A weak odd dominated (WOD) set in a graph is a subset B of vertices for which there exists a dist... more 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 size of WOD sets of a graph G, and on \kappa'(G), the minimum size of non WOD sets of G. Moreover, we prove that the corresponding decision problems are NP-complete. The study of weak odd domination is mainly motivated by the design of graph-based quantum secret sharing protocols: a graph G of order n corresponds to a secret sharing protocol which threshold is \kappa_Q(G) = max(\kappa(G), n-\kappa'(G)). These graph-based protocols are very promising in terms of physical implementation, however all such graph-based protocols studied in the literature have quasi-unanimity thresholds (i.e. \kappa_Q(G)=n-o(n) where n is the order of the graph G underlying the protocol). In this paper, we show using probabilistic methods, the existence of graphs with smaller \kappa_Q (i.e. \kappa_Q(G)< 0.811n where n is the order of G). We also prove that deciding for a given graph G whether \kappa_Q(G)< k is NP-complete, which means that one cannot efficiently double check that a graph randomly generated has actually a \kappa_Q smaller than 0.811n.
Quantum measurement is universal for quantum computation. Two models for performing measurement-b... more Quantum measurement is universal for quantum computation. Two models for performing measurement-based quantum computation exist: the one-way quantum computater was introduced by Briegel and Raussendorf and quantum computation via projective measurements only by Nielsen. The more recent development of this second model is based on state transfers instead of teleportation. From this development a finite but approximate quantum universal family of observables is exhibited which includes only one two-qubit observable while others are one-qubit observables. In this article an infinite but exact quantum universal family of observables is proposed including also only one two-qubit observable. The rest of the paper is dedicated to compare these two models of measurement-based quantum computation i.e. one-way quantum computation and quantum computation via projective measurements only. From this comparison which was initiated by Cirac and Verstraete closer and more natural connections appear between these two models. These close connections lead to a unified view of measurement-based quantum computation.
We examine the relationship between the algebraic lambda-calculus, a fragment of the differential... more 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 is closed under linear combinations. However, the two languages were built using different approaches: the former is a call-by-name language whereas the latter is call-by-value; the former considers algebraic equalities whereas the latter approaches them through rewrite rules. In this paper, we analyse how these different approaches relate to one another. To this end, we propose four canonical languages based on each of the possible choices: call-by-name versus call-by-value, algebraic equality versus algebraic rewriting. We show that the various languages simulate one another. Due to subtle interaction between beta-reduction and algebraic rewriting, to make the languages consistent some additional hypotheses such as confluence or normalisation might be required. We carefully devise the required properties for each proof, making them general enough to be valid for any sub-language satisfying the corresponding properties.
Graph states are an elegant and powerful quantum resource for measurement based quantum computati... more 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.
We introduce a new family of quantum secret sharing protocols with limited quantum resources whic... more 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 on the qubits in A; (ii) use a classical encoding to ensure the existence of a threshold. These new protocols realize ((k,n)) quantum secret sharing i.e., any set of at least k players among n can reconstruct the quantum secret, whereas any set of less than k players has no information about the secret. In the particular case where the secret is encoded on all the qubits, we explore the values of k for which there exists a graph such that the corresponding protocol realizes a ((k,n)) secret sharing. We show that for any threshold k> n-n^{0.68} there exists a graph allowing a ((k,n)) protocol. On the other hand, we prove that for any k< 79n/156 there is no graph G allowing a ((k,n)) protocol. As a consequence there exists n_0 such that the protocols introduced by Markham and Sanders admit no threshold k when the secret is encoded on all the qubits and n>n_0.
We study a simple graph-based classical secret sharing scheme: every player's share consists of a... more 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 sharing (GS-QSS) schemes when the secret is classical. When the secret is an arbitrary quantum state, a set of players is authorised for a GS-QSS scheme if and only if, for the corresponding simple classical graph-based protocol, the set is authorised and its complement set is not.
