Efficient protocol to estimate the Quantum Fisher Information Matrix for commuting-block circuits

Abstract

Abstract The Quantum Fisher Information Matrix (QFIM) quantifies how sensitive parameterized quantum states are to changes in their parameters. Recently, it has been used to improve variational quantum algorithm optimization through geometry-aware techniques. However, estimating the QFIM—particularly its off-block-diagonal elements—requires substantial resources. To address this, we introduce a novel protocol that efficiently computes these elements for commuting-block variational circuits. Our approach reduces the number of quantum state preparations from $$O(m^2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> to $$O(L^2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where m is the number of parameters and L the number of circuit layers. This also lowers classical measurement and post-processing requirements, improving computational efficiency.