Adaptive Quantum Optimized Centroid Initialization

Ajinkya Borle; Charles K. Nicholas; Nicholas R. Allgood

arxiv: 2401.11258 · v2 · submitted 2024-01-20 · 🪐 quant-ph · cs.ET

Adaptive Quantum Optimized Centroid Initialization

Nicholas R. Allgood , Ajinkya Borle , Charles K. Nicholas This is my paper

Pith reviewed 2026-05-24 04:45 UTC · model grok-4.3

classification 🪐 quant-ph cs.ET

keywords centroid initializationQUBOquantum annealingk-means clusteringV-measuremalware classificationsimulated annealingadaptive refinement

0 comments

The pith

AQOCI formulates k-means centroid initialization as a QUBO problem and recovers real-valued solutions via iterative refinement from quantum or quantum-inspired solvers.

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

The paper presents AQOCI as an extension of QOCI that turns the choice of starting cluster centers into a binary optimization task solvable by quantum annealing or similar methods. An added refinement loop, drawing on Gauss-Seidel and Jacobi ideas, adjusts scale and offset after each solver call so that the binary answers map back to usable real coordinates. Tests on controlled synthetic data and the MOTIF malware set show the resulting initializations produce clusterings that match or exceed those from k-means++ when clusters overlap or when sample sizes stay small, with V-measure gains reaching 26 percent in the latter case. The work matters because poor starting points are a known source of slow convergence and local-minima traps in prototype-based clustering, so any reliable alternative initialization route could improve practical results without changing the downstream algorithm.

Core claim

AQOCI casts centroid initialization as a QUBO whose binary variables encode candidate center locations, solves the QUBO with TABU search, simulated annealing, or D-Wave HybridBQM, then applies an adaptive iterative refinement step to convert the binary output into real-valued coordinates; on the MOTIF dataset this yields V-measure scores competitive with or superior to k-means++ at smaller sample sizes, while on heavily overlapping synthetic clusters AQOCI-SimAnn also improves over k-means++ and the method scales to at least ten dimensions.

What carries the argument

The iterative refinement mechanism that repeatedly adjusts scaling and offset parameters to map binary QUBO solver outputs onto accurate real-valued centroid coordinates.

If this is right

On data with heavily overlapping clusters AQOCI can deliver higher V-measure than k-means++ initialization.
At reduced sample sizes on real classification tasks such as MOTIF, AQOCI initializations produce clusterings up to 26 percent better in V-measure than k-means++.
The approach remains effective at least up to ten dimensions with no observed degradation.
On well-separated clusters the binary encoding introduces a performance ceiling that k-means++ does not share.

Where Pith is reading between the lines

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

The same QUBO-plus-refinement pattern could be tested on other distance-based clustering objectives that also suffer from initialization sensitivity.
If higher-resolution binary encodings or hybrid continuous-binary solvers remove the observed plateau, the method could become competitive even on cleanly separated data.
Because the refinement loop is solver-agnostic, classical heuristic QUBO solvers alone might already deliver most of the reported gains on modest hardware.

Load-bearing premise

The iterative refinement step can extract accurate real-valued centroid positions from the binary solver outputs without the limited resolution of the binary encoding itself capping solution quality on the target data.

What would settle it

A controlled experiment that increases the bit resolution of the binary encoding on well-separated synthetic data and still observes the same performance plateau relative to k-means++ would show that the refinement step cannot overcome the encoding limit.

Figures

Figures reproduced from arXiv: 2401.11258 by Ajinkya Borle, Charles K. Nicholas, Nicholas R. Allgood.

