by Damian Markham (LIP6, CNRS - Sorbonne Université)
The massive global investment in quantum technologies promises unprecedented boosts for security, computation, communication and sensing. In this article we explore the use of so-called ‘graph states’ – a family of multipartite entangled states which act as ubiquitous resources for quantum information, are easily adapted for different tasks and applications, and can be combined in ways that fuses different utilities.
Quantum computing and quantum information in general offer incredible benefits to our information society. Since the original discoveries of better than classical security in quantum key distribution and super-classical computational advantage, the field has exploded with new possibilities for exploiting quantum encoding.
In quantum cryptography we have seen new protocols for coin flipping, quantum money and secure multiparty computation offering functionality and or security that is not possible classically. For certain communication tasks, quantum techniques provide an exponential gap between what is possible quantumly and classically. Even before universal quantum computers are born, sub-universal devices will have a plethora of applications from quantum learning to simulation. In quantum metrology, quantum sensing provides precision in measurements that would simply not be possible without uniquely quantum features.
A remarkable family of multipartite entangled states acts as generic resources for almost all these applications. Graph states are described in one to one correspondence with a simple graph, where vertices represent qubits (a two dimensional quantum system which is the basic quantum information unit) and edges represent a particular entanglement preparation. They are universal resources for quantum computation, act as codes for quantum error correction and are the entangled resource for many communication and cryptographic protocols. Perhaps their most compelling strength is that they can be connected in different ways to allow the utility of one function to be combined with another in a natural way. In this sense, these “graph states” are like quantum Lego – decide what you want to make and put them together in the right way to achieve it. This capacity will be key in taking the best advantage of quantum technologies in future quantum networks. Indeed, typically, more sophisticated applications are built up by combining basic protocols.
The natural connection to graph theory has proven an additional benefit of the graph state approach. It turns out that one can understand many properties of their use for quantum information – how computation flows, where information sits – entirely in terms of the underlying graph properties. This allows many graph theory techniques to be put into play to push more what quantum advantages can be had. Examples include graphical characterisation of where information sits in secret sharing  and the application of random graph techniques for finding optimal codes .
Another consequence of their ubiquity in quantum information is that there has been a lot of effort to demonstrate them experimentally. Indeed they represent the cutting edge in what entangled states are prepared, with audacious experiments preparing graph states of thousands of qubits. Experiments routinely produce and control graph states of up to 10 qubits in different media in optics, atomics and ions.
There are now several groups across Europe and the world working on exploring graph state quantum information processing. In a series of works with the groups of Mark Tame (Durban, South Africa) and John Rarity (Bristol, UK) we have been pushing the quantum Lego aspect, in particular. In one experiment we demonstrated a graph state protocol, which combined three different protocols to enable verified secret sharing . This flexibility here proved crucial – by combining protocols we get functionality that is better than could be achieved by any single protocol in isolation. But there are many exciting things still to do, and we’re still figuring out how graph states can be used, in several directions to push the limits of quantum information. For example, graph states have proven a fertile test space to understand the resources of non-locality and contextuality and their role in many quantum advantages. The graphical notion of flow of information also lends itself to the study of exciting new directions in quantum information where the inherent ambiguity in causal order has been shown to be yet another source of quantum advantage. Recently, we have also seen that graph states are resources for the generation of quantum randomness, an almost generic resource in quantum information and key in our understanding in much of physics.
 D. Markham, B. C. Sanders: “Graph states for quantum secret sharing”, Physical Review A 78.4 (2008): 042309.
 J. Javelle, M. Mehdi, S. Perdrix: “New Protocols and Lower Bounds for Quantum Secret Sharing with Graph States”, TQC 12 (2012): 1-12.
 B. A. Bell, et al.: “Experimental demonstration of a graph state quantum error-correction code”, Nature communications, 5, 2014
CNRS, LIP6, Sorbonne Université