Showing papers by "Aleksandrs Belovs published in 2014"

Applications of Adversary Method in Quantum Query Algorithms

Abstract: In the thesis, we use a recently developed tight characterisation of quantum query complexity, the adversary bound, to develop new quantum algorithms and lower bounds. Our results are as follows: * We develop a new technique for the construction of quantum algorithms: learning graphs. * We use learning graphs to improve quantum query complexity of the triangle detection and the $k$-distinctness problems. * We prove tight lower bounds for the $k$-sum and the triangle sum problems. * We construct quantum algorithms for some subgraph-finding problems that are optimal in terms of query, time and space complexities. * We develop a generalisation of quantum walks that connects electrical properties of a graph and its quantum hitting time. We use it to construct a time-efficient quantum algorithm for 3-distinctness.

On the Power of Non-adaptive Learning Graphs

Abstract: We introduce a notion of the quantum query complexity of a certificate structure. This is a formalization of a well-known observation that many quantum query algorithms only require the knowledge of the position of possible certificates in the input string, not the precise values therein. Next, we derive a dual formulation of the complexity of a non-adaptive learning graph and use it to show that non-adaptive learning graphs are tight for all certificate structures. By this, we mean that there exists a function possessing the certificate structure such that a learning graph gives an optimal quantum query algorithm for it. For a special case of certificate structures generated by certificates of bounded size, we construct a relatively general class of functions having this property. The construction is based on orthogonal arrays and generalizes the quantum query lower bound for the k-sum problem derived recently by Belovs and ?palek (Proceeding of 4th ACM ITCS, 323---328, 2013). Finally, we use these results to show that the learning graph for the triangle problem by Lee et al. (Proceeding of 24th ACM-SIAM SODA, 1486---1502, 2013) is almost optimal in the above settings. This also gives a quantum query lower bound for the triangle sum problem.

Quantum lower bound for inverting a permutation with advice.

Abstract: Given a random permutation f : [N] → [N] as a black box and y e [N], we want to output x = f-1(y). Supplementary to our input, we are given classical advice in the form of a pre-computed data structure; this advice can depend on the permutation but not on the input y. Classically, there is a data structure of size O(S) and an algorithm that with the help of the data structure, given f(x), can invert f in time O(T), for every choice of parameters S, T, such that S ċ T ≥ N. We prove a quantum lower bound of T2 ċ S = Ω(eN) for quantum algorithms that invert a random permutation f on an e fraction of inputs, where T is the number of queries to f and S is the amount of advice. This answers an open question of De et al. We also give a Ω(√N/m) quantum lower bound for the simpler but related Yao's box problem, which is the problem of recovering a bit xj, given the ability to query an N-bit string x at any index except the j-th, and also given m bits of classical advice that depend on x but not on j.

Quantum Algorithms for Learning Symmetric Juntas via Adversary Bound

Abstract: In this paper, we study the following variant of the junta learning problem We are given oracle access to a Boolean function f on n variables that only depends on k variables, and, when restricted to them, equals some predefined function h The task is to identify the variables the function depends on This is a generalisation of the Bernstein-Vazirani problem (when h is the XOR function) and the combinatorial group testing problem (when h is the OR function) We analyse the general case using the adversary bound, and give an alternative formulation for the quantum query complexity of this problem We construct optimal quantum query algorithms for the cases when h is the OR function (complexity is square root of k) or the exact-half function (complexity is the fourth power root of k) The first algorithm resolves an open problem from For the case when h is the majority function, we prove an upper bound of the fourth power root of k We obtain a quartic improvement when compared to the randomised complexity (if h is the exact-half or the majority function), and a quadratic one when compared to the non-adaptive quantum complexity (for all functions considered in the paper)

Quantum lower bound for inverting a permutation with advice

Abstract: Given a random permutation $f: [N] \to [N]$ as a black box and $y \in [N]$, we want to output $x = f^{-1}(y)$. Supplementary to our input, we are given classical advice in the form of a pre-computed data structure; this advice can depend on the permutation but \emph{not} on the input $y$. Classically, there is a data structure of size $\tilde{O}(S)$ and an algorithm that with the help of the data structure, given $f(x)$, can invert $f$ in time $\tilde{O}(T)$, for every choice of parameters $S$, $T$, such that $S\cdot T \ge N$. We prove a quantum lower bound of $T^2\cdot S \ge \tilde{\Omega}(\epsilon N)$ for quantum algorithms that invert a random permutation $f$ on an $\epsilon$ fraction of inputs, where $T$ is the number of queries to $f$ and $S$ is the amount of advice. This answers an open question of De et al. We also give a $\Omega(\sqrt{N/m})$ quantum lower bound for the simpler but related Yao's box problem, which is the problem of recovering a bit $x_j$, given the ability to query an $N$-bit string $x$ at any index except the $j$-th, and also given $m$ bits of advice that depend on $x$ but not on $j$.