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Scalable Quantum Clustering of Renewable Profiles via a Hybrid Qubit-efficient Strategy
Yifeng Cao, Chengjun Liu, Yijun Xu, Kai Wen, Wei Gu, Lamine Mili, Shuai Lu
TL;DR
Large-scale renewable energy profile clustering is NP-hard, while quantum implementations are constrained by limited hardware qubits. The paper proposes CBVR, combining one-hot elimination and consensus-based equality constraints, and reports hardware-feasible QUBOs with clustering performance comparable to SQA and Gurobi.
Problem
Large-scale renewable energy profile clustering is computationally demanding, and current quantum hardware’s limited qubit capacity constrains practical implementation.
Source: §I · p. 1Method
CBVR reduces QUBO variables by eliminating an assignment variable per profile under one-hot constraints and using stable co-assignment relations as additional equality constraints.
Source: §I · p. 1, A. Variable Reduction Based on the… · p. 2, B. Variable Reduction Based on… · p. 3Results
CBVR reduced the 300-profile cases to 300, 550, and 549 variables for 2, 3, and 4 clusters, respectively, enabling all three to run on quantum hardware while clustering performance remained comparable to SQA and Gurobi.
Source: 2) Clustering Effectiveness · p. 5Takeaways & Limitations
The experiments show that CBVR can compress oversized QUBOs into hardware-admissible formulations while maintaining high clustering accuracy and efficiency for renewable energy profiles.
Source: §V · p. 5One-hot elimination alone removes only one variable per profile, so the resulting model may remain large when profile or cluster counts are significant.
Source: A. Variable Reduction Based on the… · p. 2
Abstract
from arXiv · showhide
Large-scale renewable energy profile clustering is a computationally challenging NP-hard problem. Recent quantum computing techniques offer a potential remedy, yet their scalability is severely constrained by the limited qubit count of current quantum hardware. To address this issue, we first cast the problem into a QUBO form and then we propose a consensus-based variable reduction (CBVR) method to drastically cut qubit usage. More specifically, we tailor the one-hot constraint to eliminate assignment variables and leverage consensus analysis to convert stable clustering relations from initial anneals into equality constraints for further reduction. Experiments on real quantum hardware confirm that CBVR compresses oversized QUBO models into feasible sizes while retaining high clustering quality and efficiency.
I. INTRODUCTION
Renewable energy profile clustering is NP-hard, and scaling it remains computationally demanding despite classical and quantum approaches. The paper proposes CBVR to reduce the QUBO variables needed for quantum clustering.
I. INTRODUCTION
- Clustering is NP-hard in general, and large-scale instances can make finding a global optimum computationally demanding despite classical sampling and ensemble methods.The cited methods include RSPCA, RSPCE, and ACEM –; the passage also cites, on computational demands.Source: §I · p. 1
- Quantum clustering studies demonstrate potential for renewable-profile clustering, but practical implementation is limited by current hardware’s qubit capacity.The passage names quantum K-means and quantum K-medoids.Source: §I · p. 1
- CBVR uses one-hot constraints to remove one assignment variable per profile, then turns stable co-assignment relations into additional equality constraints.The method targets renewable energy profile clustering on QUBO-based quantum hardware.Source: §I · p. 1
A. Renewable Energy Profile Clustering Model
- Clustering minimizes intra-group dissimilarity among profiles representing renewable-station power output over time; the paper uses wind turbine profiles as an example.The model uses Euclidean distances between profiles and is reformulated as a QUBO for quantum hardware.Source: A. Renewable Energy Profile Clustering… · p. 1
B. The Ising Model
The paper maps constrained clustering into a quantum-compatible QUBO and explains why the variable count can exceed current hardware capacity. Its Ising representation encodes candidate solutions as spin configurations.
