Graph entropy and quantum sorting problems

Let P = (X, < <inf>P</inf>) be a partial order on a set of n elements X = x<inf>1</inf>, x<inf>2</inf>,..., x<inf>n</inf>. Define the <i>quantum sorting problem</i> QSORT<inf>P</inf> as: given n distinct numbers x<inf>1</inf>, x<inf>2</inf>,..., x<inf>n</inf> consistent with P, sort them by a quantum decision tree using comparisons of the form "x<inf>i</inf>: x<inf>j</inf>". Let Q<inf>ε</inf>(P) be the minimum number of queries used by any quantum decision tree for solving QSORT<inf>P</inf> with error less than ε (where 0 < ε < 1/10 is fixed). It was proved by Hoyer, Neerbek and Shi (<i>Algorithmica</i> <b>34</b> (2002), 429--448) that, when P<inf>0</inf> is the empty partial order, Q<inf>ε</inf>(P<inf>0</inf>) ≥ Ω (n log n), i. e., the classical information lower bound holds for quantum decision trees when the input permutations are unrestricted.In this paper we show that the classical information lower bound holds, up to an additive linear term, for quantum decision trees for any partial order P. Precisely, we prove Q<inf>ε</inf>(P) ≥ c log<inf>2</inf> e(P)-c'n where c,c' > 0 are constants and e(P) is the number of linear orderings consistent with P. Our proof builds on an interesting connection between sorting and Korner's graph entropy that was first noted and developed by Kahn and Kim (<i>JCSS</i> <b>51</b>(1995), 390--399).