{"title":"Training quantum neural networks using the quantum information bottleneck method","authors":"Ahmet Burak Çatlı, Nathan Wiebe","doi":"10.1088/1751-8121/ad6daf","DOIUrl":null,"url":null,"abstract":"We provide in this paper a concrete method for training a quantum neural network to maximize the relevant information about a property that is transmitted through the network. This is significant because it gives an operationally well founded quantity to optimize when training autoencoders for problems where the inputs and outputs are fully quantum. We provide a rigorous algorithm for computing the value of the quantum information bottleneck quantity within error <italic toggle=\"yes\">ε</italic> that requires <inline-formula>\n<tex-math><?CDATA $O(\\log^2(1/\\epsilon) + 1/\\delta^2)$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>log</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo></mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>ϵ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>δ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"aad6dafieqn1.gif\"></inline-graphic></inline-formula> queries to a purification of the input density operator if its spectrum is supported on <inline-formula>\n<tex-math><?CDATA $\\{0\\}~\\bigcup ~[\\delta,1-\\delta]$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo><mml:mtext> </mml:mtext><mml:mo>⋃</mml:mo><mml:mtext> </mml:mtext><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>δ</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>δ</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"aad6dafieqn2.gif\"></inline-graphic></inline-formula> for <italic toggle=\"yes\">δ</italic> > 0 and the kernels of the relevant density matrices are disjoint. We further provide algorithms for estimating the derivatives of the QIB function, showing that quantum neural networks can be trained efficiently using the QIB quantity given that the number of gradient steps required is polynomial.","PeriodicalId":16763,"journal":{"name":"Journal of Physics A: Mathematical and Theoretical","volume":null,"pages":null},"PeriodicalIF":2.0000,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Physics A: Mathematical and Theoretical","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1088/1751-8121/ad6daf","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
We provide in this paper a concrete method for training a quantum neural network to maximize the relevant information about a property that is transmitted through the network. This is significant because it gives an operationally well founded quantity to optimize when training autoencoders for problems where the inputs and outputs are fully quantum. We provide a rigorous algorithm for computing the value of the quantum information bottleneck quantity within error ε that requires O(log2(1/ϵ)+1/δ2) queries to a purification of the input density operator if its spectrum is supported on {0}⋃[δ,1−δ] for δ > 0 and the kernels of the relevant density matrices are disjoint. We further provide algorithms for estimating the derivatives of the QIB function, showing that quantum neural networks can be trained efficiently using the QIB quantity given that the number of gradient steps required is polynomial.
期刊介绍:
Publishing 50 issues a year, Journal of Physics A: Mathematical and Theoretical is a major journal of theoretical physics reporting research on the mathematical structures that describe fundamental processes of the physical world and on the analytical, computational and numerical methods for exploring these structures.