# Minimal physical resources for the realisation of measurement-based quantum computation

## TL;DR

This paper investigates the minimal number of physical qubits needed for measurement-based quantum computation, showing that fewer qubits can be used than traditionally required, especially with flow properties, enhancing resource efficiency.

## Contribution

It establishes the minimal qubit requirements for implementing measurement patterns with flow or gflow, and analyzes the impact of removing Clifford operations on resource needs.

## Key findings

- Only min(n + 1, m) qubits needed for flow patterns.
- Up to m-2 qubits required for gflow patterns.
- Removing Clifford parts affects flow and gflow, influencing qubit requirements.

## Abstract

In measurement-based quantum computation (MBQC), a special highly-entangled state (called a resource state) allows for universal quantum computation driven by single-qubit measurements and post-measurement corrections. Physical realisations of this model have been achieved in various physical systems for low numbers of qubits. The large number of qubits necessary to construct the resource state constitutes one of the main down sides to MBQC. However, in some instances it is possible to extend the resource state on the fly, meaning that not every qubit must be realised in the devices simultaneously. We consider the question of the minimal number of physical qubits that must be present in a system to directly implement a given measurement pattern. For measurement patterns with n inputs, n outputs and m total qubits which have flow, we show that only min(n + 1, m) qubits are required, while the number of required qubits can be as high as m-2 for measurement patterns with only gflow. We discuss the implications of removing the Clifford part of a measurement pattern, using well-established transformation rules for Pauli measurements, for the presence of flow versus gflow, and hence the effect on the minimum number of physical qubits required to directly realise the measurement pattern.