scaling of non-Clifford cost with the number of copies and uncover the fundamental resource requirements for coherent symmetry projection in quantum state purification.