To prove W[2]-membership of the domination-type problems we extend the Turing-way to parameterize... more To prove W[2]-membership of the domination-type problems we extend the Turing-way to parameterized complexity by introducing a new kind of non deterministic Turing machine with the ability to perform `blind' transitions, i.e. transitions which do not depend on the content of the tapes. We prove that the corresponding problem Short Blind Multi-Tape Non-Deterministic Turing Machine is W[2]-complete. We believe that this new machine can be used to prove W[2]-membership of other problems, not necessarily related to domination
Proceedings of the 15th International Symposium on Static Analysis, Jan 28, 2008
Entanglement is a non local property of quantum states which has no classical counterpart and pla... more 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. The exact role of the entanglement is nevertheless not well understood. Since an exact analysis of entanglement evolution induces an exponential slowdown, we consider approximative analysis based on the framework of abstract interpretation. In this paper, a concrete quantum semantics based on superoperators is associated with a simple quantum programming language. The representation of entanglement, i.e. the design of the abstract domain is a key issue. A representation of entanglement as a partition of the memory is chosen. An abstract semantics is introduced, and the soundness of the approximation is proven.
An accessing set in a graph is a subset B of vertices such that there exists D subset of B, such ... more 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 perfect codes and give explicitly kappa(G) and kappa'(G) for several families of graphs. Finally, we show that the corresponding decision problems are NP-Complete.
Electronic Notes in Theoretical Computer Science, Mar 3, 2006
Quantum computations usually take place under the control of the classical world. We introduce a ... more Quantum computations usually take place under the control of the classical world. We introduce a Classically-controlled Quantum Turing Machine (CQTM) which is a Turing Machine (TM) with a quantum tape for acting on quantum data, and a classical transition function for a formalized classical control. In CQTM, unitary transformations and measurements are allowed. We show that any classical TM is simulated by a CQTM without loss of efficiency. The gap between classical and quantum computations, already pointed out in the framework of measurement-based quantum computation is confirmed. To appreciate the similarity of programming classical TM and CQTM, examples are given.
The graph state formalism offers strong connections between quantum information processing and gr... more 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 by triangular grids is a universal measurement-based model of quantum computation. These two results are in fact two sides of the same coin, the proof of which is a combination of graph theoretical and quantum information techniques.
Quantum measurement is universal for quantum computation. The model of quantum computation introd... more Quantum measurement is universal for quantum computation. The model of quantum computation introduced by Nielsen and further developed by Leung relies on a generalized form of teleportation. In order to simulate any n-qubit unitary transformation with this model, 4 auxiliary qubits are required. Moreover Leung exhibited a universal family of observables composed of 4 two-qubit measurements. We introduce a model
A weak odd dominated (WOD) set in a graph is a subset B of vertices for which there exists a dist... more 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 size of WOD sets of a graph G, and on \kappa'(G), the minimum size of non WOD sets of G. Moreover, we prove that the corresponding decision problems are NP-complete. The study of weak odd domination is mainly motivated by the design of graph-based quantum secret sharing protocols: a graph G of order n corresponds to a secret sharing protocol which threshold is \kappa_Q(G) = max(\kappa(G), n-\kappa'(G)). These graph-based protocols are very promising in terms of physical implementation, however all such graph-based protocols studied in the literature have quasi-unanimity thresholds (i.e. \kappa_Q(G)=n-o(n) where n is the order of the graph G underlying the protocol). In this paper, we show using probabilistic methods, the existence of graphs with smaller \kappa_Q (i.e. \kappa_Q(G)< 0.811n where n is the order of G). We also prove that deciding for a given graph G whether \kappa_Q(G)< k is NP-complete, which means that one cannot efficiently double check that a graph randomly generated has actually a \kappa_Q smaller than 0.811n.
Quantum measurement is universal for quantum computation. Two models for performing measurement-b... more Quantum measurement is universal for quantum computation. Two models for performing measurement-based quantum computation exist: the one-way quantum computater was introduced by Briegel and Raussendorf and quantum computation via projective measurements only by Nielsen. The more recent development of this second model is based on state transfers instead of teleportation. From this development a finite but approximate quantum universal family of observables is exhibited which includes only one two-qubit observable while others are one-qubit observables. In this article an infinite but exact quantum universal family of observables is proposed including also only one two-qubit observable. The rest of the paper is dedicated to compare these two models of measurement-based quantum computation i.e. one-way quantum computation and quantum computation via projective measurements only. From this comparison which was initiated by Cirac and Verstraete closer and more natural connections appear between these two models. These close connections lead to a unified view of measurement-based quantum computation.
