Section: New Results
Quantum Computing

On Weak Odd Domination and Graphbased Quantum Secret Sharing. In this work published in the journal Theoretical Computer Science [15] , Simon Perdrix and his coauthors Sylvain Gravier, Jérôme Javelle and Mehdi Mhalla study weak odd domination in graphs and its application in quantum secret sharing. A weak odd dominated (WOD) set in a graph is a subset B of vertices for which there exists a distinct set of vertices C such that every vertex in B has an odd number of neighbors in C. They point out the connections of weak odd domination with odd domination, [σ,ρ]domination, and perfect codes. They introduce bounds on κ(G), the maximum size of WOD sets of a graph G, and on κ′(G), the minimum size of nonWOD sets of G. Moreover, they prove that the corresponding decision problems are NPcomplete. The study of weak odd domination is mainly motivated by the design of graphbased quantum secret sharing protocols: a graph G of order n corresponds to a secret sharing protocol whose threshold is κQ(G)=max(κ(G),n−κ′(G)). These graphbased protocols are very promising in terms of physical implementation, however all such graphbased protocols studied in the literature have quasiunanimity thresholds (i.e. κQ(G)=n−o(n) where n is the order of the graph G underlying the protocol). In this paper, they show using probabilistic methods the existence of graphs with smaller κQ (i.e. κQ(G)≤0.811n where n is the order of G). They also prove that deciding for a given graph G whether κQ(G)≤k is NPcomplete, which means that one cannot efficiently double check that a graph randomly generated has actually a κQ smaller than 0.811n.

Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms. In this work presented at ISAAC [25] , David Cattaneo and Simon Perdrix introduce new upper bounds and exact algorithms for the local minimum degree. The author also prove the W[2]membership of the corresponding decision problem. The local minimum degree of a graph is the minimum degree that can be reached by means of local complementation. For any n, there exist graphs of order n which have a local minimum degree at least $0.189n$, or at least $0.110n$ when restricted to bipartite graphs. Regarding the upper bound, they show that the local minimum degree is at most $3/8n+o\left(n\right)$ for general graphs and $n/4+o\left(n\right)$ for bipartite graphs, improving the known $n/2$ upper bound. They also prove that the local minimum degree is smaller than half of the vertex cover number (up to a logarithmic term). The local minimum degree problem is NPComplete and hard to approximate. They show that this problem, even when restricted to bipartite graphs, is in W[2] and FPTequivalent to the EvenSet problem, whose W[1]hardness is a long standing open question. Finally, they show that the local minimum degree is computed by a $O\ast (1.938n)$algorithm, and a $O\ast (1.466n)$algorithm for the bipartite graphs.

The ZX Calculus is incomplete for Clifford+T quantum mechanics. The ZX calculus is a diagrammatic language for quantum mechanics and quantum information processing. In this paper[17] , Simon Perdrix and Harny Wang prove that the ZXcalculus is not complete for the Clifford+T quantum mechanics. The completeness for this fragment has been stated as one of the main current open problems in categorical quantum mechanics. The ZX calculus was known to be incomplete for quantum mechanics, on the other hand, it has been proved complete for Clifford quantum mechanics (a.k.a. stabilizer quantum mechanics), and for singlequbit Clifford+T quantum mechanics. The question of the completeness of the ZX calculus for Clifford+T is a crucial step in the development of the ZX calculus because of its (approximate) universality for quantum mechanics (i.e. any unitary evolution can be approximated using Clifford and T gates only). They exhibit a property which is know to be true in Clifford+T quantum mechanics and prove that this equation cannot be derived in the ZX calculus, by introducing a new sound interpretation of the ZX calculus in which this particular property does not hold. Finally, we propose to extend the language with a new axiom. This result has been presented as invited speakers in the conferences "Quantum Theory: from foundations to technologies" in Vaxjo Sweden, and "Higher TQFT and categorical quantum mechanics” at the Scrounger Institute in Vienna. The authors also presented these results at the workshop of the CNRS groupe de travail Informatique Quantique du GDR IM, in Grenoble.

Block Representation of Reversible Causal Graph Dynamics. In this work presented at the conference on Foundation of computer science (FCT’15) [18] , Pablo Arrighi, Simon Martiel and Simon Perdrix, consider a reversible version of the causal graph dynamics. Causal Graph Dynamics extend Cellular Automata to arbitrary, boundeddegree, timevarying graphs. The whole graph evolves in discrete time steps, and this global evolution is required to have a number of physicslike symmetries: shiftinvariance (it acts everywhere the same) and causality (information has a bounded speed of propagation). We study a further physicslike symmetry, namely reversibility. More precisely, we show that Reversible Causal Graph Dynamics can be represented as finitedepth circuits of local reversible gates.

Reversibility in the Extended Measurementbased Quantum Computation. In this work by Nidal Hamrit and Simon Perdrix has been presented at the conference on Reversible Computation in Grenoble [23] . When applied on some particular quantum entangled states, measurements are universal for quantum computing. In particular, despite the fondamental probabilistic evolution of quantum measurements, any unitary evolution can be simulated by a measurementbased quantum computer (MBQC). They consider the extended version of the MBQC where each measurement can occur not only in the X,Yplane of the Bloch sphere but also in the X,Z and Y,Zplanes. The existence of a gflow in the underlying graph of the computation is a necessary and sufficient condition for a certain kind of determinism. They extend the focused gflow (a gflow in a particular normal form) defined for the X,Yplane to the extended case, and provide necessary and sufficient conditions for the existence of such normal forms.

Quantum Circuits for the Unitary Permutation Problem. In this paper presented at TAMC’15 [20] Stefano Facchni and Simon Perdrix consider the Unitary Permutation problem which consists, given $n$ quantum gates ${U}_{1},...,{U}_{n}$ and a permutation $\sigma $ of $\{1,...,n\}$, in applying the quantum gates in the order specified by $\sigma $, i.e., in performing ${U}_{\sigma \left(n\right)}\circ ...\circ {U}_{\sigma \left(1\right)}$. This problem has been introduced and investigated in [47] where two models of computations are considered. The first is the (standard) model of query complexity: the complexity measure is the number of calls to any of the quantum gates ${U}_{i}$ in a quantum circuit which solves the problem. The second model is roughly speaking a model for higher order quantum computation, where quantum gates can be treated as objects of second order. In both model the existing bounds are improved, in particular the upper and lower bounds for the standard quantum circuit model are established by pointing out connections with the permutation as substring problem introduced by Karp.