Approximate pushforward designs and image bounds on approximations

2025/12/01 by Jakub Czartowski, Czartowski, Jakub, Adam Sawicki +3
Mathematics · Computer Science · Decision Sciences · #Mathematical Approximation and Integration #Quantum Computing Algorithms and Architecture #Optimal Experimental Design Methods

paper · pdf · doi:10.48550/arxiv.2512.01858

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.

Citations

• The non-Clifford cost of random unitaries
• Quantum Pushforward Designs
• Cyclic measurements and simplified quantum state tomography
• Duality of averaging of quantum states over arbitrary symmetry groups revealing Schur-Weyl duality
• Linear Growth of Circuit Complexity from Brownian Dynamics
• Geometry of the Welch Bounds
• Quantum t-designs: t-wise independence in the quantum world
• Induced measures in the space of mixed quantum states

Related

• Quantum Pushforward Designs
• Quantum Approximation I. Embeddings of Finite Dimensional L_p Spaces
• Best-approximation error for parametric quantum circuits
• Density theorems with applications in quantum signal processing
• Approximate Unitary k-Designs from Shallow, Low-Communication Circuits