A slow month to start 2022, as far as property testing (and myself) are concerned — “only” 3 papers, and a delay of several days in posting this. Let’s jump in with quantum testing!

**Testing matrix product states**, by Mehdi Soleimanifar and John Wright (arXiv). Suppose you are given a state \(|\psi\rangle\) of \(n\) qubits, and want to know “how entangled” this whole thing is: for instance, is \(|\psi\rangle\) a product state (no entanglement between the \(n\) qudits)? More generally, the “amount of entanglement” allowed is captured by an integer \(r\), the* bond dimension*, where product state corresponds to \(r=1\), and larger \(r\) allows for more entanglement. This paper then considers the following property testing question: how many copies of \(|\psi\rangle\) are needed to test whether it has bond dimension at most \(r\), or is \(\varepsilon\)-far from every such state (in trace distance)? While the case \(r=1\) had been previously considered, this paper considers the general case; and, in particular, shows a qualitative gap between \(r=1\) (for which a constant number of copies, \(O(1/\varepsilon^2)\), suffice) and \(r\geq 2\) (for which they show the number of states is \(\Omega(\sqrt{n}/\varepsilon^2)\), and \(O(n r^2/\varepsilon^2)\)).

**Constant-time one-shot testing of large-scale graph states**, by Hayata Yamasaki and Sathyawageeswar Subramanian (arXiv). In this paper, the authors consider the task of testing if the physical error rate of a given system is below a given threshold — namely, the threshold below which fault-tolerant measurement-based quantum computation (MBQC) becomes feasible. Casting this into the framework of property testing, the paper shows that measuring very few (a constant number!) of the input state is enough to test whether the error rate is low.

And, to conclude, a paper which escaped us in December, on private distribution testing:

**Pure Differential Privacy from Secure Intermediaries**, by Albert Cheu and Chao Yan (arXiv). Throwback to April 2020 and August 2021, which covered results on distribution testing (uniformity testing!) under the *shuffle model* of differential privacy. Namely, there was an upper bound of $$ O( k^{2/3}/(\alpha^{4/3}\varepsilon^{2/3})\log^{1/3}(1/\delta) + k^{1/2}/(\alpha\varepsilon) \log^{1/2}(1/\delta) + k^{1/2}/\alpha^2)$$ samples for testing uniformity of distributions over \([k]\), to distance \(\alpha\), under \((\varepsilon,\delta)\)*–*shuffle privacy (so, *approximate* privacy: \(\delta>0\)). A partial lower bound existed for *pure* differential privacy, i.e., when \(\delta=0\): however, no upper bound was known for pure shuffle privacy.

Until now: this new paper shows that pure DP basically comes at no cost, by providing an \((\varepsilon,0)\)-shuffle private testing algorithm with sample complexity $$ O( k^{2/3}/(\alpha^{4/3}\varepsilon^{2/3}) + k^{1/2}/(\alpha\varepsilon) + k^{1/2}/\alpha^2)$$ The paper actually does a lot more, focusing on a different problem, private summation; and the testing upper bound is a corollary of the new methods they develop in the process.