Quantum circuit synthesis of Bell and GHZ states using projective simulation in the NISQ era

Quantum computing is a promising new paradigm for computer science. Quantum algorithms have proven themselves superior to classical ones in some classes of problems. The major examples are Shor's algorithm for solving the hidden subgroup problem and Grover's algorithm for the unstructured search problem, but quantum computing is not limited to them[18]. Simulating complex atomic systems using a quantum computer[8] could profoundly impact physics research since a quantum computer can naturally simulate the effects of quantum mechanics. Quantum Computing has also been impactful in the machine learning field. Basic linear algebra subroutines, such as Fourier transform, finding eigenvectors and eigenvalues, and solving linear equations, for example, exhibit exponential quantum speedups over their best known classical counterparts[4] Also, quantum machine learning aims to implement machine learning algorithms in quantum systems, by using the quantum properties such as superposition and entanglement to solve these problems efficiently[17]. However, practical limitations still hinder the development of a universal quantum computer that could use such algorithms. Current quantum computers are referred to as NISQ (Noisy Intermediate-Scale Quantum) computers [22] because of qubits imperfections and their available limited number, currently between 50 and 100. These numbers are not sufficient to execute error-correcting codes once 9 qubits are necessary to make 1 qubit fault-tolerant [7].