# Simon Perdrix's research while affiliated with University of Lorraine and other places

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## Publications (51)

Causal Graph Dynamics extend Cellular Automata to arbitrary time-varying graphs of bounded degree. The whole graph evolves in discrete time steps, and this global evolution is required to have a number of symmetries: shift-invariance (it acts everywhere the same) and causality (information has a bounded speed of propagation). We add a further physi...

Inductive datatypes in programming languages allow users to define useful data structures such as natural numbers, lists, trees, and others. In this paper we show how inductive datatypes may be added to the quantum programming language QPL. We construct a sound categorical model for the language and by doing so we provide the first detailed semanti...

Analyzing pseudo-telepathy graph games, we propose 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 multipartite...

Causal Graph Dynamics extend Cellular Automata to arbitrary, bounded-degree, time-varying graphs. The whole graph evolves in discrete time steps, and this global evolution is required to have a number of physics-like symmetries: shift-invariance (it acts everywhere the same) and causality (information has a bounded speed of propagation). We study a...

The stabilizer ZX-calculus is a rigorous graphical language for reasoning about stabilizer quantum mechanics. This language has been proved to be complete in two steps: first in a setting where scalars (diagrams with no inputs or outputs) are ignored and then in a more general setting where a new symbol and three additional rules have been added to...

Si l’ordinateur quantique universel est encore lointain, les physiciens espèrent fabriquer bientôt un ordinateur quantique imparfait, capable de simuler des systèmes quantiques. En jeu, la compréhension de systèmes inaccessibles par les moyens habituels.

The ZX calculus is a diagrammatic language for quantum mechanics and quantum
information processing. We prove that the ZX-calculus is not complete for
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...

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, we show that for any
graph of order n, its local minimum degre...

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 measurement-based quantum computer (MBQC). We consider the
extended version of the MBQC where each measurement...

Causal Graph Dynamics extend Cellular Automata to arbitrary, bounded-degree, time-varying graphs. The whole graph evolves in discrete time steps, and this global evolution is required to have a number of physics-like symmetries: shift-invariance (it acts everywhere the same) and causality (information has a bounded speed of propagation). We study a...

We examine the relationship between the algebraic {\lambda}-calculus, a fragment of the differential {\lambda}-calculus; and the linear-algebraic {\lambda}-calculus, a candidate {\lambda}-calculus for quantum computation. Both calculi are algebraic: each one is equipped with an additive and a scalar-multiplicative structure, and their set of terms...

We consider the Unitary Permutation problem which consists, given \(n\) unitary gates \(U_1, \ldots , U_n\) and a permutation \(\sigma \) of \(\{1,\ldots , n\}\), in applying the unitary gates in the order specified by \(\sigma \), i.e. in performing \(U_{\sigma (n)}\circ \ldots \circ U_{\sigma (1)}\).
This problem has been introduced and investiga...

We show that pivoting property of graph states cannot be derived from the
axioms of the ZX-calculus, and that pivoting does not imply local
complementation of graph states. Therefore the ZX-calculus augmented with
pivoting is strictly weaker than the calculus augmented with the Euler
decomposition of the Hadamard gate. We derive an angle-free versi...

We give graphical characterisation of the access structure to both classical
and quantum information encoded onto a multigraph defined for prime dimension
$q$, as well as explicit decoding operations for quantum secret sharing based
on graph state protocols. We give a lower bound on $k$ for the existence of a
$((k,n))_q$ scheme and prove, using pro...

We study the graph-state-based quantum secret sharing protocols [24,17] which are not only very promising in terms of physical implementation, but also resource efficient since every player’s share is composed of a single qubit. The threshold of a graph-state-based protocol admits a lower bound: for any graph of order n, the threshold of the corres...

We study the parameterized complexity of domination-type problems. (σ,ρ)-domination is a general and unifying framework introduced by Telle: given σ, ρ ⊆ ℕ, a set D of vertices of a graph G is (σ,ρ)-dominating if for any v ∈ D, |N(v) ∩ D| ∈ σ and for any v ∉ D, |N(v) ∩ D| ∈ ρ. Our main result is that for any σ and ρ recursive sets, deciding whether...

The algebraic lambda calculus and the linear algebraic lambda calculus are
two extensions of the classical lambda calculus with linear combinations of
terms. They arise independently in distinct contexts: the former is a fragment
of the differential lambda calculus, the latter is a candidate lambda calculus
for quantum computation. They differ in t...

Measurement-based quantum computation (MBQC) has emerged as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model that takes unitary operations as fundamental. Among measurement-based quantum computation methods the recently introdu...

A weak odd dominated (WOD) set is a set B of vertices for which there exists
a subset C of V\B such that every vertex in B has an odd number of neighbors in
C. Given a graph G of order n, \kappa(G) denotes the size of the greatest WOD
set, and \kappa'(G) the size of the smallest non-WOD set. The maximum of
\kappa(G) and n-\kappa'(G), denoted \kappa...

