Quantum Algorithms in Measurement-Based Quantum Computing

Measurement-based quantum computing (MBQC) is a formulation of quantum computing alternative to the canonical circuit model. While equivalent in terms of universal computational power, MBQC can be advantageous in experiments and can highlight the core mechanics of quantum algorithms. The Deutsch-Jozsa and Simon algorithms are two of the most prominent quantum algorithms. The Deutsch-Jozsa algorithm determines if a function is constant or balanced and is frequently used as introduction to the realm of quantum algorithms. The Simon algorithm finds the period of a bitwise function and was the first algorithm proven to be supreme to its classical counterpart. We formulate the two- and three-qubit Deutsch-Jozsa algorithm as well as the n-qubit Simon algorithm in the language of MBQC. We employ the framework of the ZX-calculus, a graphical tensor description of quantum states and operators, to translate the circuit representations of these algorithms to MBQC and to outline the necessary single-qubit projective measurements to implement their oracles. The resulting ZX-diagrams depict the resource cluster states of the oracles, and the measurement axes can be read directly off the diagrams. Formulating these established algorithms in the framework of MBQC should aid in understanding the core mechanics and can serve as a blueprint for experimental implementation.