arxiv: 2604.27782 · v1 · submitted 2026-04-30 · 🪐 quant-ph

Recognition: unknown

Explicit Quantum Search Algorithm for the Densest k-Subgraph Problem

Yu.A. Biriukov , R.D. Morozov , I.V. Dyakonov , S.S. Straupe

Authors on Pith no claims yet

Pith reviewed 2026-05-07 06:55 UTC · model grok-4.3

classification 🪐 quant-ph

keywords densest k-subgraphGrover's algorithmDicke statesQuantum Fourier Transformquantum searchgraph optimizationquadratic speedupQUBO

0 comments

The pith

Quantum search with an explicit Dicke-state oracle solves the densest k-subgraph problem quadratically faster than classical brute force.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents two quantum methods for the NP-hard densest k-subgraph problem: mapping it to QUBO and applying Grover's search algorithm. For the search approach, it gives a detailed circuit for the oracle that uses Dicke states to prepare uniform superpositions over k-subsets and the quantum Fourier transform to count the edges inside the subset. Simulations on small graphs show the expected quadratic reduction in the number of queries compared to checking all possible subsets classically. This matters because finding dense subgraphs helps in analyzing social networks, detecting fraud, and other practical tasks where classical methods struggle with large graphs. If the circuit scales without hidden costs, it could bring quantum advantage to combinatorial graph problems.

Core claim

We present an explicit gate-based oracle circuit for Grover's quantum search algorithm that utilizes Dicke states and the Quantum Fourier Transform for edge counting in candidate k-subgraphs. Numerical simulations on small instances demonstrate a quadratic speedup over classical brute-force search for the densest k-subgraph problem.

What carries the argument

The explicit gate-based oracle circuit utilizing Dicke states to encode k-subsets and the Quantum Fourier Transform to count edges within the subset.

If this is right

The densest k-subgraph problem can be solved using a number of quantum queries that scales as the square root of the number of possible k-subsets.
The method provides a fully explicit quantum circuit implementation for the search oracle rather than relying on an abstract black-box function.
It offers a direct quantum search strategy as an alternative to formulating the problem as a QUBO for quantum annealing or optimization.
Practical applications in social network analysis and fraud detection could benefit from the reduced search time on quantum hardware.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar techniques for constructing oracles with symmetric states like Dicke states could extend to other NP-hard graph problems such as finding maximum cliques or dense communities.
The circuit may require further optimization of depth and gate count to be feasible on current quantum processors before error-corrected machines arrive.
Hybrid algorithms combining this quantum search with classical preprocessing could improve performance on real-world large graphs.

Load-bearing premise

The proposed oracle circuit implements correct edge counting for arbitrary graphs without introducing exponential overhead in the number of gates or circuit depth.

What would settle it

Constructing and simulating the oracle circuit for graphs with 20 or more vertices to verify that the total resource cost scales polynomially rather than exponentially with graph size.

Figures

Figures reproduced from arXiv: 2604.27782 by I.V. Dyakonov, R.D. Morozov, S.S. Straupe, Yu.A. Biriukov.

**Figure 1.** Figure 1: FIG. 1: General scheme of a single Grover-based search view at source ↗

**Figure 2.** Figure 2: FIG. 2: Gate scheme for oracle view at source ↗

**Figure 4.** Figure 4: FIG. 4: Benchmark graph with view at source ↗

**Figure 5.** Figure 5: FIG. 5: Convergence of the best-so-far edge count for the four algorithms on the benchmark graph from Fig. view at source ↗

**Figure 7.** Figure 7: FIG. 7: Comparison of oracle cost scaling for the view at source ↗

read the original abstract

This paper addresses the problem of finding the densest $k$-vertex subgraph in an arbitrary graph. This problem is NP-hard and has important applications in social network analysis, fraud detection, recommendation systems, and bioinformatics. We propose two quantum approaches to solve this problem: reduction to Quadratic Unconstrained Binary Optimization (QUBO) and using Grover's quantum search algorithm. For the latter approach, we present an explicit gate-based oracle circuit utilizing Dicke states and Quantum Fourier Transform for edge counting. Numerical simulations demonstrate a quadratic speedup over classical Brute-force search.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper spells out a concrete Grover oracle for densest k-subgraph using Dicke states and QFT edge counting, which is a useful implementation detail even if the quadratic speedup is the standard one.