**Figure 3.** Figure 3: Gaussian: Silhouette [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 5.** Figure 5: Gaussian: Completeness [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 7.** Figure 7: MOTIF: Inertia [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

**Figure 9.** Figure 9: MOTIF: Silhouette [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗

**Figure 11.** Figure 11: MOTIF: Completeness [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗

read the original abstract

Prototype-based clustering algorithms such as k-means are sensitive to the selection of initial cluster centroids, with poor initialization leading to slower convergence and suboptimal solutions trapped in local minima. We present Adaptive Quantum Optimized Centroid Initialization (AQOCI), a method that formulates the centroid initialization problem as a Quadratic Unconstrained Binary Optimization (QUBO) problem and solves it using quantum annealing or quantum-inspired solvers. AQOCI extends a prior method (QOCI) by introducing an iterative refinement mechanism inspired by the Gauss-Seidel and Jacobi methods, enabling the recovery of real-valued centroid coordinates from binary solver outputs through adaptive scaling and offset adjustments. We evaluate AQOCI using three solver backends: TABU search, simulated annealing, and D-Wave's HybridBQM on synthetic Gaussian data with controlled sweeps over cluster separation, cluster count, dimensionality, and sample size, as well as on the MOTIF malware classification dataset, comparing against standard k-means with random initialization and k-means++ initialization. On the MOTIF dataset, AQOCI produces clusterings that are competitive with and, at smaller sample sizes, superior to k-means++, with V-measure improvements of up to 26\%. On synthetic data with heavily overlapping clusters, AQOCI--SimAnn outperforms k-means++ in V-measure. On well-separated synthetic data, k-means++ is clearly superior, and AQOCI exhibits a consistent performance plateau attributable to the binary encoding resolution. The dimensionality sweep demonstrates scalability to at least $d = 10$ without degradation.

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

AQOCI adds an adaptive refinement step to prior QOCI work for recovering real-valued centroids from binary QUBO outputs, with reported gains on MOTIF at small sample sizes but a clear performance cap on well-separated synthetic data.

read the letter

The paper's main contribution is the adaptive refinement mechanism—drawing from Gauss-Seidel and Jacobi ideas—to adjust scaling and offsets after the binary solver step. This is presented as an extension of the earlier QOCI formulation, and the authors run controlled sweeps on synthetic Gaussians plus the MOTIF malware dataset against random and k-means++ baselines. The MOTIF results at smaller N show up to 26% V-measure lift, and the method scales to d=10 without obvious drop-off. Those are the concrete additions worth noting. The synthetic experiments are also useful because they expose the binary encoding limit directly: on well-separated clusters the method plateaus and k-means++ wins cleanly, which the abstract attributes to resolution. That honesty about the drawback is helpful. The soft spots are the missing error bars, statistical tests, and full protocol details, plus the fact that the refinement's ability to recover accurate coordinates is asserted without quantitative checks on how much discretization error remains. The circularity risk is low since the comparisons use independent classical baselines, but the central claim on MOTIF still rests on whether the same resolution issue is active there. This is incremental work aimed at people already tracking quantum or quantum-inspired clustering heuristics. A reader looking for new initialization tricks on modest-sized datasets could extract value from the MOTIF numbers and the solver comparisons. It deserves a serious referee to check the refinement implementation and add the missing statistical controls, rather than a desk reject.

Referee Report

3 major / 1 minor

Summary. The manuscript introduces Adaptive Quantum Optimized Centroid Initialization (AQOCI), an extension of prior QOCI work that formulates k-means centroid initialization as a QUBO problem solved by quantum annealing or quantum-inspired solvers (TABU, simulated annealing, HybridBQM). It adds an iterative refinement step inspired by Gauss-Seidel and Jacobi methods that recovers real-valued centroid coordinates from binary outputs via adaptive scaling and offset adjustments. The method is evaluated on synthetic Gaussian data with sweeps over cluster separation, number of clusters, dimensionality, and sample size, plus the MOTIF malware dataset, with comparisons to random and k-means++ initializations; it reports competitiveness with k-means++ on MOTIF (up to 26% V-measure gain at small N) and outperformance on overlapping synthetic clusters, while noting a plateau on well-separated data due to binary encoding resolution.

