A complexity theory for non-local quantum computation

TL;DR

This paper develops a complexity-theoretic framework for non-local quantum computation, establishing equivalences between key tasks and deriving bounds, thereby advancing understanding of resource requirements in quantum information processing.

Contribution

It introduces a resource-based reduction approach to compare NLQC tasks, proving equivalence of two main tasks and extending properties to $f$-measure, simplifying prior proofs.

Findings

$f$-measure and $f$-route are equivalent under $O(1)$ overhead reductions

Sub-exponential upper bounds on $f$-measure for all functions

Efficient protocols for functions in $ ext{Mod}_k ext{L}$

Abstract

Non-local quantum computation (NLQC) replaces a local interaction between two systems with a single round of communication and shared entanglement. Despite many partial results, it is known that a characterization of entanglement cost in at least certain NLQC tasks would imply significant breakthroughs in complexity theory. Here, we avoid these obstructions and take an indirect approach to understanding resource requirements in NLQC, which mimics the approach used by complexity theorists: we study the relative hardness of different NLQC tasks by identifying resource efficient reductions between them. Most significantly, we prove that $f$-measure and $f$-route, the two best studied NLQC tasks, are in fact equivalent under $O(1)$ overhead reductions. This result simplifies many existing proofs in the literature and extends several new properties to $f$-measure. For instance, we obtain sub-exponential upper bounds on $f$-measure for all functions, and efficient protocols for functions in the complexity class $\mathsf{Mod}_k\mathsf{L}$. Beyond this, we study a number of other examples of NLQC tasks and their relationships.