We construct efficient or query efficient quantum property testers for two existential group properties which have exponential query complexity both for their decision problem in the quantum and for their testing problem in the classical model of computing. These are periodicity in groups and the common coset range property of two functions having identical ranges within each coset of some normal subgroup. Our periodicity tester is efficient in Abelian groups and generalizes, in several aspects, previous periodicity testers. This is achieved by introducing a technique refining the majority correction process widely used for proving robustness of algebraic properties. The periodicity tester in non-Abelian groups and the common coset range tester are query efficient.We construct efficient or query efficient quantum property testers for two existential group properties which have exponential query complexity both for their decision problem in the quantum and for their testing problem in the classical model of computing. These are periodicity in groups and the common coset range property of two functions having identical ranges within each coset of some normal subgroup. Our periodicity tester is efficient in Abelian groups and generalizes, in several aspects, previous periodicity testers. This is achieved by introducing a technique refining the majority correction process widely used for proving robustness of algebraic properties. The periodicity tester in non-Abelian groups and the common coset range tester are query efficient.

arXiv:quant-ph/0208184Aug 29, 20020 citations

Quantum Testers for Hidden Group Properties

FriedlKatalin, SanthaMiklos, MagniezFrédéric +1 more

Fundamenta InformaticaeApr 1, 20090 citations

Efficient quantum algorithms for (Gapped) group testing and junta testing

Andris Ambainis, Aleksandrs Belovs, Oded Regev +1 more

In the k-junta testing problem, a tester has to efficiently decide whether a given function f: {0,1}n → {0,1} is a k-junta (i.e., depends on at most k of its input bits) or is ∈-far from any k-junta. Our main result is a quantum algorithm for this problem with query complexity Ō(√∈) and time complexity Ō(n√∈). This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atici and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of ω(k1/3) queries for junta-testing (for constant ∈).In the k-junta testing problem, a tester has to efficiently decide whether a given function f: {0,1}n → {0,1} is a k-junta (i.e., depends on at most k of its input bits) or is ∈-far from any k-junta. Our main result is a quantum algorithm for this problem with query complexity Ō(√∈) and time complexity Ō(n√∈). This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atici and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of ω(k1/3) queries for junta-testing (for constant ∈).

Jan 1, 201618 citations

Efficient Quantum Algorithms for (Gapped) Group Testing and Junta Testing

Andris Ambainis, Aleksandrs Belovs, Oded Regev +1 more

In the k-junta testing problem, a tester has to efficiently decide whether a given function f: {0, 1}n → {0, 1} is a k-junta (i.e., depends on at most k of its input bits) or is ∊-far from any k-junta. Our main result is a quantum algorithm for this problem with query complexity and time complexity . This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atıcı and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of Ω(k1/3) queries for junta-testing (for constant ∊).In the k-junta testing problem, a tester has to efficiently decide whether a given function f: {0, 1}n → {0, 1} is a k-junta (i.e., depends on at most k of its input bits) or is ∊-far from any k-junta. Our main result is a quantum algorithm for this problem with query complexity and time complexity . This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atıcı and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of Ω(k1/3) queries for junta-testing (for constant ∊).

ACM-SIAM Symposium on Discrete Algorithms (SODA)Dec 21, 201527 citations

Efficient Quantum Algorithms for (Gapped) Group Testing and Junta Testing

Andris Ambainis, Aleksandrs Belovs, Oded Regev +1 more

In the $k$ -junta testing problem, a tester has to efficiently decide whether a given function $f:\{0,1\}^n\rightarrow \{0,1\}$ is a $k$ -junta (i.e., depends on at most $k$ of its input bits) or is $ε$ -far from any $k$ -junta. Our main result is a quantum algorithm for this problem with query complexity $\tilde O(\sqrt{k/ε})$ and time complexity $\tilde O(n\sqrt{k/ε})$ . This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atıcı and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of $Ω(k^{1/3})$ queries for junta-testing (for constant $ε$ ).In the $k$-junta testing problem, a tester has to efficiently decide whether a given function $f:\{0,1\}^n\rightarrow \{0,1\}$ is a $k$-junta (i.e., depends on at most $k$ of its input bits) or is $ε$-far from any $k$-junta. Our main result is a quantum algorithm for this problem with query complexity $\tilde O(\sqrt{k/ε})$ and time complexity $\tilde O(n\sqrt{k/ε})$. This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atıcı and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of $Ω(k^{1/3})$ queries for junta-testing (for constant $ε$).

arXiv:1507.03126Jul 11, 20150 citations

Hidden translation and orbit coset in quantum computing

Katalin Friedl, Gábor Ivanyos, Frédéric Magniez +2 more

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian 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 pn, whenever p is a fixed prime. For the induction step, we introduce the problem Orbit Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Orbit Coset in a finite group G is reducible to Orbit Coset in G/N and subgroups of N, for any solvable normal subgroup N of G.We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian 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 pn, whenever p is a fixed prime. For the induction step, we introduce the problem Orbit Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Orbit Coset in a finite group G is reducible to Orbit Coset in G/N and subgroups of N, for any solvable normal subgroup N of G.

ACM Symposium on Theory of Computing (STOC)Jan 1, 20038 citations

In the

k

-junta testing problem, a tester has to efficiently decide whether a given function

f:\{0,1\}^n\rightarrow \{0,1\}

is a

k

-junta (i.e., depends on at most

k

of its input bits) or is

ε

-far from any

k

-junta. Our main result is a quantum algorithm for this problem with query complexity

\tilde O(\sqrt{k/ε})

and time complexity

\tilde O(n\sqrt{k/ε})

. This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atıcı and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of

Ω(k^{1/3})

queries for junta-testing (for constant

ε