22,897 research outputs found
Quantum Algorithms for Learning and Testing Juntas
In this article we develop quantum algorithms for learning and testing
juntas, i.e. Boolean functions which depend only on an unknown set of k out of
n input variables. Our aim is to develop efficient algorithms:
- whose sample complexity has no dependence on n, the dimension of the domain
the Boolean functions are defined over;
- with no access to any classical or quantum membership ("black-box")
queries. Instead, our algorithms use only classical examples generated
uniformly at random and fixed quantum superpositions of such classical
examples;
- which require only a few quantum examples but possibly many classical
random examples (which are considered quite "cheap" relative to quantum
examples).
Our quantum algorithms are based on a subroutine FS which enables sampling
according to the Fourier spectrum of f; the FS subroutine was used in earlier
work of Bshouty and Jackson on quantum learning. Our results are as follows:
- We give an algorithm for testing k-juntas to accuracy that uses
quantum examples. This improves on the number of examples used
by the best known classical algorithm.
- We establish the following lower bound: any FS-based k-junta testing
algorithm requires queries.
- We give an algorithm for learning -juntas to accuracy that
uses quantum examples and
random examples. We show that this learning algorithms is close to optimal by
giving a related lower bound.Comment: 15 pages, 1 figure. Uses synttree package. To appear in Quantum
Information Processin
Testing and Learning Quantum Juntas Nearly Optimally
We consider the problem of testing and learning quantum -juntas: -qubit
unitary matrices which act non-trivially on just of the qubits and as
the identity on the rest. As our main algorithmic results, we give (a) a
-query quantum algorithm that can distinguish quantum
-juntas from unitary matrices that are "far" from every quantum -junta;
and (b) a -query algorithm to learn quantum -juntas. We complement
our upper bounds for testing quantum -juntas and learning quantum -juntas
with near-matching lower bounds of and
, respectively. Our techniques are Fourier-analytic and
make use of a notion of influence of qubits on unitaries
Adaptivity Helps for Testing Juntas
We give a new lower bound on the query complexity of any non-adaptive algorithm for testing whether an unknown Boolean function is a k-junta versus epsilon-far from every k-junta. Our lower bound is that any non-adaptive algorithm must make Omega(( k * log*(k)) / ( epsilon^c * log(log(k)/epsilon^c))) queries for this testing problem, where c is any absolute constant <1. For suitable values of epsilon this is asymptotically larger than the O(k * log(k) + k/epsilon) query complexity of the best known adaptive algorithm [Blais,STOC\u2709] for testing juntas, and thus the new lower bound shows that adaptive algorithms are more powerful than non-adaptive algorithms for the junta testing problem
Partially Symmetric Functions are Efficiently Isomorphism-Testable
Given a function f: {0,1}^n \to {0,1}, the f-isomorphism testing problem
requires a randomized algorithm to distinguish functions that are identical to
f up to relabeling of the input variables from functions that are far from
being so. An important open question in property testing is to determine for
which functions f we can test f-isomorphism with a constant number of queries.
Despite much recent attention to this question, essentially only two classes of
functions were known to be efficiently isomorphism testable: symmetric
functions and juntas.
We unify and extend these results by showing that all partially symmetric
functions---functions invariant to the reordering of all but a constant number
of their variables---are efficiently isomorphism-testable. This class of
functions, first introduced by Shannon, includes symmetric functions, juntas,
and many other functions as well. We conjecture that these functions are
essentially the only functions efficiently isomorphism-testable.
To prove our main result, we also show that partial symmetry is efficiently
testable. In turn, to prove this result we had to revisit the junta testing
problem. We provide a new proof of correctness of the nearly-optimal junta
tester. Our new proof replaces the Fourier machinery of the original proof with
a purely combinatorial argument that exploits the connection between sets of
variables with low influence and intersecting families.
Another important ingredient in our proofs is a new notion of symmetric
influence. We use this measure of influence to prove that partial symmetry is
efficiently testable and also to construct an efficient sample extractor for
partially symmetric functions. We then combine the sample extractor with the
testing-by-implicit-learning approach to complete the proof that partially
symmetric functions are efficiently isomorphism-testable.Comment: 22 page
Recommended from our members
Property Testing of Boolean Function
The field of property testing has been studied for decades, and Boolean functions are among the most classical subjects to study in this area.
In this thesis we consider the property testing of Boolean functions: distinguishing whether an unknown Boolean function has some certain property (or equivalently, belongs to a certain class of functions), or is far from having this property. We study this problem under both the standard setting, where the distance between functions is measured with respect to the uniform distribution, as well as the distribution-free setting, where the distance is measured with respect to a fixed but unknown distribution.
We obtain both new upper bounds and lower bounds for the query complexity of testing various properties of Boolean functions:
- Under the standard model of property testing, we prove a lower bound of \Omega(n^{1/3}) for the query complexity of any adaptive algorithm that tests whether an n-variable Boolean function is monotone, improving the previous best lower bound of \Omega(n^{1/4}) by Belov and Blais in 2015. We also prove a lower bound of \Omega(n^{2/3}) for adaptive algorithms, and a lower bound of \Omega(n) for non-adaptive algorithms with one-sided errors that test unateness, a natural generalization of monotonicity. The latter lower bound matches the previous upper bound proved by Chakrabarty and Seshadhri in 2016, up to poly-logarithmic factors of n.
