Memory cost of quantum protocols

TL;DR

This paper investigates the minimal ancillary system resources needed for quantum protocols with classical memory assistance, introducing the concept of memory cost and analyzing its properties and bounds.

Contribution

It introduces the notion of memory cost for quantum strategies, providing conditions for its value and demonstrating its global nature, with specific bounds for covariant protocols.

Findings

Any covariant protocol for cloning a unitary transformation requires at most one ancillary qubit.

Memory cost must be determined globally, not by local optimization at each step.

Provided conditions to evaluate the memory cost for arbitrary strategies.

Abstract

In this paper we consider the problem of minimizing the ancillary systems required to realize an arbitrary strategy of a quantum protocol, with the assistance of classical memory. For this purpose we introduce the notion of memory cost of a strategy, which measures the resources required in terms of ancillary dimension. We provide a condition for the cost to be equal to a given value, and we use this result to evaluate the cost in some special cases. As an example we show that any covariant protocol for the cloning of a unitary transformation requires at most one ancillary qubit. We also prove that the memory cost has to be determined globally, and cannot be calculated by optimizing the resources independently at each step of the strategy.