ECCC-Report TR12-004https://eccc.weizmann.ac.il/report/2012/004Comments and Revisions published for TR12-004en-usFri, 12 Oct 2012 06:10:16 +0200
Revision 3
| Tensor Rank and Strong Quantum Nondeterminism in Multiparty Communication |
Marcos Villagra,
Masaki Nakanishi,
Shigeru Yamashita,
Yasuhiko Nakashima
https://eccc.weizmann.ac.il/report/2012/004#revision3In this paper we study quantum nondeterminism in multiparty communication. There are three (possibly) different types of nondeterminism in quantum computation: i) strong, ii) weak with classical proofs, and iii) weak with quantum proofs. Here we focus on the first one. A strong quantum nondeterministic protocol accepts a correct input with positive probability, and rejects an incorrect input with probability 1. In this work we relate strong quantum nondeterministic multiparty communication complexity to the rank of the communication tensor in the Number-On-Forehead and Number-In-Hand models. In particular, by extending the definition proposed by de Wolf to {\it nondeterministic tensor-rank} ($nrank$), we show that for any boolean function $f$ when there is no prior shared entanglement between the players, 1) in the Number-On-Forehead model, the cost is upper-bounded by the logarithm of $nrank(f)$; 2) in the Number-In-Hand model, the cost is lower-bounded by the logarithm of $nrank(f)$. Furthermore, we show that when the number of players is $o(\log\log n)$ we have that $NQP\nsubseteq BQP$ for Number-On-Forehead communication.Fri, 12 Oct 2012 06:10:16 +0200https://eccc.weizmann.ac.il/report/2012/004#revision3
Revision 2
| Tensor Rank and Strong Quantum Nondeterminism in Multiparty Communication |
Marcos Villagra,
Masaki Nakanishi,
Shigeru Yamashita,
Yasuhiko Nakashima
https://eccc.weizmann.ac.il/report/2012/004#revision2In this paper we study quantum nondeterminism in multiparty communication. There are three (possibly) different types of nondeterminism in quantum computation: i) strong, ii) weak with classical proofs, and iii) weak with quantum proofs. Here we focus on the first one. A strong quantum nondeterministic protocol accepts a correct input with positive probability, and rejects an incorrect input with probability 1. In this work we relate strong quantum nondeterministic multiparty communication complexity to the rank of the communication tensor in the Number-On-Forehead and Number-In-Hand models. In particular, by extending the definition proposed by de Wolf to {\it nondeterministic tensor-rank} ($nrank$), we show that for any boolean function $f$, 1) in the Number-On-Forehead model, the cost is upper-bounded by the logarithm of $nrank(f)$; 2) in the Number-In-Hand model, the cost is lower-bounded by the logarithm of $nrank(f)$. Furthermore, we show that when the number of players is $o(\log\log n^{1/4})$ we have that $NQP\nsubseteq BQP$ for Number-On-Forehead communication.Wed, 13 Jun 2012 10:28:45 +0300https://eccc.weizmann.ac.il/report/2012/004#revision2
Revision 1
| Tensor Rank and Strong Quantum Nondeterminism in Multiparty Communication |
Marcos Villagra,
Masaki Nakanishi,
Shigeru Yamashita,
Yasuhiko Nakashima
https://eccc.weizmann.ac.il/report/2012/004#revision1In this paper we study quantum nondeterminism in multiparty communication. There are three (possibly) different types of nondeterminism in quantum computation: i) strong, ii) weak with classical proofs, and iii) weak with quantum proofs. Here we focus on the first one. A strong quantum nondeterministic protocol accepts a correct input with positive probability, and rejects an incorrect input with probability 1. In this work we relate strong quantum nondeterministic multiparty communication complexity to the rank of the communication tensor in the Number-On-Forehead and Number-In-Hand models. In particular, by extending the definition proposed by de Wolf to {\it nondeterministic tensor-rank} ($nrank$), we show that for any boolean function $f$ with communication tensor $T_f$,
\begin{enumerate}
\item in the Number-On-Forehead model, the cost is upper-bounded by the logarithm of $nrank(T_f)$;
\item in the Number-In-Hand model, the cost is lower-bounded by the logarithm of $nrank(T_f)$.
\end{enumerate}
This naturally generalizes previous results in the field and relates for the first time the concept of (high-order) tensor rank to quantum communication. Furthermore, we show that strong quantum nondeterminism can be exponentially stronger than classical multiparty nondeterministic communication. We do so by applying our results to the matrix multiplication problem.Wed, 29 Feb 2012 05:57:57 +0200https://eccc.weizmann.ac.il/report/2012/004#revision1
Paper TR12-004
| Tensor Rank and Strong Quantum Nondeterminism in Multiparty Communication |
Marcos Villagra,
Masaki Nakanishi,
Shigeru Yamashita,
Yasuhiko Nakashima
https://eccc.weizmann.ac.il/report/2012/004In this paper we study quantum nondeterminism in multiparty communication. There are three (possibly) different types of nondeterminism in quantum computation: i) strong, ii) weak with classical proofs, and iii) weak with quantum proofs. Here we focus on the first one. A strong quantum nondeterministic protocol accepts a correct input with positive probability, and rejects an incorrect input with probability 1. In this work we relate strong quantum nondeterministic multiparty communication complexity to the rank of the communication tensor in the Number-On-Forehead and Number-In-Hand models. In particular, by extending the definition proposed by de Wolf to {\it nondeterministic tensor-rank} ($nrank$), we show that for any boolean function $f$ with communication tensor $T_f$,
\begin{enumerate}
\item in the Number-On-Forehead model, the cost is upper-bounded by the logarithm of $nrank(T_f)$;
\item in the Number-In-Hand model, the cost is lower-bounded by the logarithm of $nrank(T_f)$.
\end{enumerate}
This naturally generalizes previous results in the field and relates for the first time the concept of (high-order) tensor rank to quantum communication. Furthermore, we show that strong quantum nondeterminism can be exponentially stronger than classical multiparty nondeterministic communication. We do so by applying our results to the matrix multiplication problem.Fri, 13 Jan 2012 12:21:33 +0200https://eccc.weizmann.ac.il/report/2012/004