Quantum Identification of Boolean Oracles

Arxiv preprint arXiv: …, 2010

We present a systematic construction of quantum circuits implementing Grover's database search algorithm for arbitrary number of targets. We introduce a new operator which flips the sign of the targets and evaluate its circuit complexity. We find the condition under which the circuit complexity of the database search algorithm based on this operator is less than that of the conventional one.

computational complexity, 2013

We show that quantum query complexity satisfies a strong direct product theorem. This means that computing k copies of a function with less than k times the number of quantum queries needed to compute one copy of the function implies that the overall success probability will be exponentially small in k. We do this by showing that the multiplicative adversary method, which inherently satisfies a strong direct product theorem, is always at least as large as the additive adversary method, which is known to characterize quantum query complexity.

18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings., 2003

We reformulate quantum query complexity in terms of inequalities and equations for a set of positive semidefinite matrices. Using the new formulation we: 1. show that the workspace of a quantum computer can be limited to at most n + k qubits (where n and k are the number of input and output bits respectively) without reducing the computational power of the model. 2. give an algorithm that on input the truth table of a partial boolean function and an integer t runs in time polynomial in the size of the truth table and estimates, to any desired accuracy, the minimum probability of error that can be attained by a quantum query algorithm attempts to evaluate f in t queries. 3. use semidefinite programming duality to formulate a dual SDPP (f, t, ) that is feasible if and only if f can not be evaluated within error by a tstep quantum query algorithm Using this SDP we derive a general lower bound for quantum query complexity that encompasses a lower bound method of Ambainis and its generalizations. 4. Give an interpretation of a generalized form of branching in quantum computation.

2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, 2011

The general adversary bound is a semi-definite program that lower-bounds the number of input queries needed by a quantum algorithm to evaluate a function. It is known to be tight up to constant factors for functions (total or partial) with boolean output and binary input alphabet. We show that the general adversary bound is tight for any function whatsoever, with potentially non-boolean input or output alphabets. We also show that quantum query complexity exhibits a remarkable composition property: Q f (g(x 1 ), . . . , g(x n )) = O Q(f )Q(g) for any compatible functions f, g. This was previously known only in the boolean case.

2014

In this thesis we study the power of quantum query algorithms and learning graphs; the latter essentially being very specialized quantum query algorithms themselves. We almost exclusively focus on proving lower bounds for these computational models. First, we study lower bounds on learning graph complexity. We consider two types of learning graphs: adaptive and, more restricted, non-adaptive learning graphs. We express both adaptive and non-adaptive learning graph complexities of Boolean-valued functions (i.e., decision problems) as semidefinite minimization problems, and derive their dual problems. For various functions, we construct feasible solutions to these dual problems, thereby obtaining lower bounds on the learning graph complexity of the functions. Most notably, we prove an almost optimal Ω(n9/7/ √ log n) lower bound on the non-adaptive learning graph complexity of the Triangle problem. We also prove an Ω(n1−2 /(2−1)) lower bound on the adaptive learning graph complexity of...

In a fundamental paper (Phys. Rev. Lett. 78, 325 (1997)) Grover showed how a quantum computer can …nd a single marked object in a database of size N by using only O( p N) queries of the oracle that identi…es the object. His result was generalized to the case of …nding one object in a subset of marked elements. We consider the following computational problem: A subset of marked elements is given whose number of elements is either M or K, M < K, our task is to determine which is the case. We show how to solve this problem with a high probability of success using only iterations of Grover's basic step (and no other algorithm). Let m be the required number of iterations; we prove that under certain restrictions on the sizes of M and K the estimation m � 2 p N p K p M obtains. This bound sharpens previous results and is known to be optimal up to a constant factor. Our method involves simultaneous Diophantine approximations, so that Grover's algorithm is conceptualized as an or...

viXra, 2016

We develop three new quantum algorithms for searching the desired target state in the unstructured database of size N. The first algorithm requires Log N iterative steps. It constructs two quantum bags of equal size in terms of two quantum states, out of which exactly one quantum state will have nonzero overlap with the target state. This determination of overlap is done by taking the inner product, in Log N time [2], of the implicitly known target state with any one of these two quantum states. The second algorithm requires just one single step which uses a new suitable operator and the choice of this operator is problem dependent, i.e. it depends upon the number of qubits required to be used to represent an element in the index set. The third algorithm again requires only a single step and this algorithm makes use of a fixed (same) operator. It is known that algorithm for unstructured database search can be easily adaptable for solving NP-Complete problems. However, the computatio...

Fortschritte der Physik, 1998

We provide a tight analysis of Grover's recent algorithm for quantum database searching. We give a simple closed-form formula for the probability of success after any given number of iterations of the algorithm. This allows us to determine the number of iterations necessary to achieve almost certainty of finding the answer. Furthermore, we analyse the behaviour of the algorithm when the element to be found appears more than once in the table and we provide a new algorithm to find such an element even when the number of solutions is not known ahead of time. Using techniques from Shor's quantum factoring algorithm in addition to Grover's approach, we introduce a new technique for approximate quantum counting, which allows to estimate the number of solutions. Finally we provide a lower bound on the efficiency of any possible quantum database searching algorithm and we show that Grover's algorithm nearly comes within a factor 2 of being optimal in terms of the number of probes required in the table. * Département IRO, C.P. 6128, succursale centre-ville, Montréal, Canada H3C 3J7. {boyer,brassard,tappa}@iro.umontreal.ca †

2014

We prove improved quantum query complexity bounds for some graph problem. Our results are based on a new quantum algorithm in [1] that could be used to improve query complexity upper bounds. We prove a lower bound of Ω ( kn ) queries to an adjacency matrix for the k-source shortest paths problem for unweighted graphs, which matches the upper bound proved in [1]. We also prove an upper bound of O(n) queries for finding the minimum vertex cover in a bipartite graph G if we are given the maximum matching in G. In [1], Lin and Lin proved that the latter could be found in O(n) queries, which gives us an O(n) upper bound for the minimum vertex cover problem for bipartite graphs. We also discuss the implications of their results on the query complexity of other related graph problems.

Lecture Notes in Computer Science, 2008

We consider Grover's unstructured search problem in the setting where each oracle call has some small probability of failing. We show that no quantum speed-up is possible in this case.