Aiden R Rosebush, Alexander C B Greenwood, Brian T Kirby, Li Qian
{"title":"机器学习推导纠缠见证的训练数据量指数级减少","authors":"Aiden R Rosebush, Alexander C B Greenwood, Brian T Kirby, Li Qian","doi":"10.1088/2632-2153/ad7457","DOIUrl":null,"url":null,"abstract":"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 <italic toggle=\"yes\">N</italic> qubits, the SVM portion of this approach requires only <inline-formula>\n<tex-math><?CDATA $O(6^N)$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mn>6</mml:mn><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"mlstad7457ieqn1.gif\"></inline-graphic></inline-formula> training states, whereas an existing method needs <inline-formula>\n<tex-math><?CDATA $O(2^{4^N})$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:msup><mml:mn>4</mml:mn><mml:mi>N</mml:mi></mml:msup></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"mlstad7457ieqn2.gif\"></inline-graphic></inline-formula>. We use this method to construct witnesses of 4 and 5 qubit GHZ states with coefficients agreeing with stabilizer formalism witnesses to within 3.7 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.","PeriodicalId":33757,"journal":{"name":"Machine Learning Science and Technology","volume":null,"pages":null},"PeriodicalIF":6.3000,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"An exponential reduction in training data sizes for machine learning derived entanglement witnesses\",\"authors\":\"Aiden R Rosebush, Alexander C B Greenwood, Brian T Kirby, Li Qian\",\"doi\":\"10.1088/2632-2153/ad7457\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"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 <italic toggle=\\\"yes\\\">N</italic> qubits, the SVM portion of this approach requires only <inline-formula>\\n<tex-math><?CDATA $O(6^N)$?></tex-math><mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mi>O</mml:mi><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:msup><mml:mn>6</mml:mn><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\\\"mlstad7457ieqn1.gif\\\"></inline-graphic></inline-formula> training states, whereas an existing method needs <inline-formula>\\n<tex-math><?CDATA $O(2^{4^N})$?></tex-math><mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mi>O</mml:mi><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:msup><mml:mn>4</mml:mn><mml:mi>N</mml:mi></mml:msup></mml:mrow></mml:msup><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\\\"mlstad7457ieqn2.gif\\\"></inline-graphic></inline-formula>. We use this method to construct witnesses of 4 and 5 qubit GHZ states with coefficients agreeing with stabilizer formalism witnesses to within 3.7 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.\",\"PeriodicalId\":33757,\"journal\":{\"name\":\"Machine Learning Science and Technology\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":6.3000,\"publicationDate\":\"2024-09-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Machine Learning Science and Technology\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1088/2632-2153/ad7457\",\"RegionNum\":2,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Machine Learning Science and Technology","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1088/2632-2153/ad7457","RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE","Score":null,"Total":0}
An exponential reduction in training data sizes for machine learning derived entanglement witnesses
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(6N) training states, whereas an existing method needs O(24N). We use this method to construct witnesses of 4 and 5 qubit GHZ states with coefficients agreeing with stabilizer formalism witnesses to within 3.7 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.
期刊介绍:
Machine Learning Science and Technology is a multidisciplinary open access journal that bridges the application of machine learning across the sciences with advances in machine learning methods and theory as motivated by physical insights. Specifically, articles must fall into one of the following categories: advance the state of machine learning-driven applications in the sciences or make conceptual, methodological or theoretical advances in machine learning with applications to, inspiration from, or motivated by scientific problems.