read the letter

The main takeaway is that the authors give an explicit gate-level construction for the oracle in Grover search over k-subsets. They prepare the uniform superposition with Dicke states and then use controlled operations from the adjacency matrix plus QFT to compute the induced edge count and mark subgraphs above a density threshold. That level of detail on the oracle is more than most papers offer for this kind of combinatorial search, and the small-n simulations line up with the expected quadratic improvement over brute-force enumeration of subsets. The choice of Dicke states makes sense here because they avoid the overhead of post-selection or amplitude amplification just to get fixed-size subsets. The QFT step for summing the edges is a standard trick for quantum counting, and it keeps the register size logarithmic in the maximum possible edge count. Overall the circuit family stays polynomial in n once the graph is fixed, which is what you need for the Grover iteration count to deliver the quadratic gain. The soft spot is that the simulations are only on tiny instances, so they do not yet rule out depth or ancilla blow-up when the number of possible pairs scales as n squared. If the additions are done sequentially without parallelization or clever reuse of ancillas, the circuit depth could reach O(n^2 poly(log n)), which is still polynomial but might matter for fault-tolerant overhead or near-term compilation. The paper does not appear to include a full asymptotic gate-count table or a comparison against a naive enumeration oracle, so readers will have to do that arithmetic themselves. This work is aimed at people already thinking about quantum oracles for graph problems in network analysis or bioinformatics. It will not change the exponential scaling for large k, but the concrete circuit helps anyone who wants to test similar ideas or look for optimizations. I would send it to peer review; the construction is specific enough that referees can verify the oracle logic and ask for the missing complexity details if needed.

Referee Report

3 major / 2 minor

Summary. The paper proposes two quantum approaches to the NP-hard densest k-subgraph problem: a reduction to QUBO and a Grover search over k-subsets. For the latter, it gives an explicit gate-based oracle that prepares uniform superposition via Dicke states and uses QFT to count induced edges, marking dense subgraphs. Numerical simulations are claimed to exhibit quadratic speedup over classical brute-force enumeration.

Significance. If the oracle construction is a uniform polynomial-size circuit family for arbitrary graphs, the explicit use of Dicke states and QFT-based counting would be a useful concrete realization of Grover search for a combinatorial problem with applications in network analysis and bioinformatics. The approach avoids black-box oracles and supplies a falsifiable circuit-level implementation, which strengthens the quadratic-speedup claim beyond abstract Grover analysis. Small-instance simulations provide supporting evidence but do not yet establish asymptotic scaling.

major comments (3)

[Oracle construction] Oracle construction (abstract and § on circuit): the manuscript asserts an explicit poly(n)-gate oracle but supplies no gate-count or depth bound. If the QFT-based edge counter applies controls or additions sequentially over all binom(n,2) possible pairs before the inverse QFT, the depth scales as O(n^2 log n) or worse once precision and graph-dependent controls are included; this would make the oracle super-polynomial in practice and eliminate the claimed quadratic advantage over brute-force enumeration.
[Numerical simulations] Numerical simulations (abstract): the claim of quadratic speedup is unsupported by reported data. No values are given for n, k, graph density, number of trials, success probability, or error bars; it is also unclear whether the simulator used the exact unitary oracle or an approximation, and whether runtime was measured in gate depth or wall-clock time on a classical emulator.
[QFT oracle] Correctness of edge counting (QFT oracle section): the construction must embed the arbitrary adjacency matrix as controls without exponential ancillary resources or non-uniform circuit families. The manuscript does not verify that the integer sum produced by the QFT phase estimation matches the induced-edge count for every k-subset in superposition, nor does it bound the precision required to distinguish dense from non-dense subgraphs.

minor comments (2)

A table listing simulated (n,k) pairs, classical vs. quantum iteration counts, and observed speed-up ratios would make the numerical evidence easier to evaluate.
Notation for the QFT-based counter (e.g., the phase-kickback register and the precision parameter) should be introduced with an explicit equation showing how the edge sum is encoded.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive comments on our manuscript. We appreciate the recognition of the value in providing an explicit gate-based oracle construction. We address each major comment below and will revise the manuscript to include the requested clarifications, bounds, and details.

read point-by-point responses

Referee: Oracle construction (abstract and § on circuit): the manuscript asserts an explicit poly(n)-gate oracle but supplies no gate-count or depth bound. If the QFT-based edge counter applies controls or additions sequentially over all binom(n,2) possible pairs before the inverse QFT, the depth scales as O(n^2 log n) or worse once precision and graph-dependent controls are included; this would make the oracle super-polynomial in practice and eliminate the claimed quadratic advantage over brute-force enumeration.

Authors: We agree that an explicit gate count and depth bound should be provided. The oracle circuit has size O(n^2 log n) because there are O(n^2) possible edges, and for each, a controlled addition to an O(log n)-bit counter register (using QFT for efficient arithmetic perhaps). This is polynomial in n. The quadratic speedup is with respect to the number of subsets searched: Grover requires O(sqrt(binom(n,k))) oracle calls versus classical binom(n,k), and since each classical check costs O(n^2) time as well, the exponential scaling remains quadratically improved. We will include a detailed complexity analysis in the revised version, confirming the oracle is polynomial-sized. The concern about practicality for large n is noted, but the theoretical claim holds. revision: yes
Referee: Numerical simulations (abstract): the claim of quadratic speedup is unsupported by reported data. No values are given for n, k, graph density, number of trials, success probability, or error bars; it is also unclear whether the simulator used the exact unitary oracle or an approximation, and whether runtime was measured in gate depth or wall-clock time on a classical emulator.

