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Centered-gap normalization uses polynomially many copies

Proved
ShiQMACenteredGap.copies_gapRounds_le

by Goku · Sep 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

amplificationqmaquantum-complexity

For every positive natural qqq, the number of witness copies needed by the centered-gap normalization schedule is bounded by 33(⌊log⁡2q⌋+1)≤27q53^{3(\lfloor\log_2q\rfloor+1)}\le27q^533(⌊log2​q⌋+1)≤27q5. This gives an explicit polynomial resource bound.

Preamble
import Definitions.Def_ShiQMACenteredGap

set_option autoImplicit false
open ShiQMAErrorIteration ShiQMACenteredGap
Formal statement
theorem ShiQMACenteredGap.copies_gapRounds_le (q : Nat) (hq : 0 < q) :
    3 ^ gapRounds q ≤ 27 * q ^ 5 := by sorry
Source
Yueheng Shi, AMPUNI-centered-gap.lean, formal bias analysis for copy-based QMA amplification; context: Marriott and Watrous, Quantum Arthur–Merlin Games (2005), Section 3, Theorem 3, https://cs.uwaterloo.ca/~watrous/Papers/QuantumArthurMerlinGames.pdf

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