Approximate pushforward designs and image bounds on approximations

TL;DR

This paper extends quantum pushforward designs to approximate settings, deriving bounds on approximation parameters using Schatten norms and Lipschitz continuity, with numerical simulations confirming near-optimality in low dimensions.

Contribution

It introduces a framework for approximate quantum pushforward designs with new bounds, especially for mixed states, improving upon previous exact methods.

Findings

Derived bounds on approximation parameters using Schatten p-norms.

Refined bounds for mixed states exploiting symmetric subspace structure.

Numerical simulations demonstrate near-optimality in low-dimensional cases.

Abstract

We extend the framework of quantum pushforward designs to the approximate setting, where averaging is achieved only up to finite precision. Using Schatten $p$-norms and Lipschitz continuity arguments, we derive bounds on the approximation parameters of pushforward designs obtained from complex projective spaces, including simplices, mixed states, and quantum channels. In the mixed-state case, we refine the bounds by exploiting the symmetric subspace structure, leading to asymptotically tighter estimates. Numerical simulations support our theoretical results, showing near-optimality in low-dimensional scenarios.