In this paper, we propose a new way to represent graphs in quantum space. In that approach, we replace the rows of the adjacency matrix of the graph by state vectors in the occupation number representation. Unlike the traditional definition of graph states, we actually let the occupation number of a single-particle state denote the number of edges between each two adjacent vertices. This allows us to avoid taking into account the interaction between each two particles. Based on the creation and annihilation operators, we propose the edge creation and annihilation operators. With these two operators, we can implement the fundamental operation of adding and removing edges and vertices in a graph. Then all additional operations in the graph such as vertex contractions can be defined. Our method can be used to represent both simple and multigraphs. Directed and undirected graphs are also compatible with our approach. The method of representation proposed in this paper enriches the theory of graph representation in quantum space.
In this paper, we propose a new way to represent graphs in quantum space. In that approach, we replace the rows of the adjacency matrix of the graph by state vectors in the occupation number representation. Unlike the traditional definition of graph states, we actually let the occupation number of a single-particle state denote the number of edges between each two adjacent vertices. This allows us to avoid taking into account the interaction between each two particles. Based on the creation and annihilation operators, we propose the edge creation and annihilation operators. With these two operators, we can implement the fundamental operation of adding and removing edges and vertices in a graph. Then all additional operations in the graph such as vertex contractions can be defined. Our method can be used to represent both simple and multigraphs. Directed and undirected graphs are also compatible with our approach. The method of representation proposed in this paper enriches the theory of graph representation in quantum space.
The expansion of single- particle transition operator( defined as the product of creation and annihilation operator,namely a+iajin single- particle state is obtained through the form of single- operator in occupation particle number representation,and the commutation relations for transition operator and creation( annihilation) operator are also given,then based on these conclusions,the form of general two- and n- particle operators in occupation particle number representation can be derived succinctly without distinguishing the statistical property of identical particle system.
A review of the second quantization formulation of quantum mechanics is given based on the idea of a graded Hilbert space whose elements are represented by column matrices containing zero, one, two, ···, particle functions. The creation and annihilation operators are defined in terms of intuitive ideas about adding a particle to or removing a particle from the system. The (anti)commutation relations are shown to follow from the (anti)symmetry of the wavefunctions. The field operators which describe the creation or annihilation of a particle at a point are then obtained and their properties discussed. These operators are then used to express general operators in second quantization. Finally, a basis is introduced and the equations of occupation-number space are obtained.
Many quantum communication protocols involve multiple participants.With participants increasing, selecting designated legal participants in a nice way becomes more and more significant.In this paper, we develop a quantum route selection approach to the design of participant selection.Graph state is a special type of multi-particle entangled quantum state that can be represented by mathematical graph, where each vertex denotes a qubit and each edge denotes an Ising interaction.Motivated by the characteristics of graph state, we propose that each participant holds a vertex of graph state so that all legal participants are selected through a series of operations on specific vertices of graph state.
Abstract Focusing on systems of many identical particles, Chapter 2 introduces appropriate operators to describe their properties in terms of Schwinger’s “measurement symbols.” The latter are then factorized into “creation” and “annihilation” operators, whose fundamental properties and commutation/anticommutation relations are derived in conjunction with the Pauli exclusion principle. This leads to “second quantization” with the Hamiltonian, number, linear and angular momentum operators expressed in terms of the annihilation and creation operators, as well as the occupation number representation. Finally, the concept of coherent states, as eigenstates of the annihilation operator, having minimum uncertainty, is introduced and discussed in detail.
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.
In application domains such as routing, network analysis, scheduling, and planning, directed graphs are widely used as both formal models and core data structures for the development of efficient algorithmic solutions. In these areas, graphs are often evolving in time: for example, connection links may fail due to temporary technical issues, meaning that edges of the graph cannot be traversed for some time interval and alternative paths have to be followed. In classical computation graphs have been implemented both explicitly through adjacency matrices/lists and symbolically as ordered binary decision diagrams. Moreover, ad-hoc visit procedures have been developed to deal with dynamically evolving graphs. Quantum computation, exploiting interference and entanglement, has provided an exponential speed-up for specific problems, e.g., database search and integer factorization. In the quantum framework everything must be represented and manipulated using reversible operators. This poses a challenge when one has to deal with traversals of dynamically evolving directed graphs. Graph traversals are not intrinsically reversible because of converging paths. In the case of dynamically evolving graphs also the creation/destruction of paths comes into play against reversibility. In this paper we propose a novel high level graph representation in quantum computation supporting dynamic connectivity typical of real-world network applications. Our procedure allows to encode any multigraph into a unitary matrix. We devise algorithms for computing the encoding that are optimal in terms of time and space and we show the effectiveness of the proposal with some examples. We describe how to react to edge/node failures in constant time. Furthermore, we present two methods to perform quantum random walks taking advantage of this encoding: with and without projectors. We implement and test our encoding obtaining that the theoretical bounds for the running time are confirmed by the empirical results and providing more details on the behavior of the algorithms over graphs of different densities.