Performance analysis of multi-shot shadow estimation

Abstract

Shadow estimation is an efficient method for predicting many observables of a quantum state with a statistical guarantee. In the multi-shot scenario, one performs projective measurement on the sequentially prepared state for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math> times after the same unitary evolution, and repeats this procedure for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi></mml:math> rounds of random sampled unitary. As a result, there are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi><mml:mi>K</mml:mi></mml:math> times measurements in total. Here we analyze the performance of shadow estimation in this multi-shot scenario, which is characterized by the variance of estimating the expectation value of some observable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi></mml:math>. We find that in addition to the shadow-norm <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">&amp;#x2016;</mml:mo><mml:mi>O</mml:mi><mml:msub><mml:mo fence="false" stretchy="false">&amp;#x2016;</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">s</mml:mi><mml:mi mathvariant="normal">h</mml:mi><mml:mi mathvariant="normal">a</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math> introduced in \cite{huang2020predicting}, the variance is also related to another norm, and we denote it as the cross-shadow-norm <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">&amp;#x2016;</mml:mo><mml:mi>O</mml:mi><mml:msub><mml:mo fence="false" stretchy="false">&amp;#x2016;</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">X</mml:mi><mml:mi mathvariant="normal">s</mml:mi><mml:mi mathvariant="normal">h</mml:mi><mml:mi mathvariant="normal">a</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math>. For both random Pauli and Clifford measurements, we analyze and show the upper bounds of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">&amp;#x2016;</mml:mo><mml:mi>O</mml:mi><mml:msub><mml:mo fence="false" stretchy="false">&amp;#x2016;</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">X</mml:mi><mml:mi mathvariant="normal">s</mml:mi><mml:mi mathvariant="normal">h</mml:mi><mml:mi mathvariant="normal">a</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math>. In particular, we figure out the exact variance formula for Pauli observable under random Pauli measurements. Our work gives theoretical guidance for the application of multi-shot shadow estimation.

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