Pith Number

pith:YUND3VSG

pith:2026:YUND3VSGE2ZTCVOFAKS22257H6

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$\mathcal{O}(n)$ alternative to Quantum Fourier Transform with efficient neural net classical post-processing

Kaiming Bian, Oscar Dahlsten, Zujin Wen

An O(n) circuit using Hadamards and controlled-phase gates can replace the O(n squared) quantum Fourier transform in Shor's algorithm with neural-net post-processing.

arxiv:2605.16998 v1 · 2026-05-16 · quant-ph · cs.LG

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Claims

C1strongest claim

The O(n) HP-1 can replace the O(n^2) QFT in Shor's algorithm, as demonstrated numerically, with an efficient neural network implementing classical post-processing.

C2weakest assumption

That the neural network post-processing can reliably extract the hidden subgroup generator from the measurement statistics of the HP-1 circuit for arbitrary problem sizes without requiring resources that grow exponentially with n or failing under realistic noise.

C3one line summary

HP-1 circuits achieve O(n) depth while preserving shift invariance and exponentially growing Fisher information, enabling numerical replacement of the QFT in Shor's algorithm with neural net classical post-processing.

References

38 extracted · 38 resolved · 5 Pith anchors

[1] D. R. Simon, On the power of quantum computation, SIAM Journal on Computing26, 1474 (1997) 1997

[2] P. W. Shor, Algorithms for quantum computation: Dis- crete logarithms and factoring, inProceedings 35th An- nual Symposium on Foundations of Computer Science (IEEE, 1994) pp. 124–134 1994

[3] P. W. Shor, Polynomial-time algorithms for prime factor- ization and discrete logarithms on a quantum computer, SIAM Journal on Computing26, 1484 (1997) 1997

[4] Quantum measurements and the Abelian Stabilizer Problem 1995 · arXiv:quant-ph/9511026

[5] Jozsa, Quantum factoring, discrete logarithms, and the hidden subgroup problem, Computing in Science & Engineering3, 34 (2001) 2001

Formal links

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Receipt and verification

First computed	2026-05-20T00:03:35.196470Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c51a3dd64626b33155c502a5ad6bbf3f84ea4c54ff7f4c1369aa8bb24fe238dc

Aliases

arxiv: 2605.16998 · arxiv_version: 2605.16998v1 · doi: 10.48550/arxiv.2605.16998 · pith_short_12: YUND3VSGE2ZT · pith_short_16: YUND3VSGE2ZTCVOF · pith_short_8: YUND3VSG

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/YUND3VSGE2ZTCVOFAKS22257H6 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: c51a3dd64626b33155c502a5ad6bbf3f84ea4c54ff7f4c1369aa8bb24fe238dc

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "57a2fd942f280f5454e7e2f2bf30fc28173bba97086d66fac550be5a1ad00ae4",
    "cross_cats_sorted": [
      "cs.LG"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "quant-ph",
    "submitted_at": "2026-05-16T13:47:11Z",
    "title_canon_sha256": "f98b16b93984d35fa06f227a3fea493ede763b73a5624cbbb111affe38a38ae8"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16998",
    "kind": "arxiv",
    "version": 1
  }
}