Significance. If the empirical gains are shown to be robust, the work provides a concrete demonstration that quantum-inspired QUBO solvers combined with iterative refinement can yield useful initializations for prototype-based clustering in regimes with overlapping clusters or limited samples. The controlled sweeps over multiple parameters and use of independent classical baselines (random, k-means++) constitute a strength of the evaluation design.

major comments (3)

[Abstract] Abstract: the reported V-measure improvements (including up to 26% on MOTIF) are given without error bars, confidence intervals, or any statistical significance tests, despite the stochastic nature of the solvers and the multiple random seeds implicit in the sweeps; this omission makes it impossible to determine whether the claimed superiority at small sample sizes is reproducible.
[Abstract] Abstract: the performance plateau on well-separated synthetic Gaussians is explicitly attributed to binary encoding resolution, yet the manuscript supplies no quantitative assessment of how that same resolution limit affects the MOTIF results (whose cluster geometry is unknown), leaving open the possibility that the reported gains arise from solver heuristics or evaluation choices rather than the adaptive refinement mechanism.
[Method] Method: the iterative refinement procedure (adaptive scaling and offset adjustments) is described only at a high level; without an explicit equation or pseudocode showing how the real-valued centroid coordinates are recovered from the binary QUBO solution at each iteration, it is difficult to verify that the procedure reliably overcomes the discretization error that produces the observed plateau.

minor comments (1)

[Abstract] The abstract states that scalability is demonstrated to d=10 without degradation, but does not indicate whether this holds uniformly across all solver backends or only for selected ones.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and will revise the manuscript to incorporate clarifications and additional details where feasible.

read point-by-point responses

Referee: [Abstract] Abstract: the reported V-measure improvements (including up to 26% on MOTIF) are given without error bars, confidence intervals, or any statistical significance tests, despite the stochastic nature of the solvers and the multiple random seeds implicit in the sweeps; this omission makes it impossible to determine whether the claimed superiority at small sample sizes is reproducible.

Authors: We agree that error bars and statistical tests are necessary given the stochastic solvers. In the revised version we will report means and standard deviations over multiple independent runs (with different random seeds) for all V-measure results, and include paired statistical significance tests comparing AQOCI variants to k-means++ on the MOTIF dataset at small sample sizes. revision: yes
Referee: [Abstract] Abstract: the performance plateau on well-separated synthetic Gaussians is explicitly attributed to binary encoding resolution, yet the manuscript supplies no quantitative assessment of how that same resolution limit affects the MOTIF results (whose cluster geometry is unknown), leaving open the possibility that the reported gains arise from solver heuristics or evaluation choices rather than the adaptive refinement mechanism.

Authors: The attribution of the plateau to binary resolution on synthetic data is based on controlled sweeps where separation is known. For MOTIF, unknown geometry prevents a direct quantitative isolation of the resolution effect. We will expand the discussion to note this limitation explicitly and to emphasize that the comparative advantage on overlapping synthetic clusters (where resolution is not the limiting factor) supports the contribution of the adaptive refinement step. revision: partial
Referee: [Method] Method: the iterative refinement procedure (adaptive scaling and offset adjustments) is described only at a high level; without an explicit equation or pseudocode showing how the real-valued centroid coordinates are recovered from the binary QUBO solution at each iteration, it is difficult to verify that the procedure reliably overcomes the discretization error that produces the observed plateau.

Authors: We accept that the current description is insufficient for reproducibility. The revised manuscript will include explicit equations for the adaptive scaling and offset computation at each iteration, together with pseudocode for the full refinement loop, demonstrating how the procedure iteratively reduces discretization error. revision: yes

standing simulated objections not resolved

A quantitative assessment of binary encoding resolution impact specifically on the MOTIF dataset cannot be provided because the true cluster geometry is unknown.

Circularity Check

0 steps flagged

No circularity in derivation chain; empirical claims rest on external baselines

full rationale

The paper formulates centroid initialization as a QUBO problem solved by external solvers (TABU, simulated annealing, D-Wave HybridBQM) and adds an iterative refinement step inspired by Gauss-Seidel/Jacobi methods. All load-bearing claims are empirical performance comparisons (V-measure on MOTIF and synthetic Gaussians) against independent classical baselines (random k-means and k-means++). No equation or result is shown to reduce by construction to a fitted parameter defined from the target metric, nor does any central premise rest on a self-citation chain that itself lacks independent verification. The abstract explicitly notes the binary-resolution performance plateau on well-separated data, confirming the evaluation is falsifiable against external benchmarks rather than self-referential.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields a partial ledger; the method rests on the domain assumption that centroid quality can be encoded as a QUBO objective and recovered via adaptive scaling, with no explicit free parameters or invented entities named.

