In L-subset finding, a black box f maps a domain D of size N into a range R, and a property P picks out some L-element subsets by their arguments and values. The task is to find an L-subset {x₁, …, x_L} of D whose pairs ((x₁, f(x₁)), …, (x_L, f(x_L))) satisfy P, or to reject if none exists, using as few queries as possible. Unstructured search is the case L = 1 and element distinctness the case L = 2.
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In L-subset finding, a black box f maps a domain D of size N into a range R, and a property P picks out some L-element subsets by their arguments and values. The task is to find an L-subset {x₁, …, x_L} of D whose pairs ((x₁, f(x₁)), …, (x_L, f(x_L))) satisfy P, or to reject if none exists, using as few queries as possible. Unstructured search is the case L = 1 and element distinctness the case L = 2. Childs and Eisenberg rework Ambainis's discrete-time quantum walk for element distinctness. The walk runs on a bipartite graph whose vertices are the size-M and size-(M+1) subsets of D, each vertex carrying the subset together with the function values already queried for it. A shift operator costing one query moves between the two subset sizes, Grover diffusion operators act as coins on the element added or removed, and a phase flip marks any subset that already contains a solution — a check that costs no further queries once the values are stored. Running the walk interleaved with the phase flip, for iteration counts chosen from a spectral and perturbative analysis of the walk operator, rotates amplitude from the symmetric initial state onto a solution state with probability close to one. The same construction is then applied to finding an L-clique in a graph, in two variants.
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Takes A rule naming the neighbours of any vertex, a rule saying whether a vertex is marked, somewhere to start — one named vertex, or a distribution over them — and the size bounds the schedule is computed from: the number of vertices, the depth, the maximum degree, or the size of the subsets a vertex stands for. Which of those a method needs is a property of the walk, not of the slot. Returns A marked vertex, with the evidence it is marked already in hand from the work stored on it — or the report that the graph holds none, at a stated failure probability.
Childs and Eisenberg rework Ambainis's discrete-time quantum walk for element distinctness. The walk runs on a bipartite graph whose vertices are the size-M and size-(M+1) subsets of D, each vertex carrying the subset together with the function values already queried for it. A shift operator costing one query moves between the two subset sizes, Grover diffusion operators act as coins on the element added or removed, and a phase flip marks any subset that already contains a solution — a check that costs no further queries once the values are stored. Running the walk interleaved with the phase flip, for iteration counts chosen from a spectral and perturbative analysis of the walk operator, rotates amplitude from the symmetric initial state onto a solution state with probability close to one. The same construction is then applied to finding an L-clique in a graph, in two variants. This record's speedup class, "Polynomial", is a secondary source's classification of the oracular algorithms it files this under — not a claim its primary paper makes. Not stated by the primary source — the full text of arXiv:quant-ph/0311038 — abstract, section I (introduction), section II (algorithm), section III (analysis), section IV (applications), section V (open problems), the note added and the references. The word "classical" does not appear in the body text. Every comparison the paper makes is against quantum query lower bounds — the Ω(√N) bound for L = 1 and the Ω(N^(2/3)) bound for element distinctness — rather than against the cost of a classical algorithm. was read and makes no such claim. Reported cost: O(N^(L/(L+1))) queries for L-subset finding. The L-clique application is given in two variants: a simple one using O(N^(2L/(L+1))) edge queries and a recursive one using Õ(N^((5L−2)/(2L+4))), which the abstract says is an improvement for L ≤ 5..
ALGORITHM: Subset finding by quantum walk
PROBLEM: In L-subset finding, a black box f maps a domain D of size N into a range R, and a property P picks out some L-element subsets by their arguments and values. The task is to find an L-subset {x₁, …, x_L} of D whose pairs ((x₁, f(x₁)), …, (x_L, f(x_L))) satisfy P, or to reject if none exists, using as few queries as possible. Unstructured search is the case L = 1 and element distinctness the case L = 2.
IDEA: Childs and Eisenberg rework Ambainis's discrete-time quantum walk for element distinctness. The walk runs on a bipartite graph whose vertices are the size-M and size-(M+1) subsets of D, each vertex carrying the subset together with the function values already queried for it. A shift operator costing one query moves between the two subset sizes, Grover diffusion operators act as coins on the element added or removed, and a phase flip marks any subset that already contains a solution — a check that costs no further queries once the values are stored. Running the walk interleaved with the phase flip, for iteration counts chosen from a spectral and perturbative analysis of the walk operator, rotates amplitude from the symmetric initial state onto a solution state with probability close to one. The same construction is then applied to finding an L-clique in a graph, in two variants.
REPORTED COST: O(N^(L/(L+1))) queries for L-subset finding. The L-clique application is given in two variants: a simple one using O(N^(2L/(L+1))) edge queries and a recursive one using Õ(N^((5L−2)/(2L+4))), which the abstract says is an improvement for L ≤ 5.
BASIS: abstract of arXiv:quant-ph/0311038: "an O(N^{L/(L+1)})-query algorithm for finding L equal numbers… One of these algorithms uses O(N^{2L/(L+1)}) edge queries, and the other uses Õ(N^{(5L−2)/(2L+4)}), which is an improvement for L ≤ 5."; section III, Theorem 1: "The quantum query complexity of L-subset finding is O(N^{L/(L+1)})."; section IV, for the two clique variants: "choosing M = ⌊N^{L/(L+1)}⌋ gives an overall query complexity of O(N^{2L/(L+1)})" and "choosing M = ⌊N^{L/(L+2)}⌋ gives an overall query complexity of Õ(N^{(5L−2)/(2L+4)})".
PRIMARY SOURCE: Andrew M. Childs, Jason M. Eisenberg (2003), Quantum algorithms for subset finding — https://arxiv.org/abs/quant-ph/0311038
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Primary source, and the source of this record's cost claim. Consult it for the uniqueness promise the analysis assumes, for the spectral argument that fixes the two iteration counts, and for section V, where the matching lower bound for general L is left as a conjecture and the best known bound is stated as Ω(N^(2/3)) independent of L.
arxiv.org/abs/quant-ph/0311038 ↗