Which graph states are useful for quantum information processing?

TL;DR

This paper characterizes which graph states are useful for quantum information processing, introducing new classes of measurement-based quantum computation with relaxed conditions, and providing graph-based criteria for input-output configurations.

Contribution

It introduces generalized classes of MBQC with relaxed gflow conditions and provides graph characterizations for their applicability in quantum information tasks.

Findings

Uniform equiprobability and constant probability classes are defined and characterized.

Deterministic and uniform equiprobability classes coincide when input and output sizes match.

Reversibility of gflow is established for equal input and output sizes.

Abstract

Graph states are an elegant and powerful quantum resource for measurement based quantum computation (MBQC). They are also used for many quantum protocols (error correction, secret sharing, etc.). The main focus of this paper is to provide a structural characterisation of the graph states that can be used for quantum information processing. The existence of a gflow (generalized flow) is known to be a requirement for open graphs (graph, input set and output set) to perform uniformly and strongly deterministic computations. We weaken the gflow conditions to define two new more general kinds of MBQC: uniform equiprobability and constant probability. These classes can be useful from a cryptographic and information point of view because even though we cannot do a deterministic computation in general we can preserve the information and transfer it perfectly from the inputs to the outputs. We derive simple graph characterisations for these classes and prove that the deterministic and uniform equiprobability classes collapse when the cardinalities of inputs and outputs are the same. We also prove the reversibility of gflow in that case. The new graphical characterisations allow us to go from open graphs to graphs in general and to consider this question: given a graph with no inputs or outputs fixed, which vertices can be chosen as input and output for quantum information processing? We present a characterisation of the sets of possible inputs and ouputs for the equiprobability class, which is also valid for deterministic computations with inputs and ouputs of the same cardinality.