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arXiv:2610.22750

The Classical Boundary of Quantum Computation: History-Resolved Causal Stochastic Machines and the Cost of Classicalization

Kyoungho Cho, Junghee Ryu, Minsu Kim, Jeongho Bang

quant-phSubmitted Oct 15, 2026• Announced Oct 19, 2026• 0 citations
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Abstract

What separates quantum computation from classical computation when a classical model may retain every branch history and arbitrary causal memory? We study this question using history-resolved causal stochastic machines (HCSMs), which describe positive causal history laws. Probabilistic Turing machines (PTMs) embed in stochastic-ensemble models (SEnMs), which in turn embed in HCSMs. We prove that HCSM history laws coincide exactly with the fixed-basis, prefix-preserving diagonal sector of ideal quantum processes, denoted by $\mathsf{PTM}_{\rm proc}$ and distinguished here from algorithmic quantum Turing machines (QTMs). Coherent quantum processes extend strictly beyond this sector. The boundary is already visible in reversibility. Born-probability shadows of unitary gates preserve inverse composition exactly for basis permutations with phases. Outside this class, an exact positive stochastic simulation that supports universal quantum continuations must retain predictive information absent from visible probabilities. An inverse-recovery identity quantifies this phase ledger, and universal continuation forces the retained probability law to be tomographically sufficient for the quantum state. Under continuous full-state encoding assumptions, this requirement yields quantum-state dimension lower bounds. The familiar slogan $\mathsf{PTM}\subseteq\mathsf{QTM}$ is thereby sharpened into a precise classical boundary and a no-free-classicalization principle. Complete classical branching does not remove the predictive information needed to represent quantum coherence.

Comments
28 pages (including Appendix), 5 figures

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How much internal quantum memory is needed to reproduce a stationary stochastic process, and how does reducing that memory change the environmental resources required when output records are retained? This paper establishes exact quantum-source constructions and resource bounds under explicitly specified readout and retained-output conditions. A general completion theorem identifies the minimum Hilbert-space dimension for stationary triple data in the orthogonal-current-symbol source class: it equals the sum of the complex positive-semidefinite ranks of the middle-symbol slices. An attaining construction reproduces the entire canonical stationary order-two process. For the delayed trine process, the minimum classical hidden Markov model requires nine states at every non-independent parameter value. Quantum instruments reproduce the same complete output law using four-dimensional memory throughout the parameter range, with four dimensions proved necessary in a certified low-noise regime and three dimensions both sufficient and necessary on an explicit high-noise interval. The paper then separates reusable memory dimension from environmental dilation cost. On the certified high-noise interval, the exact one-step Pareto frontier consists of a three-dimensional source with recorded-channel Choi rank nine and a four-dimensional source with rank three. Here, Choi rank measures the minimum initially pure environmental remainder dimension for the channel retaining both the emitted classical symbol and final quantum memory. A separate mixed-support qutrit construction at noise parameter 0.99 improves complete-history costs and gives upper bounds close to universal lower bounds. These results show why optimizing a single step need not optimize a retained multi-step history. The analysis also distinguishes complete channels, fixed-input states and archive-only outputs, and demonstrates how access to a retained reference can distinguish generators with identical passive output statistics. Full-word proofs, algebraic complete-positivity certificates, a technical companion and reproducibility materials accompany the results. The exact intermediate memory boundary and complete-history optimum remain open. All results use standard quantum probabilities and apply to the stated operational resource models; they do not establish total hardware, energy or thermodynamic minima.
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