Strong Direct Product Theorems for Quantum Communication and Query Complexity

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 $2^{-\epsilon n}$ for a small enough constant $\epsilon>0.$ We prove that quantum communication complexity obey...

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Published in:SIAM journal on computing Vol. 41; no. 5; pp. 1122 - 1165
Main Author: Sherstov, Alexander A.
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2012
Subjects:
Communication
Complexity
Computation
Computer science
Lower bounds
Probability
Query processing
Success
Theorems
ISSN:0097-5397, 1095-7111
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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 $2^{-\epsilon n}$ for a small enough constant $\epsilon>0.$ 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. [PUBLICATION ABSTRACT]
AbstractList 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 $2-\epsilon n}$ for a small enough constant $\epsilon>0.$ 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 andthreshold direct product theorems.
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 $2^{-\epsilon n}$ for a small enough constant $\epsilon>0.$ 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. [PUBLICATION ABSTRACT]
Author Sherstov, Alexander A.
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SubjectTerms Communication
Complexity
Computation
Computer science
Lower bounds
Probability
Query processing
Success
Theorems
Title Strong Direct Product Theorems for Quantum Communication and Query Complexity
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