Quantum Factor Graphs

The natural Hilbert Space of quantum particles can implement maximum-likelihood (ML) decoding of classical information. The 'Quantum Product Algorithm' (QPA) is computed on a Factor Graph, where function nodes are unitary matrix operations followed by appropriate quantum measurement. QPA is like the Sum-Product Algorithm (SPA), but without summary, giving optimal decode with exponentially finer detail than achievable using SPA. Graph cycles have no effect on QPA performance. QPA must be repeated a number of times before successful and the ML codeword is obtained only after repeated quantum 'experiments'. ML amplification improves decoding accuracy, and Distributed QPA facilitates successful evolution.Comment: Minor modifications. 24 pages, Latex, 14 figures, Presented in part at 2nd Int. Symp. on Turbo Codes and Related Topics, Brest, France, Sept 4-7, 2000 Accepted for publication in "Annals of Telecom." 200

Generalised Bent Criteria for Boolean Functions (I)

Generalisations of the bent property of a boolean function are presented, by proposing spectral analysis with respect to a well-chosen set of local unitary transforms. Quadratic boolean functions are related to simple graphs and it is shown that the orbit generated by successive Local Complementations on a graph can be found within the transform spectra under investigation. The flat spectra of a quadratic boolean function are related to modified versions of its associated adjacency matrix.Comment: 29 pages, submitted to IEEE Trans. Inform Theor

A complementary construction using mutually unbiased bases

We present a construction for complementary pairs of arrays that exploits a set of mutually-unbiased bases, and enumerate these arrays as well as the corresponding set of complementary sequences obtained from the arrays by projection. We also sketch an algorithm to uniquely generate these sequences. The pairwise squared inner-product of members of the sequence set is shown to be $\frac{1}{2}$. Moreover, a subset of the set can be viewed as a codebook that asymptotically achieves $\sqrt{\frac{3}{2}}$ times the Welch bound.Comment: 25 pages, 1 figur

Edge Local Complementation and Equivalence of Binary Linear Codes

Orbits of graphs under the operation edge local complementation (ELC) are defined. We show that the ELC orbit of a bipartite graph corresponds to the equivalence class of a binary linear code. The information sets and the minimum distance of a code can be derived from the corresponding ELC orbit. By extending earlier results on local complementation (LC) orbits, we classify the ELC orbits of all graphs on up to 12 vertices. We also give a new method for classifying binary linear codes, with running time comparable to the best known algorithm.Comment: Presented at International Workshop on Coding and Cryptography (WCC 2007), 16-20 Apr. 2007, Versailles, France. (12 pages, 3 figures

Mixed graph states

We have generalised the concept of graph states to what we have called mixed graph states, which we define in terms of mixed graphs, that is graphs with both directed and undirected edges, as the density matrix stabilized by the associated stabilizer matrix defined by the mixed graph. We can interpret this matrix as a quantum object by making it part of a larger fully commuting matrix, i.e. where we choose the environment appropriately, and this will imply that our quantum object is a mixed state. We prove that, in the same way as (pure) graph states, the density matrix of a parent of mixed graph state can be written as sum of a few Pauli matrices, well defined from the mixed graph. We have proven that the set of matrices that appear in this sum is fully pair-wise commuting, and form a multiplicative group up to global constants, which is always of maximum size. Furthermore, the cardinality of the set depends solely of the miminum possible number of extension columns/rows, and the number of nodes of the mixed graph. We prove a formula for this cardinality. Finally, in the case of purely undirected graphs, this corresponds to the usual pure graph state. Also, we have developed a way of finding maximal commutative group of such Pauli matrices as a linear subspace problem, for any given mixed graph. We also have proven how the structure of maximal commutative groups is independent of the direction of the arrows of the mixed graph, and also of the undirected edges; this allows the simplification of the problem of finding these groups in general to finding them for a much smaller set of graphs

Generalised Bent Criteria for Boolean Functions (II)