The local minimum degree of a graph is the minimum degree reached by means of a series of local complementations. In this paper, we investigate on this quantity which plays an important role in quantum computation and quantum error correcting codes.
First, we show that the local minimum degree of the Paley graph of order p is greater than \(\sqrt{p...

The graph state formalism offers strong connections between quantum
information processing and graph theory. Exploring these connections, first we
show that any graph is a pivot-minor of a planar graph, and even a pivot minor
of a triangular grid. Then, we prove that the application of measurements in
the (X,Z)-plane over graph states represented b...

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. We point out the connections of weak odd
domination with odd domination, [sigma,rho]-domination, and perfect codes. We
introduce bounds on \kappa(G), the maximum siz...

An accessing set in a graph is a subset B of vertices such that there exists
D subset of B, such that each vertex of V\B has an even number of neighbors in
D. In this paper, we introduce new bounds on the minimal size kappa'(G) of an
accessing set, and on the maximal size kappa(G) of a non-accessing set of a
graph G. We show strong connections with...

We study a simple graph-based classical secret sharing scheme: every player's
share consists of a random key together with the encryption of the secret with
the keys of his neighbours. A characterisation of the authorised and forbidden
sets of players is given. Moreover, we show that this protocol is equivalent to
the graph state quantum secret sha...

We introduce a new family of quantum secret sharing protocols with limited
quantum resources which extends the protocols proposed by Markham and Sanders
and by Broadbent, Chouha, and Tapp. Parametrized by a graph G and a subset of
its vertices A, the protocol consists in: (i) encoding the quantum secret into
the corresponding graph state by acting...

The quantum Turing machine (QTM) has been introduced by Deutsch as an abstract model of quantum computation. The transition function of a QTM is linear, and has to be unitary to be a well-formed QTM. This well-formedness condition ensures that the evolution of the machine does not violate the postulates of quantum mechanics. How- ever, we claim in...

We study the observation of quantum Turing machines by allowing interactions between a quantum machine and its environment during the computation, whereas a quantum Turing machine -original model introduced by Deutsch- remains isolated. We show that the introduction of observations leads to a weakening of the well formedness conditions of quantum T...

We examine the relationship between the algebraic lambda-calculus Lalg, a fragment of the differential lambda-calculus, and the linear-algebraic lambda-calculus Llin, a candidate lambda-calculus for quantum computation. Both calculi are algebraic: each one is equipped with an additive and a scalar-multiplicative structure, and the set of terms is c...

We examine the relationship between the algebraic lambda-calculus λalg, a fragment of the differential lambda-calculus, and the linear-algebraic lambda-calculus λlin, a candidate lambda-calculus for quantum computation. Both calculi are algebraic: each one is equipped with an additive and a scalar-multiplicative structure, and the set of terms is c...

We present a method for verifying measurement-based quantum computations, by producing a quantum circuit equivalent to a given
deterministic measurement pattern. We define a diagrammatic presentation of the pattern, and produce a circuit via a rewriting
strategy based on the generalised flow of the pattern. Unlike other methods for translating meas...

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 exis...

We present a both simple and comprehensive graphical calculus for quantum
computing. In particular, we axiomatize the notion of an environment, which
together with the earlier introduced axiomatic notion of classical structure
enables us to define classical channels, quantum measurements and classical
control. If we moreover adjoin the earlier intr...

The entangled graph states have emerged as an elegant and powerful quantum resource, indeed almost all multiparty protocols can be written in terms of graph states including measurement based quantum computation (MBQC), error correction and secret sharing amongst others. In addition they are at the forefront in terms of implementations. As such the...

We prove that one-way quantum computations have the same computational power as quantum circuits with unbounded fan-out. It demonstrates that the one-way model is not only one of the most promising models of physical realisation, but also a very powerful model of quantum computation. It confirms and completes previous results which have pointed out...

Coecke and Duncan recently introduced a categorical formalisation of the
interaction of complementary quantum observables. In this paper we use their
diagrammatic language to study graph states, a computationally interesting
class of quantum states. We give a graphical proof of the fixpoint property of
graph states. We then introduce a new equation...

Several domains [S. Abramsky, Tributes 1, 1–18 (2005; Zbl 1279.03073); B. Coecke and K. Martin, Lect. Notes Phys. 813, 493–683 (2011; Zbl 1253.81008); P. Selinger, Math. Struct. Comput. Sci. 14, No. 4, 527–586 (2004; Zbl 1085.68014)] can be used to define the semantics of quantum programs. Among them Abramsky [loc. cit.] has introduced a semantics...

This paper contains two new results:(i)We amend the notion of abstract basis in a dagger symmetric monoidal category, as well as its corresponding graphical representation, in order to accommodate non-self-dual dagger compact structures; this is crucial for obtaining a planar diagrammatical representation of the induced dagger compact structure as...

