{
  "abstract": "Quantum circuit optimization is an indispensable component of the quantum software stack, used to reduce noise, maximize performance, alleviate costs of quantum error correction, or target specific hardware. A related problem is approximate membership: for a fixed set of unitaries $S$, decide whether a given circuit $C$ is within distance $\\varepsilon$ of a unitary in $S$ under some metric $m$. Membership is a special case of optimization, where $S$ is a set of unitaries implementable at no cost (e.g., with noise-minimal circuits, or without magic states). Janzing, Wocjan, and Beth (Int. J. Quantum Inf., 2005) showed that, for the operator metric, the case $S=\\{I\\}$ (approximate identity) is QMA-complete when $C$ is promised to be either close to the identity or far from it. We generalize this setting in three directions: (i) promise-free inputs, (ii) besides $S=\\{I\\}$, sets $S$ containing only Clifford unitaries, or only generalized permutation matrices (i.e., no superposition), and (iii) the trace-based, spectral, and operator metrics. We show that, in all cases, approximate membership is PP-hard for wide ranges of $\\varepsilon$, provided $C$'s gate-set can implement Clifford+T, as the gate-sets of standard hardware do. If $S=\\{I\\}$, or if $S$ contains only generalized permutation matrices, $I\\in S$, and $m$ is a trace-based distance, PP-hardness holds for all non-trivial $\\varepsilon$. If $S=\\{I\\}$ and $m$ is a trace-based distance, PP-hardness holds even for circuits over classical-reversible gate-sets. Our results indicate that approximate membership is harder over promise-free inputs than the QMA-complete promise version, and that this hardness persists across quantum resources (any gates, non-Clifford, superposition) and distance metrics. They directly imply PP-hardness of circuit optimization when minimizing total, non-Clifford, or superposition gate depth or count.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04381",
  "author_total": 3,
  "authors": [
    {
      "name": "Adam Husted Kjelstrøm"
    },
    {
      "name": "Jaco van de Pol"
    },
    {
      "name": "Andreas Pavlogiannis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7gqsfx8stc70j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04381",
  "fulltext_url": null,
  "id": "7gqsfx8stc70j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04381",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "quant-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04381",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7gqsfx8stc70j.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:25Z",
      "native_id": "2611.04381",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04381"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "On the Hardness of Approximate Quantum Circuit Membership and Optimization",
  "title_text": "On the Hardness of Approximate Quantum Circuit Membership and Optimization",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/7gqsfx8stc70j",
  "venue": "arXiv",
  "versions": []
}