Simon algorithm in measurement-based quantum computing

Abstract

Simon's hidden subgroup algorithm was the first quantum algorithm to prove the superiority of quantum computing over classical computing in terms of complexity. Measurement-based quantum computing (MBQC) is a formulation of quantum computing that, while equivalent in terms of computational power, can be advantageous in experiments and in displaying the core mechanics of quantum algorithms. We present a reformulation of the Simon algorithm into the language of MBQC -- in detail for two qubits and schematically for $n$ qubits. We utilize the framework of ZX-calculus, a graphical tensor description of quantum states and operators, to translate the circuit description of the algorithm into a form concordant with MBQC. The result for the two-qubit Simon algorithm is a ten-qubit cluster state on which single-qubit measurements suffice to extract the desired information. Additionally, we show that the $n$-qubit version of the Simon algorithm can be formulated in MBQC as cluster state graph with $2n$ nodes and $n^2$ edges. This is an example of the MBQC formulation of a quantum algorithm that is exponentially faster than its classical counterpart. As such, this formulation should aid in understanding the core mechanics of such an established algorithm and could serve as a blueprint for experimental implementation.