{
  "abstract": "We prove an exponential separation between standard (XOR) quantum queries and forward-erasing queries for a Boolean property of injective functions, answering a question of Aaronson~\\cite{Aaronson2021Open}. We also obtain an exponential separation for a Boolean promise problem on permutations, where forward in-place queries are unitary but inverse queries are not supplied. Our construction follows Aaronson's suggestion of appending fixed random garbage to a Simon function. Let $X=\\{0,1\\}^n$ and $N=2^n$. We use the injection $f_{h,r}(x)=(h(x),x,r_x)$, where $h:X\\to X$ is promised to be either a permutation or a Simon function, and $r:X\\to X$ is an arbitrary tag table that is reused on every query. The resulting problem is solvable with at most $n+2$ standard queries, but requires $Θ(\\sqrt N)$ forward-erasing queries, matching classical collision sampling. Embedding these instances into permutations, with the copy of $x$ replaced by a single bit that keeps the embedding injective, gives a promise problem on permutations of $L=4N^2$ elements that requires $O(\\log L)$ standard queries versus $Θ(L^{1/4})$ forward in-place queries. The lower bounds use the recording method. We replace each query by a nearby contraction that records cells of a purified tag table. After $T$ queries, tracing out the purified tag table yields a sum of positive-semidefinite contributions, each depending on $h$ at no more than $T$ addresses. A query-by-query hybrid bounds the trace-norm difference between the actual and auxiliary final operators by $O(T/\\sqrt N)$. Random permutations and random Simon functions have the same restriction distribution on such a set unless it contains a hidden Simon pair, an event of probability $O(T^2/N)$. Consequently, the actual final states in the two cases remain close unless $T=Ω(\\sqrt N)$, giving the required lower bounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03628",
  "author_total": 3,
  "authors": [
    {
      "name": "Khaled Elbassioni"
    },
    {
      "name": "Rishikesh Gajjala"
    },
    {
      "name": "Saurabh Ray"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/43nknrqjzgp3s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03628",
  "fulltext_url": null,
  "id": "43nknrqjzgp3s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03628",
  "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",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7208771364",
  "pdf_url": "https://arxiv.org/pdf/2609.03628",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/43nknrqjzgp3s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:48Z",
      "native_id": "2609.03628",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03628"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Random Garbage Separates XOR from Forward-Only Queries",
  "title_text": "Random Garbage Separates XOR from Forward-Only Queries",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/43nknrqjzgp3s",
  "venue": "arXiv",
  "versions": []
}