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ACM Symposium on Theory of Computing (STOC)

Graph entropy and quantum sorting problems

Andrew Chi-Chih Yao

Published Jun 13, 2004• 6 citations

Abstract

Let P = (X, < P) be a partial order on a set of n elements X = x1, x2,..., xn. Define the quantum sorting problem QSORTP as: given n distinct numbers x1, x2,..., xn consistent with P, sort them by a quantum decision tree using comparisons of the form "xi: xj". Let Qε(P) be the minimum number of queries used by any quantum decision tree for solving QSORTP with error less than ε (where 0 < ε < 1/10 is fixed). It was proved by Hoyer, Neerbek and Shi (Algorithmica 34 (2002), 429--448) that, when P0 is the empty partial order, Qε(P0) ≥ 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 Qε(P) ≥ c log2 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 (JCSS 51(1995), 390--399).

DOI: 10.1145/1007352.1007377

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Let P = (X, < P) be a partial order on a set of n elements X = x1, x2,..., xn. Define the quantum sorting problem QSORTP as: given n distinct numbers x1, x2,..., xn consistent with P, sort them by a quantum decision tree using comparisons of the form "xi: xj". Let Qε(P) be the minimum number of queries used by any quantum decision tree for solving QSORTP with error less than ε (where 0 < ε < 1/10 is fixed). It was proved by Hoyer, Neerbek and Shi (Algorithmica 34 (2002), 429--448) that, when P0 is the empty partial order, Qε(P0) ≥ 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 Qε(P) ≥ c log2 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 (JCSS 51(1995), 390--399).

We analyze the quantum query complexity of sorting under partial information. In this problem, we are given a partially ordered set

P

and are asked to identify a linear extension of

P

using pairwise comparisons. For the standard sorting problem, in which

P

is empty, it is known that the quantum query complexity is not asymptotically smaller than the classical information-theoretic lower bound. We prove that this holds for a wide class of partially ordered sets, thereby improving on a result from Yao (STOC'04).

We prove a general lower bound of quantum decision tree complexity in terms of some entropy notion. We regard the computation as a communication process in which the oracle and the computer exchange several rounds of messages, each round consisting of O(log(n)) bits. Let E(f) be the Shannon entropy of the random variable f(X), where X is uniformly random in f's domain. Our main result is that it takes Ω(E(f)) queries to compute any \emph{total} function f. It is interesting to contrast this bound with the Ω(E(f)/log(n)) bound, which is tight for \emph{partial} functions. Our approach is the polynomial method.

We consider the quantum complexities of searching an ordered list and sorting an un-ordered list. For searching an ordered list of N elements, we prove a lower bound of \\frac{1}{\\pi}(\\ln(N)-1) on the number of oracle queries that access the list elements. This improves the previously best lower bound of ({1/12}\\log_2(N) - O(1)) due to Ambainis. For sorting N numbers, we prove a lower bound of \\frac{N}{2\\pi}(\\ln(N)-1) on the number of binary comparisons. The previously best lower bound is \\Omega(N). Our proofs are based on a weighted all-pairs inner product argument, and our results generalize to bounded error quantum algorithms. Both results are proven in the so-called quantum black box model, a quantum analogue of classical decision trees. In addition to our lower bound results, we give an exact quantum algorithm for ordered searching using (\\log_3(N) + O(1)) queries, which is roughly 0.631 \\log_2(N). Although our algorithm is worse than that of Farhi, Goldstone, Gutmann and Sipser, which makes 0.526 \\log_2(N) queries, its philosophy is completely different. Our algorithm is a quantum version of the classical binary search algorithm, and it uses a quantum routine for traversing through a binary search tree faster than classically.

Lin and Lin \cite{LL16} have recently shown how starting with a classical query algorithm (decision tree) for a function, we may find upper bounds on its quantum query complexity. More precisely, they have shown that given a decision tree for a function f:{0,1}n→[m] whose input can be accessed via queries to its bits, and a guessing algorithm that predicts answers to the queries, there is a quantum query algorithm for f which makes at most O(GT) quantum queries where T is the depth of the decision tree and G is the maximum number of mistakes of the guessing algorithm. In this paper we give a simple proof of and generalize this result for functions f:[ℓ]n→[m] with non-binary input as well as output alphabets. Our main tool for this generalization is non-binary span program which has recently been developed for non-binary functions, and the dual adversary bound. As applications of our main result we present several quantum query upper bounds, some of which are new. In particular, we show that topological sorting of vertices of a directed graph G can be done with O(n3/2) quantum queries in the adjacency matrix model. Also, we show that the quantum query complexity of the maximum bipartite matching is upper bounded by O(n3/4m+n) in the adjacency list model.

We consider the problem where /spl pi/ is an unknown permutation on (0, 1,..., 2/sup n/-1), /spl gamma//sub 0//spl isin/(0, 1,..., 2/sup n/-1), and the goal is to determine the minimum r>0 such that /spl pi//sup r/(y/sub 0/)=y/sub 0/. Information about /spl pi/ is available only via queries that yield /spl pi//sup x/(y) from any x/spl isin/(0, 1,..., 2/sup n/-1) and /spl gamma//sub /spl isin//(0, 1,..., 2/sup n/-1) (where m is polynomial in n). The resource under consideration is the number of these queries (hence our model of computation is the decision tree). We show that the number of queries necessary to solve the problem in the classical probabilistic bounded error model is exponential in n. This contrasts sharply with the quantum bounded-error model, where a constant number of queries suffices.