Pauli Flow on Open Graphs with Unknown Measurement Labels

TL;DR

This paper introduces a randomized polynomial-time algorithm to determine the existence of Pauli flow in open graphs with unknown measurement labels, extending algebraic methods and optimizing qubit configurations for measurement-based quantum computation.

Contribution

It provides the first efficient method to decide Pauli flow existence without known measurement labels, using algebraic interpretations and matrix invertibility.

Findings

Decides Pauli flow existence in RP for unknown measurement labels.

Shows flow existence relates to matrix right-invertibility.

Reduces output qubits to match input qubits while preserving flow.

Abstract

One-way quantum computation, or measurement-based quantum computation, is a universal model of quantum computation alternative to the circuit model. The computation progresses by measurements of a pre-prepared resource state together with corrections of undesired outcomes via applications of Pauli gates to yet unmeasured qubits. The fundamental question of this model is determining whether computation can be implemented deterministically. Pauli flow is one of the most general structures guaranteeing determinism. It is also essential for polynomial time ancilla-free circuit extraction. It is known how to efficiently determine the existence of Pauli flow given an open graph together with a measurement labelling (a choice of measurements to be performed). In this work, we focus on the problem of deciding the existence of Pauli flow for a given open graph when the measurement labelling is unknown. We show that this problem is in RP by providing a random polynomial time algorithm. To do it, we extend previous algebraic interpretations of Pauli flow, by showing that, in the case of X and Z measurements only, flow existence corresponds to the right-invertibility of a matrix derived from the adjacency matrix. We also use this interpretation to show that the number of output qubits can always be reduced to match the number of input qubits while preserving the existence of flow.