- We also study the distribution-free testing of k-juntas, where a function is a k-junta if it depends on at most k out of its n input variables. The standard property testing of k-juntas under the uniform distribution has been well understood: it has been shown that, for adaptive testing of k-juntas the optimal query complexity is \Theta(k); and for non-adaptive testing of k-juntas it is \Theta(k^{3/2}). Both bounds are tight up to poly-logarithmic factors of k. However, this problem is far from clear under the more general setting of distribution-free testing. Previous results only imply an O(2^k)-query algorithm for distribution-free testing of k-juntas, and besides lower bounds under the uniform distribution setting that naturally extend to this more general setting, no other results were known from the lower bound side. We significantly improve these results with an O(k^2)-query adaptive distribution-free tester for k-juntas, as well as an exponential lower bound of \Omega(2^{k/3}) for the query complexity of non-adaptive distribution-free testers for this problem. These results illustrate the hardness of distribution-free testing and also the significant role of adaptivity under this setting.
- In the end we also study distribution-free testing of other basic Boolean functions. Under the distribution-free setting, a lower bound of \Omega(n^{1/5}) was proved for testing of conjunctions, decision lists, and linear threshold functions by Glasner and Servedio in 2009, and an O(n^{1/3})-query algorithm for testing monotone conjunctions was shown by Dolev and Ron in 2011. Building on techniques developed in these two papers, we improve these lower bounds to \Omega(n^{1/3}), and specifically for the class of conjunctions we present an adaptive algorithm with query complexity O(n^{1/3}). Our lower and upper bounds are tight for testing conjunctions, up to poly-logarithmic factors of n
Property testing of unitary operators
In this paper, we systematically study property testing of unitary operators.
We first introduce a distance measure that reflects the average difference
between unitary operators. Then we show that, with respect to this distance
measure, the orthogonal group, quantum juntas (i.e. unitary operators that only
nontrivially act on a few qubits of the system) and Clifford group can be all
efficiently tested. In fact, their testing algorithms have query complexities
independent of the system's size and have only one-sided error. Then we give an
algorithm that tests any finite subset of the unitary group, and demonstrate an
application of this algorithm to the permutation group. This algorithm also has
one-sided error and polynomial query complexity, but it is unknown whether it
can be efficiently implemented in general
On active and passive testing
Given a property of Boolean functions, what is the minimum number of queries
required to determine with high probability if an input function satisfies this
property or is "far" from satisfying it? This is a fundamental question in
Property Testing, where traditionally the testing algorithm is allowed to pick
its queries among the entire set of inputs. Balcan, Blais, Blum and Yang have
recently suggested to restrict the tester to take its queries from a smaller
random subset of polynomial size of the inputs. This model is called active
testing, and in the extreme case when the size of the set we can query from is
exactly the number of queries performed it is known as passive testing.
We prove that passive or active testing of k-linear functions (that is, sums
of k variables among n over Z_2) requires Theta(k*log n) queries, assuming k is
not too large. This extends the case k=1, (that is, dictator functions),
analyzed by Balcan et. al.
We also consider other classes of functions including low degree polynomials,
juntas, and partially symmetric functions. Our methods combine algebraic,
combinatorial, and probabilistic techniques, including the Talagrand
concentration inequality and the Erdos--Rado theorem on Delta-systems.Comment: 16 page
New Lower Bounds for Adaptive Tolerant Junta Testing
We prove a lower bound for
adaptively testing whether a Boolean function is -close to or
-far from -juntas. Our results provide the first
superpolynomial separation between tolerant and non-tolerant testing for a
natural property of boolean functions under the adaptive setting. Furthermore,
our techniques generalize to show that adaptively testing whether a function is
-close to a -junta or -far from -juntas cannot be done with queries. This is in contrast to an algorithm by Iyer, Tal
and Whitmeyer [CCC 2021] which uses queries to test whether a function is
-close to a -junta or -far from
-juntas.Comment: 22 page
Junta Distance Approximation with Sub-Exponential Queries
Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two
different results pertaining to the \emph{tolerant testing} of juntas. Given
black-box access to a Boolean function , we give a
query algorithm that distinguishes between
functions that are -close to -juntas and -far
from -juntas, where .
In the non-relaxed setting, we extend our ideas to give a
(adaptive) query algorithm that
distinguishes between functions that are -close to -juntas and
-far from -juntas. To the best of our knowledge, this
is the first subexponential-in- query algorithm for approximating the
distance of to being a -junta (previous results of Blais, Canonne, Eden,
Levi, and Ron [SODA, 2018] and De, Mossel, and Neeman [FOCS, 2019] required
exponentially many queries in ).
Our techniques are Fourier analytical and make use of the notion of
"normalized influences" that was introduced by Talagrand [AoP, 1994].Comment: To appear in CCC 202
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