Classical shadows based on locally-entangled measurements
Matteo Ippoliti · Quantum · 2024
We study classical shadows protocols based on randomized measurements in n -qubit entangled bases, generalizing the random Pauli measurement protocol ( n=1 ). We show that entangled measurements ( n≥2 ) 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 k improves quadratically (from ∼3k down to ∼3k/2 ) 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 n , we show that randomized measurements in n -qubit GHZ bases further improve the best scaling to ∼(3/2)k , 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.