dorsal/arxiv
View SchemaQuantum and Classical Communication-Space Tradeoffs from Rectangle Bounds
| Authors | Hartmut Klauck |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0412088 |
| URL | https://arxiv.org/abs/quant-ph/0412088 |
Abstract
We derive lower bounds for tradeoffs between the communication C and space S for communicating circuits. The first such bound applies to quantum circuits. If for any function f with image Z the multicolor discrepancy of the communication matrix of f is 1/2^d, then any bounded error quantum protocol with space S, in which Alice receives some l inputs, Bob r inputs, and they compute f(x_i,y_j) for the lr pairs of inputs (x_i,y_j) needs communication C=\Omega(lrd \log |Z|/S). In particular, n\times n-matrix multiplication over a finite field F requires C=\Theta(n^3\log^2 |F|/S). We then turn to randomized bounded error protocols, and derive the bound C=\Omega(n^3/S^2) for Boolean matrix multiplication, utilizing a new direct product result for the one-sided rectangle lower bound on randomized communication complexity. This implies a separation between quantum and randomized protocols.
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"abstract": "We derive lower bounds for tradeoffs between the communication C and space S\nfor communicating circuits. The first such bound applies to quantum circuits.\nIf for any function f with image Z the multicolor discrepancy of the\ncommunication matrix of f is 1/2^d, then any bounded error quantum protocol\nwith space S, in which Alice receives some l inputs, Bob r inputs, and they\ncompute f(x_i,y_j) for the lr pairs of inputs (x_i,y_j) needs communication\nC=\\Omega(lrd \\log |Z|/S). In particular, n\\times n-matrix multiplication over a\nfinite field F requires C=\\Theta(n^3\\log^2 |F|/S). We then turn to randomized\nbounded error protocols, and derive the bound C=\\Omega(n^3/S^2) for Boolean\nmatrix multiplication, utilizing a new direct product result for the one-sided\nrectangle lower bound on randomized communication complexity. This implies a\nseparation between quantum and randomized protocols.",
"arxiv_id": "quant-ph/0412088",
"authors": [
"Hartmut Klauck"
],
"categories": [
"quant-ph",
"cs.CC"
],
"title": "Quantum and Classical Communication-Space Tradeoffs from Rectangle Bounds",
"url": "https://arxiv.org/abs/quant-ph/0412088"
},
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