No-iteration of unknown quantum gates

TL;DR

This paper proves a no-go theorem for deterministic quantum circuit iteration of unknown gates with a single call, explores approximate schemes, and finds that for many iterations, simple strategies perform optimally regardless of additional resources.

Contribution

It introduces a new no-go theorem for single-call quantum gate iteration and analyzes optimal approximate schemes, revealing that simple strategies suffice for large iterations.

Findings

Deterministic iteration of unknown gates with one call is impossible.

Simple strategies like identity perform optimally for many iterations.

Additional resources do not improve performance in large iteration regimes.

Abstract

We propose a new no-go theorem by proving the impossibility of constructing a deterministic quantum circuit that iterates a unitary oracle by calling it only once. Different schemes are provided to bypass this result and to approximately realize the iteration. The optimal scheme is also studied. An interesting observation is that for large number of iterations, a trivial strategy like using the identity channel has the optimal performance, and preprocessing, postprocessing, or using resources like entanglement does not help at all. Intriguingly, the number of iterations, when being large enough, does not affect the performance of the proposed schemes.