Testing and Learning Quantum Juntas Nearly Optimally

We consider the problem of testing and learning quantum $k$-juntas: $n$-qubit unitary matrices which act non-trivially on just $k$ of the $n$ qubits and as the identity on the rest. As our main algorithmic results, we give (a) a $\widetilde{O}(\sqrt{k})$-query quantum algorithm that can distinguish quantum $k$-juntas from unitary matrices that are "far" from every quantum $k$-junta; and (b) a $O(4^k)$-query algorithm to learn quantum $k$-juntas. We complement our upper bounds for testing quantum $k$-juntas and learning quantum $k$-juntas with near-matching lower bounds of $\Omega(\sqrt{k})$ and $\Omega(\frac{4^k}{k})$, respectively. Our techniques are Fourier-analytic and make use of a notion of influence of qubits on unitaries

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

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

Testing Booleanity and the Uncertainty Principle

Let f:{-1,1}^n -> R be a real function on the hypercube, given by its discrete Fourier expansion, or, equivalently, represented as a multilinear polynomial. We say that it is Boolean if its image is in {-1,1}. We show that every function on the hypercube with a sparse Fourier expansion must either be Boolean or far from Boolean. In particular, we show that a multilinear polynomial with at most k terms must either be Boolean, or output values different than -1 or 1 for a fraction of at least 2/(k+2)^2 of its domain. It follows that given oracle access to f, together with the guarantee that its representation as a multilinear polynomial has at most k terms, one can test Booleanity using O(k^2) queries. We show an \Omega(k) queries lower bound for this problem. Our proof crucially uses Hirschman's entropic version of Heisenberg's uncertainty principle.Comment: 15 page

A Strong Composition Theorem for Junta Complexity and the Boosting of Property Testers

We prove a strong composition theorem for junta complexity and show how such theorems can be used to generically boost the performance of property testers. The $\varepsilon$-approximate junta complexity of a function $f$ is the smallest integer $r$ such that $f$ is $\varepsilon$-close to a function that depends only on $r$ variables. A strong composition theorem states that if $f$ has large $\varepsilon$-approximate junta complexity, then $g \circ f$ has even larger $\varepsilon'$-approximate junta complexity, even for $\varepsilon' \gg \varepsilon$. We develop a fairly complete understanding of this behavior, proving that the junta complexity of $g \circ f$ is characterized by that of $f$ along with the multivariate noise sensitivity of $g$. For the important case of symmetric functions $g$, we relate their multivariate noise sensitivity to the simpler and well-studied case of univariate noise sensitivity. We then show how strong composition theorems yield boosting algorithms for property testers: with a strong composition theorem for any class of functions, a large-distance tester for that class is immediately upgraded into one for small distances. Combining our contributions yields a booster for junta testers, and with it new implications for junta testing. This is the first boosting-type result in property testing, and we hope that the connection to composition theorems adds compelling motivation to the study of both topics.Comment: 44 pages, 1 figure, FOCS 202

Local Correction of Juntas

A Boolean function f over n variables is said to be q-locally correctable if, given a black-box access to a function g which is "close" to an isomorphism f_sigma of f, we can compute f_sigma(x) for any x in Z_2^n with good probability using q queries to g. We observe that any k-junta, that is, any function which depends only on k of its input variables, is O(2^k)-locally correctable. Moreover, we show that there are examples where this is essentially best possible, and locally correcting some k-juntas requires a number of queries which is exponential in k. These examples, however, are far from being typical, and indeed we prove that for almost every k-junta, O(k log k) queries suffice.Comment: 6 page