The quantum query complexity of the hidden subgroup problem is polynomial

We present a quantum algorithm which identifies with certainty a hidden subgroup of an arbitrary finite group G in only a polynomial (in log |G|) number of calls to the oracle. This is exponentially better than the best classical algorithm. However our quantum algorithm requires exponential time, as in the classical case. Our algorithm utilizes a new technique for constructing error-free algorithms for non-decision problems on quantum computers.Comment: To appear in Information Processing Letters (IPL

Quantum algorithm for the Boolean hidden shift problem

The hidden shift problem is a natural place to look for new separations between classical and quantum models of computation. One advantage of this problem is its flexibility, since it can be defined for a whole range of functions and a whole range of underlying groups. In a way, this distinguishes it from the hidden subgroup problem where more stringent requirements about the existence of a periodic subgroup have to be made. And yet, the hidden shift problem proves to be rich enough to capture interesting features of problems of algebraic, geometric, and combinatorial flavor. We present a quantum algorithm to identify the hidden shift for any Boolean function. Using Fourier analysis for Boolean functions we relate the time and query complexity of the algorithm to an intrinsic property of the function, namely its minimum influence. We show that for randomly chosen functions the time complexity of the algorithm is polynomial. Based on this we show an average case exponential separation between classical and quantum time complexity. A perhaps interesting aspect of this work is that, while the extremal case of the Boolean hidden shift problem over so-called bent functions can be reduced to a hidden subgroup problem over an abelian group, the more general case studied here does not seem to allow such a reduction.Comment: 10 pages, 1 figur

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

Quantum algorithms for highly non-linear Boolean functions

Attempts to separate the power of classical and quantum models of computation have a long history. The ultimate goal is to find exponential separations for computational problems. However, such separations do not come a dime a dozen: while there were some early successes in the form of hidden subgroup problems for abelian groups--which generalize Shor's factoring algorithm perhaps most faithfully--only for a handful of non-abelian groups efficient quantum algorithms were found. Recently, problems have gotten increased attention that seek to identify hidden sub-structures of other combinatorial and algebraic objects besides groups. In this paper we provide new examples for exponential separations by considering hidden shift problems that are defined for several classes of highly non-linear Boolean functions. These so-called bent functions arise in cryptography, where their property of having perfectly flat Fourier spectra on the Boolean hypercube gives them resilience against certain types of attack. We present new quantum algorithms that solve the hidden shift problems for several well-known classes of bent functions in polynomial time and with a constant number of queries, while the classical query complexity is shown to be exponential. Our approach uses a technique that exploits the duality between bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of the paper contains a new exponential separation between classical and quantum query complexit

Quantum algorithms for hidden nonlinear structures

Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples of two such problems that can be solved efficiently by a quantum computer, but not by a classical computer. We also give some positive results on the quantum query complexity of finding hidden nonlinear structures.Comment: 13 page

Hidden Translation and Translating Coset in Quantum Computing

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in $\Z_{p}^{n}$, whenever $p$ is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Translating Coset in a finite solvable group $G$ is reducible to instances of Translating Coset in $G/N$ and $N$, for appropriate normal subgroups $N$ of $G$. Our self-reducibility framework combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.Comment: Journal version: change of title and several minor update

Improved Low-qubit Hidden Shift Algorithms

Hidden shift problems are relevant to assess the quantum security of various cryptographic constructs. Multiple quantum subexponential time algorithms have been proposed. In this paper, we propose some improvements on a polynomial quantum memory algorithm proposed by Childs, Jao and Soukharev in 2010. We use subset-sum algorithms to significantly reduce its complexity. We also propose new tradeoffs between quantum queries, classical time and classical memory to solve this problem