An Overview of Quantum Algorithms: From Quantum Supremacy to Shor Factorization

Recently, a team of scientists from Google claims to have carried a computation on their noisy, intermediate-scale quantum (NISQ) computer which no regular computer can achieve. A feat that is sometimes referred as quantum supremacy. In the first part of this work, we explain their approach, their randomised circuit construction and the consequences of their work. Having achieved this milestone, the natural question becomes: what else can we do with a quantum computer? We answer this question in the second part of this work and give an overview of the most widely used quantum primitives. We will see how one can implement them using quantum circuits and how these primitive circuits are composed to create fast algorithms capable of solving commercially relevant problems like simulation of complicated quantum systems for chemistry and material science, Monte Carlo simulation, solving large systems of linear equations, or breaking widely used cryptography like RSA.