Classical shadows based on locally-entangled measurements

Abstract

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 ($n\ge 2$) 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.

8 pages, 3 figures. v2: added discussion of operators with non-contiguous support in 1D, reference to Shor-Laflamme distributions, minor improvements. v3: fixed typos, added some references and clarified some derivations. Accepted in Quantum

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Cited by in corpus (9)

• Error-mitigated fermionic classical shadows on noisy quantum devices
• Many-body entropies and entanglement from polynomially-many local measurements
• Efficient Local Classical Shadow Tomography with Number Conservation
• Approximate inverse measurement channel for shallow shadows
• Holographic Classical Shadow Tomography
• Monte Carlo Simulation of Operator Dynamics and Entanglement in Dual-Unitary Circuits
• Improving shadow estimation with locally-optimal dual frames
• Classical Shadows with Improved Median-of-Means Estimation
• Selective and efficient quantum state tomography for multi-qubit systems