The Quantum Communication Complexity of Sampling

Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function $f: X \times Y \rightarrow \{0,1\}$ and a probability distribution ${\cal D}$ over $X \times Y$, we define the sampling complexity of $(f, {\cal D})$ as the minimum number of bits that Alice and Bob must communicate for Alice to pick $x \in X$ and Bob to pick $y \in Y$ as well as a value $z$ such that the resulting distribution of $(x,y,z)$ is close to the distribution $({\cal D}, f({\cal D}))$. In this paper we initiate the study of sampling complexity, in both the classical and quantum models. We give several variants of a definition. We completely characterize some of these variants and give upper and lower bounds on others. In particular, this allows us to establish an exponential gap between quantum and classical sampling complexity for the set-disjointness function.

[1]  A. Razborov Quantum communication complexity of symmetric predicates , 2002, quant-ph/0204025.

[2]  Lov K. Grover A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[3]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[4]  Noam Nisan,et al.  On rank vs. communication complexity , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[5]  Andrew Chi-Chih Yao,et al.  Quantum Circuit Complexity , 1993, FOCS.

[6]  Andrew Chi-Chih Yao,et al.  Some complexity questions related to distributive computing(Preliminary Report) , 1979, STOC.

[7]  Charles H. Bennett,et al.  Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. , 1992, Physical review letters.

[8]  Rae Baxter,et al.  Acknowledgments.-The authors would like to , 1982 .

[9]  Andris Ambainis,et al.  Quantum search of spatial regions , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[10]  Ran Raz,et al.  Exponential separation of quantum and classical communication complexity , 1999, STOC '99.

[11]  Ashwin Nayak,et al.  Optimal lower bounds for quantum automata and random access codes , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[12]  Ilan Newman,et al.  Private vs. Common Random Bits in Communication Complexity , 1991, Inf. Process. Lett..

[13]  Noam Nisan,et al.  Quantum circuits with mixed states , 1998, STOC '98.

[14]  Peter Frankl,et al.  Complexity classes in communication complexity theory , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[15]  Avi Wigderson,et al.  Quantum vs. classical communication and computation , 1998, STOC '98.

[16]  Andris Ambainis,et al.  Dense quantum coding and quantum finite automata , 2002, JACM.

[17]  A. Holevo Bounds for the quantity of information transmitted by a quantum communication channel , 1973 .

[18]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[19]  Ran Raz,et al.  On the "log rank"-conjecture in communication complexity , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[20]  D. Deutsch,et al.  Rapid solution of problems by quantum computation , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.