free parameters (1)

adaptive scaling and offset parameters
Parameters adjusted iteratively to map binary solver outputs to real-valued centroids; their functional form is not specified in the abstract.

axioms (1)

domain assumption The centroid initialization objective can be expressed as a QUBO without essential loss of information about cluster quality.
This mapping is the prerequisite for using quantum or quantum-inspired solvers.

pith-pipeline@v0.9.0 · 5799 in / 1408 out tokens · 29396 ms · 2026-05-24T04:45:25.892657+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Relation between the paper passage and the cited Recognition theorem.

formulates the centroid initialization problem as a Quadratic Unconstrained Binary Optimization (QUBO) problem... iterative refinement mechanism inspired by the Gauss-Seidel and Jacobi methods, enabling the recovery of real-valued centroid coordinates from binary solver outputs through adaptive scaling and offset adjustments

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 2 internal anchors

[1]

Allgood, Ajinkya Borle & Charles K

Nicholas R. Allgood, Ajinkya Borle & Charles K. Nicholas (2023): Quantum Optimized Centroid Initial- ization. In Kohei Arai, editor: Proceedings of the Future Technologies Conference (FTC) 2023, V olume 2, Springer Nature Switzerland, Cham, pp. 71–85

work page 2023
[2]

Christian Bauckhage (2015): k-Means Clustering Is Matrix Factorization, doi:10.48550/ARXIV .1512.07548

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv 2015
[3]

Analyzing the Quantum Annealing Approach for Solving Linear Least Squares Problems

Ajinkya Borle & Samuel J. Lomonaco (2018): Analyzing the Quantum Annealing Approach for Solving Linear Least Squares Problems. arXiv 1809.07649

work page internal anchor Pith review Pith/arXiv arXiv 2018
[4]

Cryptology ePrint Archive, Report 2021/620

El ˙zbieta Burek, Michał Misztal & Michał Wro´nski (2021): Algebraic attacks on block ciphers using quantum annealing. Cryptology ePrint Archive, Report 2021/620. https://ia.cr/2021/620

work page 2021
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Casaña-Eslava, Paulo J.G

Raúl V . Casaña-Eslava, Paulo J.G. Lisboa, Sandra Ortega-Martorell, Ian H. Jarman & José D. Martín- Guerrero (2020): Probabilistic quantum clustering . Knowledge-Based Systems 194, p. 105567, doi:https://doi.org/10.1016/j.knosys.2020.105567. Available at https://www.sciencedirect.com/ science/article/pii/S0950705120300587

work page doi:10.1016/j.knosys.2020.105567 2020
[6]

Humble & Shigetoshi Sota (2019):Quantum annealing for sys- tems of polynomial equations

Chia Cheng Chang, Arjun Gambhir, Travis S. Humble & Shigetoshi Sota (2019):Quantum annealing for sys- tems of polynomial equations. Scientific Reports9(1), p. 10258, doi:10.1038/s41598-019-46729-0. Available at https://doi.org/10.1038/s41598-019-46729-0

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Joyce, Dev Amlani, Charles Nicholas & Edward Raff (2023): MOTIF: A Large Malware Reference Dataset with Ground Truth Family Labels

Robert J. Joyce, Dev Amlani, Charles Nicholas & Edward Raff (2023): MOTIF: A Large Malware Reference Dataset with Ground Truth Family Labels. Computers and Security 124, Issue C

work page 2023
[10]

Jain A. K. & Dubes R. C. (1988): Algorithms for Clustering Data. Prentice-Hall

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Rousseeuw PJ Kaufman L (1987): Clustering by means of medoids. Proc. Statistical Data Analysis Based on the L1 Norm Conference, Neuchatel, 1987, pp. 405–416