We examine the relationship between the algebraic lambda-calculus, a fragment of the differential... more 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 is closed under linear combinations. However, the two languages were built using different approaches: the former is a call-by-name language whereas the latter is call-by-value; the former considers algebraic equalities whereas the latter approaches them through rewrite rules. In this paper, we analyse how these different approaches relate to one another. To this end, we propose four canonical languages based on each of the possible choices: call-by-name versus call-by-value, algebraic equality versus algebraic rewriting. We show that the various languages simulate one another. Due to subtle interaction between beta-reduction and algebraic rewriting, to make the languages consistent some additional hypotheses such as confluence or normalisation might be required. We carefully devise the required properties for each proof, making them general enough to be valid for any sub-language satisfying the corresponding properties.
Graph states are an elegant and powerful quantum resource for measurement based quantum computati... more 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.
We introduce a new family of quantum secret sharing protocols with limited quantum resources whic... more 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 on the qubits in A; (ii) use a classical encoding to ensure the existence of a threshold. These new protocols realize ((k,n)) quantum secret sharing i.e., any set of at least k players among n can reconstruct the quantum secret, whereas any set of less than k players has no information about the secret. In the particular case where the secret is encoded on all the qubits, we explore the values of k for which there exists a graph such that the corresponding protocol realizes a ((k,n)) secret sharing. We show that for any threshold k> n-n^{0.68} there exists a graph allowing a ((k,n)) protocol. On the other hand, we prove that for any k< 79n/156 there is no graph G allowing a ((k,n)) protocol. As a consequence there exists n_0 such that the protocols introduced by Markham and Sanders admit no threshold k when the secret is encoded on all the qubits and n>n_0.
We study a simple graph-based classical secret sharing scheme: every player's share consists of a... more 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 sharing (GS-QSS) schemes when the secret is classical. When the secret is an arbitrary quantum state, a set of players is authorised for a GS-QSS scheme if and only if, for the corresponding simple classical graph-based protocol, the set is authorised and its complement set is not.
To prove W[2]-membership of the domination-type problems we extend the Turing-way to parameterize... more To prove W[2]-membership of the domination-type problems we extend the Turing-way to parameterized complexity by introducing a new kind of non deterministic Turing machine with the ability to perform `blind' transitions, i.e. transitions which do not depend on the content of the tapes. We prove that the corresponding problem Short Blind Multi-Tape Non-Deterministic Turing Machine is W[2]-complete. We believe that this new machine can be used to prove W[2]-membership of other problems, not necessarily related to domination
Proceedings of the 15th International Symposium on Static Analysis, Jan 28, 2008
Entanglement is a non local property of quantum states which has no classical counterpart and pla... more 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. The exact role of the entanglement is nevertheless not well understood. Since an exact analysis of entanglement evolution induces an exponential slowdown, we consider approximative analysis based on the framework of abstract interpretation. In this paper, a concrete quantum semantics based on superoperators is associated with a simple quantum programming language. The representation of entanglement, i.e. the design of the abstract domain is a key issue. A representation of entanglement as a partition of the memory is chosen. An abstract semantics is introduced, and the soundness of the approximation is proven.
An accessing set in a graph is a subset B of vertices such that there exists D subset of B, such ... more 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 perfect codes and give explicitly kappa(G) and kappa'(G) for several families of graphs. Finally, we show that the corresponding decision problems are NP-Complete.
Electronic Notes in Theoretical Computer Science, Mar 3, 2006
Quantum computations usually take place under the control of the classical world. We introduce a ... more Quantum computations usually take place under the control of the classical world. We introduce a Classically-controlled Quantum Turing Machine (CQTM) which is a Turing Machine (TM) with a quantum tape for acting on quantum data, and a classical transition function for a formalized classical control. In CQTM, unitary transformations and measurements are allowed. We show that any classical TM is simulated by a CQTM without loss of efficiency. The gap between classical and quantum computations, already pointed out in the framework of measurement-based quantum computation is confirmed. To appreciate the similarity of programming classical TM and CQTM, examples are given.
The graph state formalism offers strong connections between quantum information processing and gr... more 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 by triangular grids is a universal measurement-based model of quantum computation. These two results are in fact two sides of the same coin, the proof of which is a combination of graph theoretical and quantum information techniques.
Quantum measurement is universal for quantum computation. The model of quantum computation introd... more Quantum measurement is universal for quantum computation. The model of quantum computation introduced by Nielsen and further developed by Leung relies on a generalized form of teleportation. In order to simulate any n-qubit unitary transformation with this model, 4 auxiliary qubits are required. Moreover Leung exhibited a universal family of observables composed of 4 two-qubit measurements. We introduce a model
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