Classical shadows based on locally-entangled measurements

Quantum 8, 1293 (2024).

https://doi.org/10.22331/q-2024-03-21-1293

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 ($ngeq 2$) 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 $sim 3^k$ down to $sim 3^{k/2}$) 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 $sim (3/2)^k$, 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.