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Jingu Kim & Haesun Park (2008): Sparse Nonnegative Matrix Factorization for Clustering. Technical Report GT-CSE-08-01, Georgia Institute of Technology. Available athttps://smartech.gatech.edu/handle/ 1853/20058. Allgood, Borle, and Nicholas 11

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Daniel D. Lee & H. Sebastian Seung (2000): Algorithms for Non-Negative Matrix Factorization . In: Pro- ceedings of the 13th International Conference on Neural Information Processing Systems , NIPS’00, MIT Press, Cambridge, MA, USA, p. 535–541

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J. B. MacQueen (1967): Some Methods for classification and Analysis of Multivariate Observations . In: Proceedings of 5th Berkeley Symposium on Mathematical Statistics and Probability: 281-297, University of California Press

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Michael A. Nielsen & Isaac L. Chuang (2000): Quantum Computation and Quantum Information. Cambridge University Press

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In: Annals of Operations Research

Gintaras Palubeckis (2004): Multistart Tabu Search Strategies for the Unconstrained Binary Quadratic Op- timization Problem. In: Annals of Operations Research . Available at https://doi.org/10.1023/B: ANOR.0000039522.58036.68

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Pollachini, Juan P

Giovani G. Pollachini, Juan P. L. C. Salazar, Caio B. D. Goes, Thiago O. Maciel & Eduardo I. Duzzioni (2021): Hybrid classical-quantum approach to solve the heat equation using quantum annealers . Phys- ical Review A 104(3), doi:10.1103/physreva.104.032426. Available at https://doi.org/10.1103% 2Fphysreva.104.032426

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Expert Systems with Applications 37(7), pp

Jing Xiao, YuPing Yan, Jun Zhang & Yong Tang (2010): A quantum-inspired genetic al- gorithm for k-means clustering . Expert Systems with Applications 37(7), pp. 4966–4973, doi:https://doi.org/10.1016/j.eswa.2009.12.017. Available at https://www.sciencedirect.com/ science/article/pii/S095741740901063X

work page doi:10.1016/j.eswa.2009.12.017 2010

[1] [1]

Allgood, Ajinkya Borle & Charles K

Nicholas R. Allgood, Ajinkya Borle & Charles K. Nicholas (2023): Quantum Optimized Centroid Initial- ization. In Kohei Arai, editor: Proceedings of the Future Technologies Conference (FTC) 2023, V olume 2, Springer Nature Switzerland, Cham, pp. 71–85

work page 2023

[2] [2]

Christian Bauckhage (2015): k-Means Clustering Is Matrix Factorization, doi:10.48550/ARXIV .1512.07548

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv 2015

[3] [3]

Analyzing the Quantum Annealing Approach for Solving Linear Least Squares Problems

Ajinkya Borle & Samuel J. Lomonaco (2018): Analyzing the Quantum Annealing Approach for Solving Linear Least Squares Problems. arXiv 1809.07649

work page internal anchor Pith review Pith/arXiv arXiv 2018

[4] [4]

Cryptology ePrint Archive, Report 2021/620

El ˙zbieta Burek, Michał Misztal & Michał Wro´nski (2021): Algebraic attacks on block ciphers using quantum annealing. Cryptology ePrint Archive, Report 2021/620. https://ia.cr/2021/620

work page 2021

[5] [5]

Casaña-Eslava, Paulo J.G

Raúl V . Casaña-Eslava, Paulo J.G. Lisboa, Sandra Ortega-Martorell, Ian H. Jarman & José D. Martín- Guerrero (2020): Probabilistic quantum clustering . Knowledge-Based Systems 194, p. 105567, doi:https://doi.org/10.1016/j.knosys.2020.105567. Available at https://www.sciencedirect.com/ science/article/pii/S0950705120300587

work page doi:10.1016/j.knosys.2020.105567 2020

[6] [6]

Humble & Shigetoshi Sota (2019):Quantum annealing for sys- tems of polynomial equations

