Matthew Aldridge

Matthew Aldridge, Oliver Johnson, Jonathan Scarlett

The group testing problem concerns discovering a small number of defective items within a large population by performing tests on pools of items. A test is positive if the pool contains at least one defective, and negative if it contains no defectives. This is a sparse inference problem with a combinatorial flavour, with applications in medical testing, biology, telecommunications, information technology, data science, and more. In this monograph, we survey recent developments in the group testing problem from an information-theoretic perspective. We cover several related developments: efficient algorithms with practical storage and computation requirements, achievability bounds for optimal decoding methods, and algorithm-independent converse bounds. We assess the theoretical guarantees not only in terms of scaling laws, but also in terms of the constant factors, leading to the notion of the {\em rate} of group testing, indicating the amount of information learned per test. For the noiseless setting, we present a series of results leading to optimal rates, which in turn imply optimality and suboptimality results of various algorithms depending on the sparsity regime. We also survey analogous developments in noisy settings. In addition, we survey results concerning a number of variations on the standard group testing problem, including approximate recovery criteria, adaptive algorithms with a limited number of stages, sublinear-time algorithms, and settings with additional prior information, among others.

Duong The Do, Matthew Aldridge, Duong Tung Nguyen

We propose a projected variational quantum extragradient (VQEG) framework for computing approximate Nash equilibria in two-player zero-sum matrix games. Mixed strategies are parameterized as Born distributions of parameterized quantum circuits (PQCs), transforming the classical bilinear saddle point problem into a smooth but generally minmax optimization in circuit-parameter space. The expected payoff is expressed as the expectation of a diagonal observable, enabling gradient evaluation via the parameter shift rule and compatibility with shot based quantum hardware. To support arbitrary game sizes, we introduce a dominated embedding that maps (m,n) games to qubit-compatible power-of-two dimensions while preserving equilibrium structure. We then develop a projected extragradient method using stochastic gradient estimates derived from finite measurement shots, and establish variance bounds scaling as O(1/S) with respect to the number of measurement shots S, along with convergence to approximate first-order stationarity under standard assumptions. Since stationarity does not guarantee equilibrium optimality, we evaluate performance using the game-space Nash gap. Numerical results demonstrate high-precision solutions on structured instances up to 32x32, while highlighting challenges in unstructured settings.