Section: New Results

Quantum Computing

Participants : Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart.

• ZX-calculus

  The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language – i.e. the ability to derive any true equation – is a crucial question. In the quest for a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning [73]. Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles. In [22], we introduce a generalised supplementarity – called cyclotomic supplementarity – which consists in merging $n$ subdiagrams at once, when the $n$ angles divide the circle into equal parts. We show that when $n$ is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning. We exhibit another new simple axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity. This work has been presented at MFCS 2017 [22].

  The ZX-Calculus is devoted to represent complex quantum evolutions. But the advantages of quantum computing still exist when working with rebits, and evolutions with real coefficients. Some models explicitly use rebits, but the ZX-Calculus cannot handle these evolutions as it is. Hence, in [21], we define an alternative language solely dealing with real matrices, with a new set of rules. We show that three of its non-trivial rules are not derivable from the other ones and we prove that the language is complete for the $\pi$/2-fragment. We define a generalisation of the Hadamard node, and exhibit two interpretations from and to the ZX-Calculus, showing the consistency between the two languages. This work has been presented at QPL 2017 [21].

• Causality and Quantum Computing

  Since the classic no-go theorems by [43] and [65], contextuality has gained great importance in the development of quantum information and computation. This key characteristic feature of quantum mechanics represents one of the most valuable resources at our disposal to break through the limits of classical computation and information processing, with various concrete application

  An important class of contextuality arguments in quantum foundations are the All-versus-Nothing (AvN) proofs, generalising a construction originally due to Mermin. In [11], we present a general formulation of All-versus-Nothing arguments, and a complete characterisation of all such arguments which arise from stabiliser states. We show that every AvN argument for an n-qubit stabiliser state can be reduced to an AvN proof for a three-qubit state which is local Clifford-equivalent to the tripartite GHZ state. This is achieved through a combinatorial characterisation of AvN arguments, the AvN triple Theorem, whose proof makes use of the theory of graph states. This result enables the development of a computational method to generate all the AvN arguments in Z2 on n-qubit stabiliser states. We also present new insights into the stabiliser formalism and its connections with logic. This work has been presented at QPL 2017 [25] and published in the Philosophical Transactions of the Royal Society A [11].

  Analyzing pseudo-telepathy graph games, we propose in [15] a way to build contextuality scenarios exhibiting the quantum supremacy using graph states. We consider the combinatorial structures generating equivalent scenarios. We investigate which scenarios are more multipartite and show that there exist graphs generating scenarios with a linear multipartiteness width. This work has been presented at FCT 2017 [15].

• Measurement-based Quantum Computing

  Measurement-based quantum computing (MBQC) is a universal model for quantum computation [74]. The combinatorial characterisation of determinism in this model [51], [48], [69], powered by measurements, and hence, fundamentally probabilistic, is the cornerstone of most of the breakthrough results in this field. The most general known sufficient condition for a deterministic MBQC to be driven is that the underlying graph of the computation has a particular kind of flow called Pauli flow. The necessity of the Pauli flow was an open question [48]. In [23], we show that the Pauli flow is necessary for real-MBQC, and not in general providing counterexamples for (complex) MBQC. We explore the consequences of this result for real MBQC and its applications. Real MBQC and more generally real quantum computing is known to be universal for quantum computing. Real MBQC has been used for interactive proofs by McKague. The two-prover case corresponds to real-MBQC on bipartite graphs. While (complex) MBQC on bipartite graphs are universal, the universality of real MBQC on bipartite graphs was an open question. We show that real bipartite MBQC is not universal proving that all measurements of real bipartite MBQC can be parallelised leading to constant depth computations. As a consequence, McKague techniques cannot lead to two-prover interactive proofs. This work has been presented at FCT 2017 [23].