9 research outputs found
Quantum algorithms for testing 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
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Quantum pattern matching fast on average
The -dimensional pattern matching problem is to find an occurrence of a
pattern of length within a text of length , with . This task models various problems in text and
image processing, among other application areas. This work describes a quantum
algorithm which solves the pattern matching problem for random patterns and
texts in time . For
large this is super-polynomially faster than the best possible classical
algorithm, which requires time . The
algorithm is based on the use of a quantum subroutine for finding hidden shifts
in dimensions, which is a variant of algorithms proposed by Kuperberg.Comment: 22 pages, 2 figures; v3: further minor changes, essentially published
versio
Hidden Symmetry Subgroup Problems
We advocate a new approach for addressing hidden structure problems and finding efficient quantum algorithms. We introduce and investigate the hidden symmetry subgroup problem (HSSP), which is a generalization of the well-studied hidden subgroup problem (HSP). Given a group acting on a set and an oracle whose level sets define a partition of the set, the task is to recover the subgroup of symmetries of this partition inside the group. The HSSP provides a unifying framework that, besides the HSP, encompasses a wide range of algebraic oracle problems, including quadratic hidden polynomial problems. While the HSSP can have provably exponential quantum query complexity, we obtain efficient quantum algorithms for various interesting cases. To achieve this, we present a general method for reducing the HSSP to the HSP, which works efficiently in several cases related to symmetries of polynomials. The HSSP therefore connects in a rather surprising way certain hidden polynomial problems with the HSP. Using this connection, we obtain the first efficient quantum algorithm for the hidden polynomial problem for multivariate quadratic polynomials over fields of constant characteristic. We also apply the new methods to polynomial function graph problems and present an efficient quantum procedure for constant degree multivariate polynomials over any field. This result improves in several ways the currently known algorithms