The first month of 2020 is behind us, and it’s already looking good! Four papers last month, spanning quantum, graphs, and probability distributions.

**On Efficient Distance Approximation for Graph Properties**, by Nimrod Fiat and Dana Ron (arXiv). In the dense graph (i.e., adjacency-matrix) model, one is given a distance parameter \(\varepsilon\) and granted query access to the adjacency matrix of a graph \(G\), and seeks to determine something about the distance of \(G\) to a prespecified property \(\mathcal{P}\) of interest (i.e., the fraction of the matrix that needs to be changed for \(G\) to satisfy the property). *Testing* requires to distinguish whether that distance is zero, or at least \(\varepsilon\); many results over the past years have shown that many properties can be tested with a number of queries depending only on \(\varepsilon\) (and not on \(n=|G|\). This work focuses on the harder problem of *distance estimation*, or, equivalently, *tolerant* testing: that is, estimate this distance up to \(\pm\varepsilon\). The authors introduce a general framework to get distance approximation algorithms from a “complicated” property by decomposing it into simpler properties and using algorithms for those. Applying this framework to a few flagship properties, they then show that \(P_3\)-freeness, induced \(P_4\)-freeness, and cordiality are properties which have *efficient* distance estimation algorithms (independent of \(n\), and polynomial in \(1/\varepsilon\)).

**Minimax Optimal Conditional Independence Testing**, by Matey Neykov, Sivaraman Balakrishnan, and Larry Wasserman (arXiv). Given i.i.d. draws from a triple \((X,Y,Z)\), how hard is it to check whether \(X \perp Y \mid Z\), that is, “\(X\) and \(Y\) are independent conditioned on \(Z\) (or far from it)?” This is the problem on *conditional independence testing*, which was covered back in the days for the case where \(X,Y,Z\) are discrete. Well, this new work takes the fight out of the discrete world: extending the results and techniques from the discrete case, it provides optimal bound for the *continuous* case: where \(Z\) is on \([0,1]\), and then when all three \(X,Y, Z\) are continuous.

**How symmetric is too symmetric for large quantum speedups?**, by Shalev Ben-David and Supartha Podder (arXiv); and **Can graph properties have exponential quantum speedup?**, by Andrew M. Childs and Daochen Wang (arXiv). Both these works independently study the relation between the (bounded-error) randomized and quantum query complexities of any graph property \(f\), in the dense graph (adjacency-matrix) model. In particular, how much advantage do quantum algorithms provide for those?

As it turns out, not so much: for those functions, both papers show the two quantities are always polynomially related (\(R(f) \leq Q(f) \leq O(R(f)^6))\)) in the dense-graph model. As a corollary, this implies that *testing* any such property won’t benefit much from quantum (that is, at most polynomially)…. at least in this model. In the adjacency *list* model (also known as the bounded-degree graph model), the first paper conjectures that exponential query complexity improvements are possible; and the second paper provides an example, establishing it. Altogether, this fully settles an open problem of Ambainis, Childs, and Liu, and Montanaro and de Wolf.