{
  "abstract": "Inequality constraints in combinatorial optimization are conventionally handled by introducing slack variables and quadratic penalties, which increase the qubit count and distort the optimization landscape. Slack-free formulations instead apply a nonlinear custom penalty function directly to the constraint, which are difficult to be implemented in quantum approximate optimization algorithm (QAOA). We show that the Walsh--Hadamard transform (WHT) removes this obstruction for any penalty function: a penalty applied to a linear constraint is exactly a weighted sum of Pauli-$Z$ strings. We propose this framework as the Walsh custom penalty QAOA (WCP-QAOA). Truncating the expansion at Walsh degree $k$ costs $\\mathcal{O}(n^k)$ Pauli terms and leaves the qubit count untouched. The truncation is the optimal degree-$k$ approximation of the penalty in $L^2$, and we give a condition on the discarded Walsh coefficients under which the truncated Hamiltonian still ranks every feasible solution below every infeasible one. Since evaluating a Walsh coefficient from its definition requires all $2^n$ penalty values, we give an algorithm that returns any coefficient in sub-exponential time, so that the degree-$k$ Hamiltonian is obtained in time polynomial in $n$ for fixed $k$. We show that for the exponential penalty, there exists a closed product form of WHT, and hence a closed form expression to calculate the Walsh coefficients. We identify structural shortcuts that shorten the exact spectrum. We also tested WCP-QAOA in solving a 6-qubit knapsack problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04908",
  "author_total": 3,
  "authors": [
    {
      "name": "Xin Wei LEE"
    },
    {
      "name": "Siong Thye GOH"
    },
    {
      "name": "Hoong Chuin LAU"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1kjx5xk2be653.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04908",
  "fulltext_url": null,
  "id": "1kjx5xk2be653",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04908",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7220660875",
  "pdf_url": "https://arxiv.org/pdf/2610.04908",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1kjx5xk2be653.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2610.04908",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04908"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Custom Penalty QAOA via Walsh-Hadamard Transform",
  "title_text": "Custom Penalty QAOA via Walsh-Hadamard Transform",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:01Z",
  "url": "https://research.draconitic.com/paper/1kjx5xk2be653",
  "venue": "arXiv",
  "versions": []
}