A lower bound for the integer element distinctness problem

AbstractA lower bound of Ω(n log n) is proved for the integer element distinctness problem—Given (x1, …, xn) ϵ Zn, are the xi's distinct—on the bounded-order algebraic decision tree model

The quantum complexity of approximating the frequency moments

The $k$'th frequency moment of a sequence of integers is defined as $F_k = \sum_j n_j^k$, where $n_j$ is the number of times that $j$ occurs in the sequence. Here we study the quantum complexity of approximately computing the frequency moments in two settings. In the query complexity setting, we wish to minimise the number of queries to the input used to approximate $F_k$ up to relative error $\epsilon$. We give quantum algorithms which outperform the best possible classical algorithms up to quadratically. In the multiple-pass streaming setting, we see the elements of the input one at a time, and seek to minimise the amount of storage space, or passes over the data, used to approximate $F_k$. We describe quantum algorithms for $F_0$, $F_2$ and $F_\infty$ in this model which substantially outperform the best possible classical algorithms in certain parameter regimes.Comment: 22 pages; v3: essentially published versio

Quantum complexities of ordered searching, sorting, and element distinctness

We consider the quantum complexities of the following three problems: searching an ordered list, sorting an un-ordered list, and deciding whether the numbers in a list are all distinct. Letting N be the number of elements in the input list, we prove a lower bound of \frac{1}{\pi}(\ln(N)-1) accesses to the list elements for ordered searching, a lower bound of \Omega(N\log{N}) binary comparisons for sorting, and a lower bound of \Omega(\sqrt{N}\log{N}) binary comparisons for element distinctness. The previously best known lower bounds are {1/12}\log_2(N) - O(1) due to Ambainis, \Omega(N), and \Omega(\sqrt{N}), respectively. Our proofs are based on a weighted all-pairs inner product argument. In addition to our lower bound results, we give a quantum algorithm for ordered searching using roughly 0.631 \log_2(N) oracle accesses. Our algorithm uses a quantum routine for traversing through a binary search tree faster than classically, and it is of a nature very different from a faster algorithm due to Farhi, Goldstone, Gutmann, and Sipser.Comment: This new version contains new results. To appear at ICALP '01. Some of the results have previously been presented at QIP '01. This paper subsumes the papers quant-ph/0009091 and quant-ph/000903

Element Distinctness, Frequency Moments, and Sliding Windows

We derive new time-space tradeoff lower bounds and algorithms for exactly computing statistics of input data, including frequency moments, element distinctness, and order statistics, that are simple to calculate for sorted data. We develop a randomized algorithm for the element distinctness problem whose time T and space S satisfy T in O (n^{3/2}/S^{1/2}), smaller than previous lower bounds for comparison-based algorithms, showing that element distinctness is strictly easier than sorting for randomized branching programs. This algorithm is based on a new time and space efficient algorithm for finding all collisions of a function f from a finite set to itself that are reachable by iterating f from a given set of starting points. We further show that our element distinctness algorithm can be extended at only a polylogarithmic factor cost to solve the element distinctness problem over sliding windows, where the task is to take an input of length 2n-1 and produce an output for each window of length n, giving n outputs in total. In contrast, we show a time-space tradeoff lower bound of T in Omega(n^2/S) for randomized branching programs to compute the number of distinct elements over sliding windows. The same lower bound holds for computing the low-order bit of F_0 and computing any frequency moment F_k, k neq 1. This shows that those frequency moments and the decision problem F_0 mod 2 are strictly harder than element distinctness. We complement this lower bound with a T in O(n^2/S) comparison-based deterministic RAM algorithm for exactly computing F_k over sliding windows, nearly matching both our lower bound for the sliding-window version and the comparison-based lower bounds for the single-window version. We further exhibit a quantum algorithm for F_0 over sliding windows with T in O(n^{3/2}/S^{1/2}). Finally, we consider the computations of order statistics over sliding windows.Comment: arXiv admin note: substantial text overlap with arXiv:1212.437

Claw Finding Algorithms Using Quantum Walk

The claw finding problem has been studied in terms of query complexity as one of the problems closely connected to cryptography. For given two functions, f and g, as an oracle which have domains of size N and M (N<=M), respectively, and the same range, the goal of the problem is to find x and y such that f(x)=g(y). This paper describes an optimal algorithm using quantum walk that solves this problem. Our algorithm can be generalized to find a claw of k functions for any constant integer k>1, where the domains of the functions may have different size.Comment: 12 pages. Introduction revised. A reference added. Weak lower bound delete

Quantum query complexity of entropy estimation

Estimation of Shannon and R\'enyi entropies of unknown discrete distributions is a fundamental problem in statistical property testing and an active research topic in both theoretical computer science and information theory. Tight bounds on the number of samples to estimate these entropies have been established in the classical setting, while little is known about their quantum counterparts. In this paper, we give the first quantum algorithms for estimating $\alpha$-R\'enyi entropies (Shannon entropy being 1-Renyi entropy). In particular, we demonstrate a quadratic quantum speedup for Shannon entropy estimation and a generic quantum speedup for $\alpha$-R\'enyi entropy estimation for all $\alpha\geq 0$, including a tight bound for the collision-entropy (2-R\'enyi entropy). We also provide quantum upper bounds for extreme cases such as the Hartley entropy (i.e., the logarithm of the support size of a distribution, corresponding to $\alpha=0$) and the min-entropy case (i.e., $\alpha=+\infty$), as well as the Kullback-Leibler divergence between two distributions. Moreover, we complement our results with quantum lower bounds on $\alpha$-R\'enyi entropy estimation for all $\alpha\geq 0$.Comment: 43 pages, 1 figur