Strong direct product theorems for quantum communication and query complexity

TL;DR

This paper establishes strong direct product theorems for quantum communication and query complexity, showing that solving multiple instances simultaneously requires proportionally more resources, under certain well-known lower bound techniques.

Contribution

It proves SDPTs for quantum communication and query complexity when using the generalized discrepancy and polynomial methods, respectively, extending the understanding of resource scaling in quantum computation.

Findings

Quantum communication complexity obeys SDPTs with generalized discrepancy bounds.

Quantum query complexity obeys SDPTs with polynomial method bounds.

Includes XOR lemmas and threshold direct product theorems for both models.

Abstract

A strong direct product theorem (SDPT) states that solving n instances of a problem requires Omega(n) times the resources for a single instance, even to achieve success probability exp(-Omega(n)). We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound for a single instance is proved by the generalized discrepancy method, the strongest technique in that model. We prove that quantum query complexity obeys an SDPT whenever the query lower bound for a single instance is proved by the polynomial method, one of the two main techniques in that model. In both models, we prove the corresponding XOR lemmas and threshold direct product theorems.