Higher-order quantum computing with known input states

TL;DR

The paper investigates higher-order quantum computing tasks when the input state is known, showing that classical knowledge of the input can dramatically boost performance for certain transformations, notably unitary storage-and-retrieval (SAR) with repeat-until-success, while some tasks like conjugation retain no advantage from knowledge. It develops a unified framework based on Choi representations and quantum supermaps, and leverages group-theoretic symmetries to transform the optimization into tractable SDPs. The main contributions include explicit probabilistic and deterministic results for single and multi-call scenarios across transposition, conjugation, and inversion, plus thorough numerical evidence supporting the analytical findings and a detailed comparison with port-based state preparation. The work highlights when known-state HOQC provides practical and theoretical benefits, and outlines open questions about composition, universality, and non-universal input handling in higher-order quantum frameworks.

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.

Figures (25)

• Figure 1: A protocol implementing a unitary transformation via the action of a supermap. Elements shown in purple are optimized; dashed elements represent unknown and universal components; green indicates fixed behaviour determined by the other elements.
• Figure 2: A protocol implementing unitary transformation via action of a supermap with delayed input state.
• Figure 3: Storage-and-retrieval protocol using $k$ parallel queries to the unknown unitary $U$, and known input state $\rho$. Note that during the storage phase, the input state $\rho$ is not known, and hence, the encoder operator $E$ cannot depend on $\rho$ (otherwise, the problem would be trivial, as we could simply take $E^\rho = \rho$). However, differently from the standard SAR protocol Sedl_k_2019Bisio_2010, in the retrieval step, the information of the input state $\rho$ is available, hence the decoder operation $D_\rho$ has an explicit dependence on $\rho$.
• Figure 4: A superchannel acting on a universal input state and a universal single-call operation. Any superchannel can be represented by an encoder operation, an ancillary system that serves as a memory, and a decoder operation. The lines corresponding to the subsystems on which the supermap acts are labeled as follows: '$\textup{P}$' for the past, '$\textup{I}$' for the operation input, "$\textup{O}$' for the operation output, '$\textup{M}$' for the ancillary memory system and '$\textup{F}$' for future.
• Figure 5: A universal supermap with a delayed input state, realising transformation on an arbitrary unknown input state (a) approximately or (b) with a certain probability of success $p$.

Theorems & Definitions (75)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 10: Independence of the choice of known state
• Theorem 11: Covariance properties of deterministic superchannel
• Theorem 12: Covariance properties of probabilistic bipartite supermap
• Theorem 13