Worst-Case Sample Complexity Bounds for Distributed Inner Product Estimation with Local Randomized Measurements

TL;DR

This paper establishes worst-case sample complexity bounds for distributed inner product estimation of n-qubit states using local randomized measurements, comparing Clifford and Haar sampling methods.

Contribution

It derives sharp bounds for Clifford and Haar sampling methods, introduces a conjecture for improved Haar scaling, and analyzes Pauli shadows for large n.

Findings

Clifford sampling achieves $oxed{ ext{O}( ext{sqrt}(4.5^n))}$ sample complexity.

Haar sampling has the same upper bound as Clifford, with a conjectured sharper bound of $oxed{ ext{O}( ext{sqrt}(3.6^n))}$.

Independent single-qubit Pauli shadows have worst-case scaling $oxed{ ext{O}( ext{sqrt}(7.5^n))}$.

Abstract

We study distributed inner product estimation for $n$-qubit states using local randomized measurements, for which rigorous worst-case guarantees are less understood. We first reduce the minimax kernel optimization to Hamming-distance kernels. Within this class, unbiasedness fixes a unique kernel. For this kernel under local Clifford sampling, we prove a sharp fourth-moment bound using the single-qubit Clifford commutant. This yields worst-case sample complexity $\mathcal{O}(\sqrt{4.5^n})$, attained by identical pure product stabilizer states. For the same kernel under local Haar sampling, we prove a local twirling identity that compares its fourth moment with the Clifford fourth moment. This gives the same rigorous upper bound as in the Clifford case, but the comparison is lossy. This motivates the conjectured sharper Haar scaling $\mathcal{O}(\sqrt{3.6^n})$ attained by product states, and verify it for several important classes of states. We also show that independent single-qubit Pauli shadows have worst-case scaling $\mathcal{O}(\sqrt{7.5^n})$ for large $n$.