Classical shadows based on locally-entangled measurements

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 (n2) 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.