# Coreset Clustering on Small Quantum Computers

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## Abstract

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## 1. Introduction

Algorithm 1: 2-means clustering via coresets+QAOA. |

Input : A data set ${\mathit{x}}_{1},...,{\mathit{x}}_{n}\in {\mathbb{R}}^{d}$Output : Cluster centers ${\mathit{\mu}}_{-1}$ and ${\mathit{\mu}}_{+1}$ which approximately minimize
$$\sum _{i\in \left[n\right]}\underset{j\in \{-1,+1\}}{min}{\left(\right)}^{{\mathit{x}}_{i}}2$$
Algorithm: 1. Construct a coreset $({\mathit{X}}^{\prime},w)$ of size m. 2. Construct a jth order m-qubit Hamiltonian for the coreset. 3. Use QAOA to variationally approximate an energy-maximizing eigenstate of the Hamiltonian. 4. Treat the 0/1 assignment of the eigenstate as the $k=2$ clustering. |

- We implemented algorithms for coresets and evaluated their performance on real data sets.
- We cast coreset clustering to a Hamiltonian optimization problem that can be solved with QAOA, and herein we demonstrate how to break past the assumption of equal cluster weights.
- We benchmarked the performance of Algorithm 1 across six different data sets, including real and synthetic data, comparing the 2-means clusterings found by quantum and classical means. We found that some data sets are better suited to coreset summarization than others, which can play a large role in the quality of the clustering solutions.

## 2. $\mathit{k}$-Means Clustering

## 3. Coresets for $\mathit{k}$-Means

## 4. Coreset $\mathit{k}$-Means via QAOA

#### 4.1. QAOA

#### 4.2. Hamiltonians for k-Means Clustering: Equal Cluster Weights

#### 4.3. Hamiltonians for k-Means: Unequal Cluster Weights

## 5. Results

#### 5.1. Data Sets

#### 5.2. Evaluation Methodology

Data Set | Description |
---|---|

CIFAR-10 | 10,000 images (32 × 32 pixels) from CIFAR-10 data set [28]. 1000 images per category. |

COCO | 5000 images from Common Objects in Context validation data set [29]. Images translated into feature vectors of dimension 512. |

Epilepsy | Epileptic seizure recognition data set from [30]. 11,500 vectors of dimension 179. |

Pulsars | Pulsar candidates from HTRU2 data set [31]. 1600/17,900 of 9-dimensional feature vectors are pulsars. |

Yeast | Localization sites of proteins [32]. 1500 8-dimensional vectors. |

Synthetic | 40,000 512-dimensional points drawn from 11 random Gaussian clusters. Ten clusters contribute 5 points each, last cluster has majority. |

#### 5.3. Coreset and QAOA Bound Results

#### 5.4. Experimental QAOA Results

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Weighted Max-Cut for a coreset consisting of five points. Given an assignment of vertices to two colors, i.e., a cut, we are interested in the sum of $-{w}_{i}{w}_{j}{\mathit{x}}_{i}\xb7{\mathit{x}}_{j}$ on edges crossing the cut. By interpreting these terms as edge weights, we seek a weighted Max-Cut.

**Figure 3.**Evaluation of quantum and classical coreset clustering on six different data sets. The green and orange bars were obtained by running classical 2-means on $m=5,10,15,$ and 20 random and BFL16 coresets, respectively. The blue bars express the cost of the highest-eigenstate of the $m=5$ and 10 Hamiltonians using a j-th order Taylor expansion. These can be interpreted as bounds on QAOA’s performance with m qubits. We report the best of 10 results for all data sets, except for the synthetic one, which shows the means and min-max error bars. All costs are scaled with respect to the cost achieved by running 2-means over the full data set.

**Figure 4.**An example of a QAOA circuit used to solve the weighted Max-Cut problem on an $m=5$ coreset implemented using the swap networks proposed in [33,34]. Here, $p=1$, $\alpha $, and $\beta $ are the variational parameters and the ${w}_{ij}$’s are the edge weights of the constructed graph (see Figure 1). Using the swap network, each qubit is able to interact with every other qubit, $O\left({n}^{2}\right)$ interactions, in linear depth.

**Figure 5.**An experimental evaluation of a QAOA circuit implemented with and without the swap network [33,34]. Each distribution consists of 8192 individual shots. The noisy execution of the quantum hardware becomes apparent when comparing the experimental results with the noiseless simulation. However, by utilizing the swap network, one of the optimal bitstrings (01100) can still be identified in the output distribution with high probability.

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**MDPI and ACS Style**

Tomesh, T.; Gokhale, P.; Anschuetz, E.R.; Chong, F.T.
Coreset Clustering on Small Quantum Computers. *Electronics* **2021**, *10*, 1690.
https://doi.org/10.3390/electronics10141690

**AMA Style**

Tomesh T, Gokhale P, Anschuetz ER, Chong FT.
Coreset Clustering on Small Quantum Computers. *Electronics*. 2021; 10(14):1690.
https://doi.org/10.3390/electronics10141690

**Chicago/Turabian Style**

Tomesh, Teague, Pranav Gokhale, Eric R. Anschuetz, and Frederic T. Chong.
2021. "Coreset Clustering on Small Quantum Computers" *Electronics* 10, no. 14: 1690.
https://doi.org/10.3390/electronics10141690