Simon algorithm in measurement-based quantum computing

TL;DR

This paper reformulates Simon's quantum algorithm within measurement-based quantum computing using ZX-calculus, demonstrating its implementation with cluster states and highlighting its potential for experimental realization.

Contribution

It provides a detailed MBQC reformulation of Simon's algorithm, including explicit cluster state constructions for two and n qubits, aiding understanding and experimental development.

Findings

Two-qubit Simon algorithm implemented with a ten-qubit cluster state.

n-qubit Simon algorithm represented with 2n-node cluster states and n^2 edges.

MBQC formulation preserves exponential speedup over classical algorithms.

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.