Higher-order quantum computing with known input states

TL;DR

This paper investigates higher-order quantum computing with known input states, demonstrating exponential advantages and new capabilities in deterministic implementation for certain mixed states, which enhances practical quantum protocols.

Contribution

It introduces a variant of HOQC where input states are known, showing significant performance improvements and new distinctions between state classes.

Findings

Exponential advantage in the SAR protocol with known input states

Ability to distinguish protocols for pure, bipartite, and mixed states

Identification of mixed states allowing deterministic, exact implementation

Abstract

In higher-order quantum computing (HOQC), one typically considers the universal transformation of unknown quantum operations, treated as blackboxes. It is also implicitly assumed that the resulting operation must act on arbitrary, and thus unknown, input states. In this work, we explore a variant of this framework in which the operation remains unknown, but the input state is fixed and known. We argue that this assumption is well-motivated in certain practical contexts, such as unitary programming, and show that classical knowledge of the input state can significantly enhance performance. We demonstrate that in the SAR protocol, this knowledge leads to an exponential advantage through a repeat-until-success strategy, highlighting the operational power of known-state higher-order transformations. Moreover, this assumption allows us to distinguish between protocols designed for pure, bipartite, and mixed states, which enables us to identify the class of mixed states for which deterministic and exact implementation becomes possible.