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arXiv:2601.00859

Sample Complexity for Embedded Multipartite Entanglement Witness via Pauli and Clifford Classical Shadows

Ziran Zhang

quant-phSubmitted Dec 30, 2025• Announced Dec 31, 2025• 0 citations
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Abstract

Detecting multipartite entanglement in many qubit systems is measurement-intensive, motivating protocols that estimate only selected observables with provable efficiency. In this work we use the classical shadow protocol to study the sample complexity required to estimate a family of subsystem $n$-partite entanglement witness embedded in an larger $N$-qubit system. We derive ensemble dependent variance bounds that lead to qualitatively distinct scaling for the snapshots cost at fixed additive error $ε$ with numerical simulations confirm these trends, exhibiting a clear crossover from Pauli favorable performance for local witness to Clifford favorable performance as the witness becomes more global.

Comments
15 pages, 5 figures

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Detecting multipartite entanglement in many qubit systems is measurement-intensive, motivating protocols that estimate only selected observables with provable efficiency. In this work we use the classical shadow protocol to study the sample complexity required to estimate a family of subsystem nnn-partite entanglement witness embedded in an larger NNN-qubit system. We derive ensemble dependent variance bounds that lead to qualitatively distinct scaling for the snapshots cost at fixed additive error εεε with numerical simulations confirm these trends, exhibiting a clear crossover from Pauli favorable performance for local witness to Clifford favorable performance as the witness becomes more global.
Quantum metrology relies on observables that maximize information about the parameter of interest, but are often hard to access. We develop a protocol that combines Pauli moments estimated from a fixed ensemble of randomized local measurements into an optimized method-of-moments estimator. For local Pauli classical shadows, we derive the shadow information which determines estimator variance and sample-complexity guarantees. Identifying the separable-state sensitivity limit, we show that entanglement can lead to a constant-factor sensitivity enhancement, as illustrated with one-axis-twisted probes. This protocol allows for efficient circuit-based metrology without choosing an observable.
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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.
Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis W after it has evolved under a unitary transformation randomly sampled from a chosen distribution U. In this work we study the case where U corresponds to either local or global orthogonal Clifford gates, and W consists of real-valued vectors. Our results show that for various situations of interest, this ``real'' classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords decrease the required number of samples by a factor of two. More dramatically, for k-local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in-k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.
Quantum platforms can realize many-body dynamics beyond classical simulation yet complete readout remains intractable: the cost of extracting accessible information scales exponentially with system size. Classical shadows and Bell sampling offer scalable, multi-observable estimation from randomized or entanglement-assisted measurements. Here we aim to push these ideas beyond static snapshots to dynamical correlators, including out-of-time-ordered correlators (OTOCs) and two-point functions. In particular, we introduce the notion of the shadow of an operator, defined as the classical shadow of the vectorized time-evolved operator. Pauli operator-shadows enable simultaneous estimation of all local OTOCs, while Clifford operator-shadows enable efficient simultaneous estimation of all two-point correlators. Alternatively, Bell sampling allows one to simultaneously compute all diagonal OTOCs. We also prove information-theoretic lower bounds for learning OTOCs, fully characterizing their query complexities in many cases, and yielding exponential separations that formalize when the vectorized approach provides measurement-efficiency advantages.
Abstract Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis W after it has evolved under a unitary transformation randomly sampled from a chosen distribution U . In this work we study the case where U corresponds to either local or global orthogonal Clifford gates, and W consists of real-valued vectors. Our results show that for various situations of interest, this ‘real’ classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords typically decrease the required number of samples by a factor of two. More dramatically, for k -local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in- k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.