dorsal/arxiv
View SchemaQuantum algorithms for a set of group theoretic problems
| Authors | Stephen Fenner, Yong Zhang |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0408150 |
| URL | https://arxiv.org/abs/quant-ph/0408150 |
Abstract
We study two group theoretic problems, GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP, in the setting of black-box groups, where DOUBLE COSET MEMBERSHIP generalizes a set of problems, including GROUP MEMBERSHIP, GROUP FACTORIZATION, and COSET INTERSECTION. No polynomial-time classical algorithms are known for these problems. We show that for solvable groups, there exist efficient quantum algorithms for GROUP INTERSECTION if one of the underlying solvable groups has a smoothly solvable commutator subgroup, and for DOUBLE COSET MEMBERSHIP if one of the underlying solvable groups is smoothly solvable. We also study the decision versions of STABILIZER and ORBIT COSET, which generalizes GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP, respectively. We show that they reduce to ORBIT COSET under certain conditions. Finally, we show that DOUBLE COSET MEMBERSHIP and DOUBLE COSET NONMEMBERSHIP have zero knowledge proof systems.
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"date_created": "2026-03-02T18:02:10.223000Z",
"date_modified": "2026-03-02T18:02:10.223000Z",
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"abstract": "We study two group theoretic problems, GROUP INTERSECTION and DOUBLE COSET\nMEMBERSHIP, in the setting of black-box groups, where DOUBLE COSET MEMBERSHIP\ngeneralizes a set of problems, including GROUP MEMBERSHIP, GROUP FACTORIZATION,\nand COSET INTERSECTION. No polynomial-time classical algorithms are known for\nthese problems. We show that for solvable groups, there exist efficient quantum\nalgorithms for GROUP INTERSECTION if one of the underlying solvable groups has\na smoothly solvable commutator subgroup, and for DOUBLE COSET MEMBERSHIP if one\nof the underlying solvable groups is smoothly solvable. We also study the\ndecision versions of STABILIZER and ORBIT COSET, which generalizes GROUP\nINTERSECTION and DOUBLE COSET MEMBERSHIP, respectively. We show that they\nreduce to ORBIT COSET under certain conditions. Finally, we show that DOUBLE\nCOSET MEMBERSHIP and DOUBLE COSET NONMEMBERSHIP have zero knowledge proof\nsystems.",
"arxiv_id": "quant-ph/0408150",
"authors": [
"Stephen Fenner",
"Yong Zhang"
],
"categories": [
"quant-ph"
],
"title": "Quantum algorithms for a set of group theoretic problems",
"url": "https://arxiv.org/abs/quant-ph/0408150"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "4da80620-cdba-48f0-9238-f2dbdaa9ab40",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
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