Classical shadows based on locally-entangled measurements
Matteo Ippoliti
Abstract
We study classical shadows protocols based on randomized measurements in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-qubit entangled bases, generalizing the random Pauli measurement protocol (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math>). We show that entangled measurements (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>&#x2265;</mml:mo><mml:mn>2</mml:mn></mml:math>) enable nontrivial and potentially advantageous trade-offs in the sample complexity of learning Pauli expectation values. This is sharply illustrated by shadows based on two-qubit Bell measurements: the scaling of sample complexity with Pauli weight <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> improves quadratically (from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x223C;</mml:mo><mml:msup><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:math> down to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x223C;</mml:mo><mml:msup><mml:mn>3</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>k</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>) for many operators, while others become impossible to learn. Tuning the amount of entanglement in the measurement bases defines a family of protocols that interpolate between Pauli and Bell shadows, retaining some of the benefits of both. For large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>, we show that randomized measurements in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-qubit GHZ bases further improve the best scaling to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x223C;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mi>k</mml:mi></mml:msup></mml:math>, albeit on an increasingly restricted set of operators. Despite their simplicity and lower hardware requirements, these protocols can match or outperform recently-introduced "shallow shadows" in some practically-relevant Pauli estimation tasks.