B. The Ising Model
- Each Ising spin configuration encodes a potential solution, and its energy reflects the objective function.The model is expressed using a linear coefficient vector, an upper-triangular coupling matrix, and spins taking values −1 or +1.Source: B. The Ising Model · p. 2
C. Quantum-Compatible Reformulated QUBO Model
- The clustering formulation’s equality constraints are incorporated into a QUBO objective using a positive penalty coefficient.The standard QUBO formulation has no constraints, so the equality-constrained model must be transformed.Source: C. Quantum-Compatible Reformulated… · p. 2
- The QUBO is represented by a binary decision vector and its coefficient matrix, yielding a form that can be solved directly on quantum hardware.The paper states that the Ising form is equivalent to the QUBO formulation.Source: C. Quantum-Compatible Reformulated… · p. 2, B. The Ising Model · p. 2
- The QUBO uses NG binary variables, which can exceed practical hardware capacity because each variable typically maps to one physical qubit.The variable count grows with both the number of profiles N and clusters G.Source: C. Quantum-Compatible Reformulated… · p. 2
III. PROPOSED METHOD
CBVR combines one-hot variable elimination with consensus analysis of stable co-assignment relations. One-hot elimination alone removes one variable per profile, so the model can remain large for substantial profile or cluster counts.
III. PROPOSED METHOD
- The proposed workflow applies one-hot variable reduction, identifies stable co-assignment relations through consensus analysis, and converts them into additional equality constraints.The section also summarizes the overall CBVR procedure.Source: §III · p. 2
A. Variable Reduction Based on the One-Hot Constraints
- One-hot constraints let the method select one assignment variable per profile as dependent and remove it from the QUBO.The transformation reduces dimensionality from NG to N(G−1).Source: A. Variable Reduction Based on the… · p. 2
- One-hot elimination removes only one variable per profile, a linear reduction in cluster count that can leave large models when N or G is significant.The passage motivates further reduction through co-assignment constraints.Source: A. Variable Reduction Based on the… · p. 2
B. Variable Reduction Based on Consensus Analysis
CBVR identifies stable co-assignment relations across annealing solutions and turns selected relations into equality constraints. It groups linked profiles into super profiles, reducing the QUBO to N′(G−1) binary variables.
B. Variable Reduction Based on Consensus Analysis
- Consensus analysis compares feasible candidate partitions generated by R annealing runs to identify pairs repeatedly assigned to the same cluster.For each run, the assignment value indicates whether a profile belongs to a given cluster.Source: B. Variable Reduction Based on… · pp. 2–3
- CBVR ranks profile pairs by co-assignment consistency and selects the top K pairs as reliable relations for additional equality constraints.A larger C_i,j means the pair was more consistently assigned to the same cluster across annealing runs.Source: B. Variable Reduction Based on… · pp. 2–3
- Selected pairs are tied to share variables, and overlapping relations are grouped into merged profile sets alongside unmerged profiles.The resulting updated set treats each merged group as a super profile representing its original profiles.Source: B. Variable Reduction Based on… · p. 3
- The reduced QUBO uses N′(G−1) binary variables, where N′ is the number of merged groups and unmerged profiles.The algorithm constructs the reduced QUBO, performs annealing for consensus analysis, optimizes the further-reduced model, and recovers original profile labels.Source: B. Variable Reduction Based on… · p. 3
IV. CASE STUDY
The case study evaluates clustering on wind-turbine profiles from the SDWPF dataset and compares methods across profile and cluster-count settings. It also evaluates CBVR across datasets of different sizes.
IV. CASE STUDY
- The experiments use 300 randomly selected valid profiles from the SDWPF wind-farm dataset, with hourly daily profiles containing 24 data points.The source SCADA measurements were collected at 10-minute intervals and averaged to hourly resolution.Source: §IV · pp. 3–4
- The first experiment compares methods for clustering 300 profiles into 2, 3, or 4 groups, while the second evaluates CBVR on 150–300 profiles with four groups.The corresponding original QUBO models range from 600 to 1200 binary variables.Source: §IV · p. 4
- Clustering performance is evaluated using the silhouette coefficient, Davies–Bouldin index, and Calinski–Harabasz index.The penalty coefficient λ is set to fifteen times the maximum entry of the coefficient matrix.Source: §IV · p. 4
A. Comparison of Clustering Methods
The initial experiment compares variable-reduction requirements across three clustering sizes. CBVR brings the 3- and 4-group models within the 550-qubit hardware limit.