Authors: We acknowledge that the simulation details were insufficiently reported in the current version. The simulations were conducted using an exact unitary simulator for the oracle on small instances. We will revise the manuscript to include the specific values for n, k, graph density, number of trials, success probabilities with error bars, and clarify the use of the exact oracle and measurement in terms of circuit depth and number of oracle calls. revision: yes
Referee: Correctness of edge counting (QFT oracle section): the construction must embed the arbitrary adjacency matrix as controls without exponential ancillary resources or non-uniform circuit families. The manuscript does not verify that the integer sum produced by the QFT phase estimation matches the induced-edge count for every k-subset in superposition, nor does it bound the precision required to distinguish dense from non-dense subgraphs.

Authors: The construction uses O(log n) ancillary qubits for the edge count register, which is polynomial and sufficient to hold the exact integer count (0 to binom(n,2)). The circuit is graph-dependent but can be constructed in polynomial time from the adjacency matrix by including controlled-add operations only for existing edges; this is standard for oracles in quantum algorithms for specific instances and does not require exponential resources or non-uniformity beyond the input. We will add a section verifying the correctness: for any computational basis state corresponding to a k-subset, the counter accumulates exactly the number of induced edges via the controlled increments, and the QFT is used to enable phase estimation or marking based on the count. Precision is exact since we use sufficient bits for integer representation, allowing clear distinction via a threshold comparator in the oracle. A formal argument will be included in the revision. revision: partial

Circularity Check

0 steps flagged

No circularity; speedup follows from standard Grover on explicitly constructed oracle

full rationale

The paper constructs an explicit gate-based oracle for edge counting via Dicke states and QFT, then invokes Grover search over k-subsets. The quadratic speedup is the direct, external consequence of Grover's algorithm once any marking oracle is supplied; it is not derived from or equivalent to the paper's own inputs by construction. No fitted parameters, self-definitional equations, or load-bearing self-citations appear in the abstract or described chain. Numerical simulations on small instances serve only as validation and do not substitute for the general circuit claim. The derivation remains self-contained against external benchmarks (Grover, Dicke preparation, QFT).

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on standard quantum computing primitives (Grover iteration, Dicke states for k-subsets, QFT for counting) plus the assumption that an efficient oracle for edge counting exists in this encoding. No new physical entities or fitted constants are introduced.

axioms (2)

domain assumption Dicke states can be prepared and used to represent exactly k-vertex subsets of an n-vertex graph.
Central to the oracle design for selecting candidate subgraphs.
domain assumption Quantum Fourier transform can be used to count the number of edges within a selected subset.
Used to implement the marking function inside the Grover oracle.

pith-pipeline@v0.9.0 · 5400 in / 1365 out tokens · 65722 ms · 2026-05-07T06:55:28.056754+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 2 canonical work pages

[1]

B. Hooi, H. A. Song, A. Beutel, N. Shah, K. Shin, and C. Faloutsos, Fraudar: Bounding graph fraud in the face of camouflage, Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining , 895 (2016)

2016
[2]

Y. Li, J. Wu, H. Wang, and W. Wang, Fraud detection in online consumer reviews, Decision Support Systems132, 113283 (2020)

2020
[3]

S. Ji, Y. Li, J. Wu, and W. Wang, A survey on graph- based fraud detection, ACM Computing Surveys55, 1 (2022)

2022
[4]

Angel, N

A. Angel, N. Sarkas, N. Koudas, and D. Srivastava, Dense subgraph maintenance under streaming edge weight up- dates for real-time story identification, VLDB Journal 23, 175 (2014)

2014
[5]

B. Saha, A. Hoch, S. Khuller, L. Raschid, and X.-N. Zhang, Dense subgraphs with restrictions and applica- tions to gene annotation graphs, inResearch in Compu- tational Molecular Biology(Springer, 2010) pp. 456–472

2010
[6]

C. S. Calude, M. J. Dinneen, and R. Hua, Quantum solu- tions for densest k-subgraph problems, Journal of Mem- brane Computing2, 26 (2020)

2020
[7]

N. Huang, A qubo formulation for the k-densest common subgraph isomorphism problem via quantum annealing, in2020 IEEE Asia-Pacific Conference on Computer Sci- ence and Data Engineering (CSDE)(2020) pp. 1–7

2020
[8]

Pelofske, G

E. Pelofske, G. Hahn, and H. Djidjev, Decomposition algorithms for solving np-hard problems on a quantum annealer, Journal of Signal Processing Systems93, 405 (2021)

2021
[9]

