We study the computation complexity of Boolean functions in the quantum black box model.In this model our task is to compute a function f : {0, 1} → {0, 1} on an input x ∈ {0, 1} n that can be accessed by querying the black box.Quantum algorithms are inherently probabilistic; we are interested in the lowest possible probability that the algorithm outputs incorrect answer (the error probability) for a fixed number of queries.We show that the lowest possible error probability for ANDn and EQUALITY n+1 is 1 2 -n n 2 +1 .
We study the computation complexity of Boolean functions in the quantum black box model.In this model our task is to compute a function f : {0, 1} → {0, 1} on an input x ∈ {0, 1} n that can be accessed by querying the black box.Quantum algorithms are inherently probabilistic; we are interested in the lowest possible probability that the algorithm outputs incorrect answer (the error probability) for a fixed number of queries.We show that the lowest possible error probability for ANDn and EQUALITY n+1 is 1 2 -n n 2 +1 .
We consider the quantum query model for computing Boolean functions. The definition of the function is known, but a black box contains the input X = (x1, x2, ..., xn). Black box can be accessed by querying xivalues. The goal is to develop an algorithm, which would compute the function value for arbitrary input using as few queries to the black box as possible. We present two different quantum query algorithms for computing the basic Boolean function - logical AND of two bits. Both algorithms use only one query to determine the function value. Correct answer probability for the first algorithm is 80%, but for the second algorithm it is 90%. To compute this function with the same probability in the classical model, two queries are necessary. We analyze algorithms implementation differences and propose a way to extend the first algorithm to compute AND of two functions.
Quantum algorithms can be analyzed in a query model to compute Boolean functions. Function input is provided in a black box, and the aim is to compute the function value using as few queries to the black box as possible. The complexity of the algorithm is measured by the number of queries on the worst-case input. In this paper we consider computing AND Boolean function. First, we present a quantum algorithm for AND of two bits. Our algorithm uses one quantum query and correct result is obtained with a probability p=4/5, that improves previous results. The main result is generalization of our approach to design efficient quantum algorithms for computing composite function AND(f1,f2) where fi is a Boolean function. Finally, we demonstrate another kind of an algorithm for AND of two variables, that has a correct answer probability p=9/10, but cannot be extended to compute AND(f1,f2).
The query model (or black-box model) has attracted much attention from the communities of both classical and quantum computing. Usually, quantum advantages are revealed by presenting a quantum algorithm that has a better query complexity than its classical counterpart. For example, the well-known quantum algorithms including Deutsch-Jozsa algorithm, Simon algorithm and Grover algorithm all show a considerable advantage of quantum computing from the viewpoint of query complexity. Recently we have considered in (Phys. Rev. A. {\bf 101}, 02232 (2020)) the problem: what functions can be computed by an exact one-query quantum algorithm? This problem has been addressed for total Boolean functions but still open for partial Boolean functions. Thus, in this paper we continue to characterize the computational power of exact one-query quantum algorithms for partial Boolean functions by giving several necessary and sufficient conditions. By these conditions, we construct some new functions that can be computed exactly by one-query quantum algorithms but have essential difference from the already known ones. Note that before our work, the known functions that can be computed by exact one-query quantum algorithms are all symmetric functions, whereas the ones constructed in this papers are generally asymmetric.
A quantum algorithm is exact if, on any input data, it outputs the correct answer with certainty (probability 1). A key question is: how big is the advantage of exact quantum algorithms over their classical counterparts: deterministic algorithms. For total Boolean functions in the query model, the biggest known gap was just a factor of 2: PARITY of N inputs bits requires N queries classically but can be computed with N/2 queries by an exact quantum algorithm. We present the first example of a Boolean function f(x_1, ..., x_N) for which exact quantum algorithms have superlinear advantage over the deterministic algorithms. Any deterministic algorithm that computes our function must use N queries but an exact quantum algorithm can compute it with O(N^{0.8675...}) queries.
Given a Boolean function f:{0,1}n→{0,1}, the goal in the usual query model is to compute f on an unknown input x∈{0,1}n while minimizing the number of queries to x. One can also consider a "distinguishing" problem denoted by fsab: given an input x∈f−1(0) and an input y∈f−1(1), either all differing locations are replaced by a ∗, or all differing locations are replaced by †, and an algorithm's goal is to identify which of these is the case while minimizing the number of queries. Ben-David and Kothari [ToC'18] introduced the notion of randomized sabotage complexity of a Boolean function to be the zero-error randomized query complexity of fsab. A natural follow-up question is to understand Q(fsab), the quantum query complexity of fsab. In this paper, we initiate a systematic study of this. The following are our main results: ∙ If we have additional query access to x and y, then Q(fsab)=O(min{Q(f),n}). ∙ If an algorithm is also required to output a differing index of a 0-input and a 1-input, then Q(fsab)=O(min{Q(f)1.5,n}). ∙Q(fsab)=Ω(fbs(f)), where fbs(f) denotes the fractional block sensitivity of f. By known results, along with the results in the previous bullets, this implies that Q(fsab) is polynomially related to Q(f). ∙ The bound above is easily seen to be tight for standard functions such as And, Or, Majority and Parity. We show that when f is the Indexing function, Q(fsab)=Θ(fbs(f)), ruling out the possibility that Q(fsab)=Θ(fbs(f)) for all f.
Quantum algorithms can be analyzed in a query model to compute Boolean functions. Function input is provided in a black box, and the aim is to compute the function value using as few queries to the black box as possible. A repetition code is an error detection scheme that repeats each bit of the original message r times. After a message with redundant bits is transmitted via a communication channel, it must be verified. If the received message consists of r-size blocks of equal bits, the conclusion is that there were no errors. The verification procedure can be interpreted as an application of a query algorithm, where input is a message to be checked. Classically, for N-bit message, values of all N variables must be queried. We demonstrate an exact quantum algorithm that uses only N/2 queries.