A Noise-Invariant Determinism Theorem for Multi-Base Post-Processing in Shor's Order Finding (v0.5.0: nonstabilizerness / magic across quantum speedups)

v0.5.0 adds a new direction to the multi-base order-finding work: nonstabilizerness ("magic") across quantum speedups, measured and proven on a numpy statevector simulation. Retained unchanged: six theorems on classical post-processing of Shor's order finding (T1 noise-invariant determinism, T2 logarithmic coverage time, T3 exact noise scaling 1/g_M(eta), T4 Regev-2023 compatibility, T5 hybrid (C)+Regev b-trick factoring, T6 stochastic-resonance no-go) and the closed-form sigma-curve verified across five algorithm classes. New (magic): using a verified stabilizer 2-Renyi entropy (SRE) tool via the XOR fast Walsh-Hadamard transform (cross-checked to 1e-15), the amount/density of nonstabilizerness separates the TYPE of quantum speedup. (i) Grover's search magic peaks at exactly 3 bits as a function of problem size and returns to zero at the answer (density M2/n -> 0; closed form, peak = max of -log2(a^8+(1-a^2)^4) at a^2=1/2). (ii) The function graph state (1/sqrt N) sum_x |x>|f(x)> has M2=0 iff f is F2-affine, so Simon with a linear (Clifford) oracle generates zero magic despite an exponential query speedup, while Shor's modular exponentiation is nonlinear and forces magic — formalizing how an oracle/FFT black box hides the nonlinearity that is magic's source. (iii) Shor's post-measurement comb magic grows with problem size, completing a speedup ladder: Clifford/affine -> 0, Grover (quadratic) -> bounded density 0, Shor (exponential) -> growing toward maximal. Includes a 42-assertion proposition checker and a prior-art survey vs arXiv:2605.05347 (Shor magic), Phys. Rev. Research 10.1103/7rwg-lhpv and Phys. Rev. B 10.1103/nzrp-49mr (quantum-walk magic). numpy + Python stdlib only.

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Publication Details

Journal
Open MIND
Published
2026-06-17
DOI
https://doi.org/10.5281/zenodo.20725965
Primary Topic
Quantum Computing Algorithms and Architecture
Type
preprint
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preprint

A Noise-Invariant Determinism Theorem for Multi-Base Post-Processing in Shor's Order Finding (v0.5.0: nonstabilizerness / magic across quantum speedups)

Hashevolution
Open MIND
Quantum Computing Algorithms and Architecture
preprint

A Noise-Invariant Determinism Theorem for Multi-Base Post-Processing in Shor's Order Finding (v0.5.0: nonstabilizerness / magic across quantum speedups)

Hashevolution
preprint en

Abstract

v0.5.0 adds a new direction to the multi-base order-finding work: nonstabilizerness ("magic") across quantum speedups, measured and proven on a numpy statevector simulation. Retained unchanged: six theorems on classical post-processing of Shor's order finding (T1 noise-invariant determinism, T2 logarithmic coverage time, T3 exact noise scaling 1/g_M(eta), T4 Regev-2023 compatibility, T5 hybrid (C)+Regev b-trick factoring, T6 stochastic-resonance no-go) and the closed-form sigma-curve verified across five algorithm classes. New (magic): using a verified stabilizer 2-Renyi entropy (SRE) tool via the XOR fast Walsh-Hadamard transform (cross-checked to 1e-15), the amount/density of nonstabilizerness separates the TYPE of quantum speedup. (i) Grover's search magic peaks at exactly 3 bits as a function of problem size and returns to zero at the answer (density M2/n -> 0; closed form, peak = max of -log2(a^8+(1-a^2)^4) at a^2=1/2). (ii) The function graph state (1/sqrt N) sum_x |x>|f(x)> has M2=0 iff f is F2-affine, so Simon with a linear (Clifford) oracle generates zero magic despite an exponential query speedup, while Shor's modular exponentiation is nonlinear and forces magic — formalizing how an oracle/FFT black box hides the nonlinearity that is magic's source. (iii) Shor's post-measurement comb magic grows with problem size, completing a speedup ladder: Clifford/affine -> 0, Grover (quadratic) -> bounded density 0, Shor (exponential) -> growing toward maximal. Includes a 42-assertion proposition checker and a prior-art survey vs arXiv:2605.05347 (Shor magic), Phys. Rev. Research 10.1103/7rwg-lhpv and Phys. Rev. B 10.1103/nzrp-49mr (quantum-walk magic). numpy + Python stdlib only.

Open MIND
Quantum Computing Algorithms and Architecture
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