Entanglement is a non local property of quantum states which has no classical counterpart and plays a decisive role in quantum information theory. Several protocols, like the teleportation, are based on quantum entangled states. Moreover, any quantum algorithm which does not create entanglement can be efficiently simulated on a classical computer....

Among the models of quantum computation, the One-way Quantum Computer [11,12] is one of the most promising proposals of physical realisation [13], and opens new perspectives for parallelisation by taking advantage of quantum entanglement [2]. Since a One-way QC is based on quantum measurement, which is a fundamentally nondeterministic evolution, a...

We extend the notion of quantum information flow defined by Danos and Kashefi for the one-way model and present a necessary and sufficient condition for the deterministic computation in this model. The generalized flow also applied in the extended model with measurements in the X-Y, X-Z and Y-Z planes. We apply both measurement calculus and the sta...

We improve the upper bound on the minimal resources required for measurement-only quantum computation (M A Nielsen 2003 Phys. Rev. A 308 96-100; D W Leung 2004 Int. J. Quantum Inform. 2 33; S Perdrix 2005 Int. J. Quantum Inform. 3 219-23). Minimizing the resources required for this model is a key issue for experimental realization of a quantum comp...

As a first step toward a notion of quantum data structures, we introduce a typing system for reflecting entanglement and separability. This is presented in the context of classically controlled quantum computation where a classical program controls a sequence of quantum operations, i.e. unitary transformations and measurements acting on a quantum m...

Graph states have become a key class of states within quantum computation. They form a basis for universal quantum computation,
capture key properties of entanglement, are related to quantum error correction, establish links to graph theory, violate
Bell inequalities, and have elegant and short graph-theoretical descriptions. We give here a rigorou...

Prix de thèse INP Grenoble 2008

The graph state formalism is a useful abstraction of entanglement. It is used in some multipartite purification schemes and it adequately represents universal resources for measurement-only quantum computation. We focus in this paper on the complexity of graph state preparation. We consider the number of ancillary qubits, the size of the primitive...

Since quantum measurement is universal for quantum computation (Nielsen [3]), the minimization of the resources required for measurement-based quantum computation is a crucial point ([3, 1, 4, 5, 6]). We have shown in [6] that a set of observables composed of one two-qubit measurement and three one-qubit measurements form a universal set for quantu...

L’étude des structures fondamentales du traitement de l’information quantique est un défi majeur, dont l’un des objectifs est de mieux cerner les capacités et les limites de l’ordinateur quantique, tout en contribuant à sa réalisation physique notamment en s’intéressant aux ressources du calcul quantique. Les ressources d’un calcul quantique inclue...

## Citations

... Still, vertex names can be cumbersome. In the classical regime, and in a variety of different early formalisms, it was shown that their presence leads to vertex-preservation, i.e. the forbidding of vertex creation/destruction [10] . This was a somewhat uncomfortable situation, because the informally defined model of Hassalcher and Meyer [29] did seem to feature reversibility, vertex creation/destruction, and non-signalling causality. ...

Reference: Quantum networks theory

... This journal paper is based upon two conference proceedings (Arrighi et al. 2016(Arrighi et al. , 2015. It organized as follows. ...

... Finally, the set of axioms we provide here is probably not minimal. It would be nice to see if a simplified version can be obtained, as was done in [BPW17] for the qubit case. Proof. ...

... This journal paper is based upon two conference proceedings (Arrighi et al. 2016(Arrighi et al. , 2015. It organized as follows. ...

... Several domains for quantum computation have been introduced [12,1,14]. Among them, the domain of superoperators over density matrices, introduced by Selinger [18] turns out to be one of the most adapted to quantum semantics. ...

... This right distributivity can alternatively be seen as the one of function sum: (f + g)(x) is defined pointwise as f(x) + g(x). This is the approach of the algebraic lambda-calculi [3,26], two independently introduced algebraic extensions which resulted strongly related afterwards [4,15]. In these algebraic calculi, a scalar pondering each 'choice' is considered in addition to the sum of terms. ...

Reference: Non determinism through type isomophism

... Graph theory may quite properly be regarded as an area of applied mathematics [28] and has many applications in quantum information [29][30][31][32][33]. For example, it has been applied in qubit information systems of extremal black branes [29], no-signalling assisted zero-error capacity of quantum channels [30] and quantum secret sharing [31]. ...

... For example, as Perdrix and Wang recently showed, making the zx-calculus complete for multi-qubit Clifford+T group requires the addition of at least one new rule [60], the supplementarity rule first introduced in [24] and given here in correctly scaled form: ...

Reference: Completeness and the ZX-calculus

... Suiteà l'introduction de ces notions essentielles, le reste de ce chapitre sera consacré aux nouveaux résultats obtenus et publiés dans [13]. ...

... In particular, we adapt the definitions of focused gflow [44] and maximally delayed gflow [42] to the extended gflow case. Our generalisation of focused gflow differs from the three generalisations suggested by Hamrit and Perdrix [29]; indeed, the desired applications naturally lead us to one unique generalisation (see Section 3.3). ...