Chia Cheng Chang, Arjun Gambhir, Travis S. Humble & Shigetoshi Sota (2019):Quantum annealing for sys- tems of polynomial equations. Scientific Reports9(1), p. 10258, doi:10.1038/s41598-019-46729-0. Available at https://doi.org/10.1038/s41598-019-46729-0

work page doi:10.1038/s41598-019-46729-0 2019

[7] [7]

Available at https://www.dwave.com

D-Wave: D-wave. Available at https://www.dwave.com

work page

[8] [8]

Forgy (1965): Cluster analysis of multivariate data: efficiency versus interpretability of classifi- cations

Edward W. Forgy (1965): Cluster analysis of multivariate data: efficiency versus interpretability of classifi- cations. Biometrics 21(3), pp. 768–769

work page 1965

[9] [9]

Joyce, Dev Amlani, Charles Nicholas & Edward Raff (2023): MOTIF: A Large Malware Reference Dataset with Ground Truth Family Labels

Robert J. Joyce, Dev Amlani, Charles Nicholas & Edward Raff (2023): MOTIF: A Large Malware Reference Dataset with Ground Truth Family Labels. Computers and Security 124, Issue C

work page 2023

[10] [10]

Jain A. K. & Dubes R. C. (1988): Algorithms for Clustering Data. Prentice-Hall

work page 1988

[11] [11]

Rousseeuw PJ Kaufman L (1987): Clustering by means of medoids. Proc. Statistical Data Analysis Based on the L1 Norm Conference, Neuchatel, 1987, pp. 405–416

work page 1987

[12] [12]

Technical Report GT-CSE-08-01, Georgia Institute of Technology

Jingu Kim & Haesun Park (2008): Sparse Nonnegative Matrix Factorization for Clustering. Technical Report GT-CSE-08-01, Georgia Institute of Technology. Available athttps://smartech.gatech.edu/handle/ 1853/20058. Allgood, Borle, and Nicholas 11

work page 2008

[13] [13]

Daniel D. Lee & H. Sebastian Seung (2000): Algorithms for Non-Negative Matrix Factorization . In: Pro- ceedings of the 13th International Conference on Neural Information Processing Systems , NIPS’00, MIT Press, Cambridge, MA, USA, p. 535–541

work page 2000

[14] [14]

IEEE Transactions on Information Theory 28(2), pp

S Lloyd (1982): Least squares quantization in PCM . IEEE Transactions on Information Theory 28(2), pp. 129–137

work page 1982

[15] [15]

J. B. MacQueen (1967): Some Methods for classification and Analysis of Multivariate Observations . In: Proceedings of 5th Berkeley Symposium on Mathematical Statistics and Probability: 281-297, University of California Press

work page 1967

[16] [16]

Nielsen & Isaac L

Michael A. Nielsen & Isaac L. Chuang (2000): Quantum Computation and Quantum Information. Cambridge University Press

work page 2000

[17] [17]

In: Annals of Operations Research

Gintaras Palubeckis (2004): Multistart Tabu Search Strategies for the Unconstrained Binary Quadratic Op- timization Problem. In: Annals of Operations Research . Available at https://doi.org/10.1023/B: ANOR.0000039522.58036.68

work page doi:10.1023/b: 2004

[18] [18]

Pollachini, Juan P

Giovani G. Pollachini, Juan P. L. C. Salazar, Caio B. D. Goes, Thiago O. Maciel & Eduardo I. Duzzioni (2021): Hybrid classical-quantum approach to solve the heat equation using quantum annealers . Phys- ical Review A 104(3), doi:10.1103/physreva.104.032426. Available at https://doi.org/10.1103% 2Fphysreva.104.032426

work page doi:10.1103/physreva.104.032426 2021

[19] [19]

Expert Systems with Applications 37(7), pp

Jing Xiao, YuPing Yan, Jun Zhang & Yong Tang (2010): A quantum-inspired genetic al- gorithm for k-means clustering . Expert Systems with Applications 37(7), pp. 4966–4973, doi:https://doi.org/10.1016/j.eswa.2009.12.017. Available at https://www.sciencedirect.com/ science/article/pii/S095741740901063X

work page doi:10.1016/j.eswa.2009.12.017 2010