In the first part of this paper [16], some results on how to compute the flat spectra of Boolean constructions w.r.t. the transforms {I,H}^n, {H,N}^n and {I,H,N}^n were presented, and the relevance of Local Complementation to the quadratic case was indicated. In this second part, the results are applied to develop recursive formulae for the numbers of flat spectra of some structural quadratics. Observations are made as to the generalised Bent properties of boolean functions of algebraic degree greater than two, and the number of flat spectra w.r.t. {I,H,N}^n are computed for some of them.Comment: 18 pages, submitted to IEEE Trans. Inform. Theor

From Graph States to Two-Graph States

The name graph state is used to describe a certain class of pure quantum state which models a physical structure on which one can perform measurement-based quantum computing, and which has a natural graphical description. We present the two-graph state, this being a generalisation of the graph state and a two-graph representation of a stabilizer state. Mathematically, the two-graph state can be viewed as a simultaneous generalisation of a binary linear code and quadratic Boolean function. It describes precisely the coefficients of the pure quantum state vector resulting from the action of a member of the local Clifford group on a graph state, and comprises a graph which encodes the magnitude properties of the state, and a graph encoding its phase properties. This description facilitates a computationally efficient spectral analysis of the graph state with respect to operations from the local Clifford group on the state, as all operations can be realised graphically. By focusing on the so-called local transform group, which is a size 3 cyclic subgroup of the local Clifford group over one qubit, and over $n$ qubits is of size $3^n$, we can efficiently compute spectral properties of the graph state

On Pivot Orbits of Boolean Functions

We derive a spectral interpretation of the pivot operation on a graph and generalise this operation to hypergraphs. We establish lower bounds on the number of flat spectra of a Boolean function, depending on internal structures, with respect to the {I,H}^n and {I,H,N}^n sets of transforms. We also construct a family of Boolean functions of degree higher than two with a large number of flat spectra with respect to {I,H}^n, and compute a lower bound on this number. The relationship between pivot orbits and equivalence classes of error-correcting codes is then highlighted. Finally, an enumeration of pivot orbits of various types of graphs is given, and it is shown that the same technique can be used to classify codes.Comment: 1 figure, 20 page

Device-independent quantum key distribution based on measurement inputs

We provide an analysis of a new family of device independent quantum key distribution (QKD) protocols with several novel features: (a) The bits used for the secret key do not come from the results of the measurements on an entangled state but from the choices of settings; (b) Instead of a single security parameter (a violation of some Bell inequality) a set of them is used to estimate the level of trust in the secrecy of the key. The main advantage of these protocols is a smaller vulnerability to imperfect random number generators made possible by feature (a). We prove the security and the robustness of such protocols. We show that using our method it is possible to construct a QKD protocol which retains its security even if the source of randomness used by communicating parties is strongly biased. As a proof of principle, an explicit example of a protocol based on the Hardy's paradox is presented. Moreover, in the noiseless case, the protocol is secure in a natural way against any type of memory attack, and thus allows to reuse the device in subsequent rounds. We also analyse the robustness of the protocol using semi-definite programming methods. Finally, we present a post-processing method, and observe a paradoxical property that rejecting some random part of the private data can increase the key rate of the protocol.Comment: 10 pages, 5 figure: In this modified version of the manuscript we have added a new section to show fact that our protocol is much better than the standard ones when the random number generators used by the parties are imperfec

From vortices to solitonic vortices in trapped atomic Bose-Einstein condensates

Motivated by recent experiments we study theoretically the dynamics of vortices in the crossover from two to one-dimension in atomic condensates in elongated traps. We explore the transition from the dynamics of a vortex to that of a dark soliton as the one-dimensional limit is approached, mapping this transition out as a function of the key system parameters. Moreover, we probe this transition dynamically through the hysteresis under time-dependent deformation of the trap at the dimensionality crossover. When the solitonic regime is probed during the hysteresis, significant angular momentum is lost from the system but, remarkably, the vortex can re-emerge.Comment: 16 pages, 4 figure