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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.18162 |
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| _version_ | 1866917599830343680 |
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| author | Rosebush, Aiden R. Greenwood, Alexander C. B. Kirby, Brian T. Qian, Li |
| author_facet | Rosebush, Aiden R. Greenwood, Alexander C. B. Kirby, Brian T. Qian, Li |
| contents | We propose a support vector machine (SVM) based approach for generating an entanglement witness that requires exponentially less training data than previously proposed methods. SVMs generate hyperplanes represented by a weighted sum of expectation values of local observables whose coefficients are optimized to sum to a positive number for all separable states and a negative number for as many entangled states as possible near a specific target state. Previous SVM-based approaches for entanglement witness generation used large amounts of randomly generated separable states to perform training, a task with considerable computational overhead. Here, we propose a method for orienting the witness hyperplane using only the significantly smaller set of states consisting of the eigenstates of the generalized Pauli matrices and a set of entangled states near the target entangled states. With the orientation of the witness hyperplane set by the SVM, we tune the plane's placement using a differential program that ensures perfect classification accuracy on a limited test set as well as maximal noise tolerance. For $N$ qubits, the SVM portion of this approach requires only $O(6^N)$ training states, whereas an existing method needs $O(2^{4^N})$. We use this method to construct witnesses of 4 and 5 qubit GHZ states with coefficients agreeing with stabilizer formalism witnesses to within 6.5 percent and 1 percent, respectively. We also use the same training states to generate novel 4 and 5 qubit W state witnesses. Finally, we computationally verify these witnesses on small test sets and propose methods for further verification. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_18162 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An Exponential Reduction in Training Data Sizes for Machine Learning Derived Entanglement Witnesses Rosebush, Aiden R. Greenwood, Alexander C. B. Kirby, Brian T. Qian, Li Quantum Physics We propose a support vector machine (SVM) based approach for generating an entanglement witness that requires exponentially less training data than previously proposed methods. SVMs generate hyperplanes represented by a weighted sum of expectation values of local observables whose coefficients are optimized to sum to a positive number for all separable states and a negative number for as many entangled states as possible near a specific target state. Previous SVM-based approaches for entanglement witness generation used large amounts of randomly generated separable states to perform training, a task with considerable computational overhead. Here, we propose a method for orienting the witness hyperplane using only the significantly smaller set of states consisting of the eigenstates of the generalized Pauli matrices and a set of entangled states near the target entangled states. With the orientation of the witness hyperplane set by the SVM, we tune the plane's placement using a differential program that ensures perfect classification accuracy on a limited test set as well as maximal noise tolerance. For $N$ qubits, the SVM portion of this approach requires only $O(6^N)$ training states, whereas an existing method needs $O(2^{4^N})$. We use this method to construct witnesses of 4 and 5 qubit GHZ states with coefficients agreeing with stabilizer formalism witnesses to within 6.5 percent and 1 percent, respectively. We also use the same training states to generate novel 4 and 5 qubit W state witnesses. Finally, we computationally verify these witnesses on small test sets and propose methods for further verification. |
| title | An Exponential Reduction in Training Data Sizes for Machine Learning Derived Entanglement Witnesses |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2311.18162 |