We discuss quantum algorithms based on the Bernstein–Vazirani algorithm for finding which input variables a Boolean function depends on. There are 2 n possible linear Boolean functions of n input variables; given a linear Boolean function, the Bernstein–Vazirani quantum algorithm can deterministically identify which one of these Boolean functions we are given using just one single function query. We show how the same quantum algorithm can also be used to learn which input variables any other type of Boolean function} depends on. The success probability of learning that the function depends on a particular input variable depends on} the form of the Boolean function that is tested, but does not depend on the total number of input variables. We also outline a procedure based on another quantum algorithm, the Grover search, to amplify further the success probability. Finally, we discuss quantum algorithms for learning the exact form of certain quadratic and cubic Boolean functions.
We discuss quantum algorithms based on the Bernstein–Vazirani algorithm for finding which input variables a Boolean function depends on. There are 2 n possible linear Boolean functions of n input variables; given a linear Boolean function, the Bernstein–Vazirani quantum algorithm can deterministically identify which one of these Boolean functions we are given using just one single function query. We show how the same quantum algorithm can also be used to learn which input variables any other type of Boolean function} depends on. The success probability of learning that the function depends on a particular input variable depends on} the form of the Boolean function that is tested, but does not depend on the total number of input variables. We also outline a procedure based on another quantum algorithm, the Grover search, to amplify further the success probability. Finally, we discuss quantum algorithms for learning the exact form of certain quadratic and cubic Boolean functions.
We discuss quantum algorithms, based on the Bernstein-Vazirani algorithm, for finding which variables a Boolean function depends on. There are 2^n possible linear Boolean functions of n variables; given a linear Boolean function, the Bernstein-Vazirani quantum algorithm can deterministically identify which one of these Boolean functions we are given using just one single function query. The same quantum algorithm can also be used to learn which input variables other types of Boolean functions depend on, with a success probability that depends on the form of the Boolean function that is tested, but does not depend on the total number of input variables. We also outline a procedure to futher amplify the success probability, based on another quantum algorithm, the Grover search.
Bernstein-Vazirani algorithm (the one-query algorithm) can identify a completely specified linear Boolean function using a single query to the oracle with certainty. The first aim of the paper is to show that if the provided Boolean function is affine, then one more query to the oracle (the two-query algorithm) is required to identify the affinity of the function with certainty. The second aim of the paper is to show that if the provided Boolean function is incompletely defined, then the one-query and the two-query algorithms can be used as bounded-error quantum polynomial algorithms to identify certain classes of incompletely defined linear and affine Boolean functions respectively with probability of success at least 2/3.
Abstract The problem of learning Boolean linear functions from quantum examples w.r.t. the uniform distribution can be solved on a quantum computer using the Bernstein–Vazirani algorithm (Bernstein and Vazirani, in: Kosaraju (ed) Proceedings of the twenty-fifth annual ACM symposium on theory of computing, ACM, New York, 1993. 10.1145/167088.167097 ). A similar strategy can be applied in the case of noisy quantum training data, as was observed in Grilo et al. (Learning with errors is easy with quantum samples, 2017). However, extensions of these learning algorithms beyond the uniform distribution have not yet been studied. We employ the biased quantum Fourier transform introduced in Kanade et al. (Learning dnfs under product distributions via μ μ -biased quantum Fourier sampling, 2018) to develop efficient quantum algorithms for learning Boolean linear functions on n bits from quantum examples w.r.t. a biased product distribution. Our first procedure is applicable to any (except full) bias and requires O(ln(n)) O(ln(n)) quantum examples. The number of quantum examples used by our second algorithm is independent of n, but the strategy is applicable only for small bias. Moreover, we show that the second procedure is stable w.r.t. noisy training data and w.r.t. faulty quantum gates. This also enables us to solve a version of the learning problem in which the underlying distribution is not known in advance. Finally, we prove lower bounds on the classical and quantum sample complexities of the learning problem. Whereas classically, Ω(n) Ω(n) examples are necessary independently of the bias, we are able to establish a quantum sample complexity lower bound of Ω(ln(n)) Ω(ln(n)) only under an assumption of large bias. Nevertheless, this allows for a discussion of the performance of our suggested learning algorithms w.r.t. sample complexity. With our analysis, we contribute to a more quantitative understanding of the power and limitations of quantum training data for learning classical functions.
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. When h is the XOR or the OR function, this gives a restricted variant of the Bernstein-Vazirani or the combinatorial group testing problem, respectively. 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 Θ(k)) or the exact-half function (complexity is Θ(k1/4)). The first algorithm resolves an open problem from arXiv:1210.1148. For the case when h is the majority function, we prove an upper bound of O(k1/4). All these algorithms can be made exact. 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).
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 [1] (when h is the XOR function) and the combinatorial group testing problem [2] (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 algorithms can be analyzed in a query model to compute Boolean functions where input is given in a black box and the aim is to compute function value for arbitrary input using as few queries as possible. We concentrate on quantum query algorithm designing tasks in this paper. The main aim of the research was to find new efficient algorithms and develop general algorithm designing techniques. First, we present several exact quantum query algorithms for certain problems that are better than classical counterparts. Next, we introduce algorithm transformation methods that allow significant enlarging of exactly computable functions sets. Finally, we propose quantum algorithm designing methods. Given algorithms for the set of sub-functions, our methods use them to design a more complex one, based on algorithms described before. Methods are applicable for input algorithms with specific properties and preserve acceptable error probability and number of queries. Methods offer constructions for computing AND, OR and MAJORITY kinds of Boolean functions.