Training quantum neural networks using the quantum information bottleneck method

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" 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> queries to a purification of the input density operator if its spectrum is supported on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" 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> for δ &gt; 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.