Calude, M

C. Calude, M. Dinneen, and R. Hua, Quantum solutions for densest k-subgraph problems, Journal of Membrane Computing2, 26 (2020)

2020
[10]

Golden, A

J. Golden, A. B¨ artschi, D. O’Malley, and S. Eidenbenz, Numerical evidence for exponential speed-up of qaoa over unstructured search for approximate constrained op- timization, in2023 IEEE International Conference on Quantum Computing and Engineering (QCE)(2023) pp. 496–505. 8

2023
[11]

Ohno, Grover’s search with learning oracle for con- strained binary optimization problems, Quantum Ma- chine Intelligence , 12 (2024)

H. Ohno, Grover’s search with learning oracle for con- strained binary optimization problems, Quantum Ma- chine Intelligence , 12 (2024)

2024
[12]

L. K. Grover, A fast quantum mechanical algorithm for database search, inProceedings of the twenty-eighth an- nual ACM symposium on Theory of computing(1996) pp. 212–219

1996
[13]

B¨ artschi and S

A. B¨ artschi and S. Eidenbenz, Short-depth circuits for dicke state preparation, in2022 IEEE International Con- ference on Quantum Computing and Engineering (QCE) (2022) pp. 87–96

2022
[14]

A Quantum Algorithm for Finding the Minimum

C. Durr and P. Hoyer, A quantum algorithm for finding the minimum, arXiv preprint quant-ph/9607014 (1996)

work page Pith review arXiv 1996
[15]

Renyi, On random graph, Publicationes Mathemati- cate6, 290 (1959)

E. Renyi, On random graph, Publicationes Mathemati- cate6, 290 (1959)

1959
[16]

V. S. Denchev, S. Boixo, S. V. Isakov, N. Ding, R. Bab- bush, V. N. Smelyanskiy, J. M. Martinis, and H. Neven, What is the computational value of finite-range tunnel- ing?, inProceedings of the National Academy of Sciences (2016)

2016
[17]

Ambainis, Quantum algorithms for tree and graph problems, arXiv preprint arXiv:1912.07403 (2019)

A. Ambainis, Quantum algorithms for tree and graph problems, arXiv preprint arXiv:1912.07403 (2019)

work page arXiv 1912
[18]

Montanaro, Quantum walks and graph optimization: a review, Quantum5, 459 (2021)

A. Montanaro, Quantum walks and graph optimization: a review, Quantum5, 459 (2021)

2021
[19]

Choi, Minor-embedding in adiabatic quantum com- putation: I

V. Choi, Minor-embedding in adiabatic quantum com- putation: I. the parameter setting problem, Quantum Information Processing7, 193 (2008)

2008
[20]

Kinget al., Scaling advantage in quantum simulation of frustrated magnets, Nature Physics17, 732 (2021)

A. Kinget al., Scaling advantage in quantum simulation of frustrated magnets, Nature Physics17, 732 (2021)

2021
[21]

B¨ artschi and S

A. B¨ artschi and S. Eidenbenz, Deterministic preparation of dicke states, inInternational Symposium on Funda- mentals of Computation Theory(2020) pp. 126–139

2020
[22]

T. G. Draper, Addition on a quantum computer, inPro- ceedings of the 2nd NASA International Conference on Quantum Computing(2000)

2000
[23]

Gidney and M

C. Gidney and M. Eker˚ a, How to factor 2048-bit rsa in- tegers in 8 hours using 20 million noisy qubits, Quantum 5, 433 (2021)

2048
[24]

Zoufal, A

C. Zoufal, A. Lucchi, and S. Woerner, Variational quan- tum algorithms for approximate dicke state preparation, Physical Review A105, 062423 (2022)

2022
[25]

Thapliyal and N

H. Thapliyal and N. Ranganathan, Reversible logic syn- thesis of optimized adder circuits, IEEE Transactions on Computers63, 349 (2013)

2013
[26]

H¨ aner and D

T. H¨ aner and D. Steiger, Optimizing quantum circuits for arithmetic, Quantum Science and Technology3, 020501 (2018)

2018
[27]

Brassard, P

G. Brassard, P. Høyer, M. Mosca, and A. Tapp, Quan- tum amplitude amplification and estimation, inQuantum Computation and Information(2002) pp. 53–74

2002
[28]

Naderipour and A

M. Naderipour and A. Montakhab, Improving quan- tum minimum finding through adaptive grover iterations, Quantum Information Processing22, 15 (2023)

2023
[29]

D. J. Egger, J. Marecek, and S. Woerner, Warm-starting quantum optimization, Quantum5, 479 (2021)

2021
[30]

Harriganet al., Quantum approximate optimization of non-planar ising problems on a planar graph, Nature Physics17, 332 (2021)

M. Harriganet al., Quantum approximate optimization of non-planar ising problems on a planar graph, Nature Physics17, 332 (2021)

2021