Sublinear Quantum Computation

The rapid rise in the scale of data is a shaping force in the evolution of computer science. With the ubiquity of massive datasets in recent years, there is an urgent need for ultra-fast algorithms that run in sublinear time, i.e., much faster than it takes to even read their input.In the quantum era, this need is even more imperative, since the dimension of a quantum system scales exponentially in the number of quantum bits. Indeed, the description size of a system with merely 127 quantum bits exceeds 10^{76} real numbers, which is close to the number of atoms in the observable universe. Hence, sublinear computation is essential in the quantum regime.This proposal is a high-risk high-gain research programme, which targets long-standing open problems in sublinear quantum computation via new and unique technical and conceptual approaches, traversing quantum learning theory, property testing, and complexity theory. This includes new paradigms for designing quantum algorithms, delegating quantum computation, and making progress on the quantum PCP conjecture.The proposed research will pioneer new notions of quantum computation, motivated by real-world needs, and establish fundamental connections between quantum computing and deep areas of mathematics such as harmonic analysis and additive combinatorics.