Resource-efficient shadow tomography using equatorial stabilizer measurements

Abstract

We propose a resource-efficient shadow-tomography scheme using equatorial-stabilizer measurements generated from subsets of Clifford unitaries. For n-qubit systems, equatorial-stabilizer-based shadow-tomography schemes can estimate M observables (up to an additive error ɛ) using O(log(M),poly(n),1/ɛ^{2}) sampling copies for a large class of observables, including those with traceless parts possessing polynomially bounded Frobenius norms. For arbitrary quantum-state observables with a constant Frobenius norm, sampling complexity becomes n independent. Our scheme only requires an n-depth controlled-Z (CZ) circuit [O(n^{2}) CZ gates] and Pauli measurements per sampling copy. Alternatively, our scheme is realizable with 2n-depth circuits comprising n^{2} nearest-neighboring cnot gates, exhibiting a smaller maximal gate count relative to previously known randomized-Clifford-based proposals. We numerically confirm our theoretically derived shadow-tomographic sampling complexities with random pure states and multiqubit graph states. Finally, we demonstrate that equatorial-stabilizer-based shadow tomography is more noise tolerant than randomized-Clifford-based schemes in terms of fidelity estimation for the Greenberger–Horne–Zeilinger state and W state.