A. Comparison of Clustering Methods
- 275×2=550 and 183×3=549 are the reduced QUBO sizes for the 300×3 and 300×4 cases, respectively, both within the 550-qubit limit.CBVR selects the top 25 co-assignment relations for the 300×3 case and 117 for the 300×4 case.Source: A. Comparison of Clustering Methods · p. 4
- 300 variables remain in the 300×2 case after spin-variable reduction using only the one-hot constraints.All original QUBO models exceed the 550-variable limit, so variable reduction is required before hardware execution.Source: A. Comparison of Clustering Methods · p. 4
- CBVR uses 100 preliminary annealing runs, each with a nominal 1-second budget, to generate candidate solutions for consensus analysis.Each run uses a simulated-annealing sampler with one sample and 200 sweeps until the time budget is reached.Source: A. Comparison of Clustering Methods · p. 4
1) Computational Performance:
SQA and Gurobi runtimes rise markedly with QUBO variable dimension, while CBVR combines lower total overhead than their optimization time with a favorable efficiency–accuracy trade-off.
1) Computational Performance:
- SQA and Gurobi runtimes increase markedly as QUBO variable dimensions grow across the 300 × 2, 300 × 3, and 300 × 4 cases.Fig. 1(a) compares computing time across algorithms and QUBO variable sizes.Source: Figure 1 · p. 4, 1) Computational Performance · p. 4
- The figure evaluates computing time alongside silhouette coefficient, Calinski–Harabasz index, and Davies–Bouldin index.Source: Figure 1 · p. 4
- CBVR’s total overhead remains substantially below the optimization time required by SQA and Gurobi, while offering a favorable efficiency–accuracy trade-off.CBVR overhead includes quantum hardware execution and classical preprocessing to identify stable co-assignment relations.Source: 1) Computational Performance · p. 4
2) Clustering Effectiveness:
CBVR reduces QUBO sizes to hardware-compatible levels while maintaining clustering performance comparable to SQA and Gurobi across the three tested cases.
2) Clustering Effectiveness:
- CBVR’s clustering performance remains comparable to SQA and Gurobi across all three cases on silhouette coefficient, CHI, and DBI.RSQ consistently has lower silhouette coefficient and CHI and higher DBI; in the 300 × 2 case, SVR shows similar quality to CBVR, SQA, and Gurobi.Source: 2) Clustering Effectiveness · p. 5
- CBVR reduces the 300 × 2, 300 × 3, and 300 × 4 cases to 300, 550, and 549 variables, enabling all three to run on quantum hardware.SVR reduces the 300 × 2 case to 300 variables but does not make the larger cases hardware-compatible.Source: 2) Clustering Effectiveness · p. 5
B. Effect of Dataset Size
For four-cluster datasets, CBVR selects ranked co-assignment relations to bring larger QUBOs within the 550-qubit limit.
B. Effect of Dataset Size
- For 200 × 4, 250 × 4, and 300 × 4 datasets, CBVR selects the top 17, 67, and 117 ranked co-assignment relations, respectively.Preliminary annealing and SA sampler settings match those of the first experiment set.Source: B. Effect of Dataset Size · p. 5
- The selected relations reduce each larger QUBO to 183 × 3 = 549 variables, within the 550-qubit hardware constraint.Source: B. Effect of Dataset Size · p. 5
- One-hot constraints alone reduce the 150 × 4 case to 450 variables and the 200 × 4, 250 × 4, and 300 × 4 cases to 600, 750, and 900 variables.Source: B. Effect of Dataset Size · p. 5
1) Clustering Effectiveness:
Across dataset sizes, CBVR maintains similar clustering quality while accommodating larger datasets within a fixed hardware limit; its reduction depends on reliable co-assignment relations from preliminary anneals.
1) Clustering Effectiveness:
- As profiles increase from 150 to 300, silhouette coefficient stays within 0.588–0.622 and DBI within 0.688–0.711, while CHI trends upward.The larger cases requiring additional consensus analysis have silhouette and DBI values comparable to the smaller case using only SVR.Source: Table II · p. 5, 1) Clustering Effectiveness · p. 5
- The findings support accommodating larger datasets within the same 550-variable hardware limit while achieving quality comparable to the smaller dataset.Source: 1) Clustering Effectiveness · p. 5
V. CONCLUSION AND FUTURE WORK
- Experiments demonstrate that CBVR reduces oversized QUBOs to hardware-admissible formulations while maintaining high clustering accuracy and efficiency for renewable energy profiles.Source: §V · p. 5
- CBVR’s variable-reduction performance depends on the reliability of co-assignment relations from preliminary anneals, which may vary with problem characteristics and annealing quality.Future work will investigate adaptive selection, larger datasets and hardware, and handling bad or missing data.Source: §V · p. 5