Optimal Nested Test Plan for Combinatorial Quantitative Group Testing

We consider the quantitative group testing problem where the objective is to identify defective items in a given population based on results of tests performed on subsets of the population. Under the quantitative group testing model, the result of each test reveals the number of defective items in the tested group. The minimum number of tests achievable by nested test plans was established by Aigner and Schughart in 1985 within a minimax framework. The optimal nested test plan offering this performance, however, was not obtained. In this work, we establish the optimal nested test plan in closed form. This optimal nested test plan is also order optimal among all test plans as the population size approaches infinity. Using heavy-hitter detection as a case study, we show via simulation examples orders of magnitude improvement of the group testing approach over two prevailing sampling-based approaches in detection accuracy and counter consumption. Other applications include anomaly detection and wideband spectrum sensing in cognitive radio systems

Quantum leakage detection using a model-independent dimension witness

Users of quantum computers must be able to confirm they are indeed functioning as intended, even when the devices are remotely accessed. In particular, if the Hilbert space dimension of the components are not as advertised -- for instance if the qubits suffer leakage -- errors can ensue and protocols may be rendered insecure. We refine the method of delayed vectors, adapted from classical chaos theory to quantum systems, and apply it remotely on the IBMQ platform -- a quantum computer composed of transmon qubits. The method witnesses, in a model-independent fashion, dynamical signatures of higher-dimensional processes. We present evidence, under mild assumptions, that the IBMQ transmons suffer state leakage, with a $p$ value no larger than $5{\times}10^{-4}$ under a single qubit operation. We also estimate the number of shots necessary for revealing leakage in a two-qubit system.Comment: 11 pages, 5 figure

Constraining the Number of Positive Responses in Adaptive, Non-Adaptive, and Two-Stage Group Testing

Group testing is a well known search problem that consists in detecting the defective members of a set of objects O by performing tests on properly chosen subsets (pools) of the given set O. In classical group testing the goal is to find all defectives by using as few tests as possible. We consider a variant of classical group testing in which one is concerned not only with minimizing the total number of tests but aims also at reducing the number of tests involving defective elements. The rationale behind this search model is that in many practical applications the devices used for the tests are subject to deterioration due to exposure to or interaction with the defective elements. In this paper we consider adaptive, non-adaptive and two-stage group testing. For all three considered scenarios, we derive upper and lower bounds on the number of "yes" responses that must be admitted by any strategy performing at most a certain number t of tests. In particular, for the adaptive case we provide an algorithm that uses a number of "yes" responses that exceeds the given lower bound by a small constant. Interestingly, this bound can be asymptotically attained also by our two-stage algorithm, which is a phenomenon analogous to the one occurring in classical group testing. For the non-adaptive scenario we give almost matching upper and lower bounds on the number of "yes" responses. In particular, we give two constructions both achieving the same asymptotic bound. An interesting feature of one of these constructions is that it is an explicit construction. The bounds for the non-adaptive and the two-stage cases follow from the bounds on the optimal sizes of new variants of d-cover free families and (p,d)-cover free families introduced in this paper, which we believe may be of interest also in other contexts