Real classical shadows

• URL: http://arxiv.org/abs/2410.23481v1
• Date: Wed, 30 Oct 2024 22:15:39 GMT
• Title: Real classical shadows
• Authors: Maxwell West, Antonio Anna Mele, Martin Larocca, M. Cerezo,
• Abstract summary: We study the case where $mathcalU$ corresponds to either local or global Clifford gates, and $mathcalW$ 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.
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• 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 $\mathcal{W}$ after it has evolved under a unitary transformation randomly sampled from a chosen distribution $\mathcal{U}$. In this work we study the case where $\mathcal{U}$ corresponds to either local or global orthogonal Clifford gates, and $\mathcal{W}$ 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.

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