A unified picture of Balance puzzles and Group testing: Some lessons from quantum mechanics for the pandemic

Abstract

Balance (Counterfeit coin) puzzles have been part of recreational mathematics for a few decades.A particular type of Counterfeit coin puzzle is known in the literature as the "Beam balance puzzle".An abstract solution to it is provided by Iwama et.al as a modification of the Bernstein-Vazirani algorithm, making use of quantum parallelism and entanglement. Moreover, during this pandemic, group testing has proved to be an efficient algorithm to save time and cost of testing specimens for the presence of infection. In this article, we propose a "Binary Spring Balance" (BSB) puzzle, to facilitate a unified picture of the counterfeit coin problem and the testing for infection problem, as both aim to reduce the number of queries. We then showcase two solutions to the BSB problem, one using bits and other using classical-qubits ('cebits") for querying. Both solutions are demonstrated using circuits. In this pursuit, we develop a modified optical implementation of Bernstein-Vazirani algorithm using only polarizers (no need of beam splitters), which has sur-prisingly not yet been proposed earlier. Under the pretext of this demonstration we question why we have not yet developed testing mechanisms inspired by Bernstein-Vazirani algorithm for the pandemic, as they solve the problem in single query, they have no issues related to prevalence of infection in the population, nor are they plagued by the issue of dilution of samples due to pooling. The modified implementation of Bernstein-Vazirani algorithm using polarizers can also be a cost-effective demonstration in an undergraduate lab.