Abstract Given a unitary operator in a finite dimensional complex vector space, its unitary reduction to a subspace is defined. The application to quantum graphs is discussed. It is shown how the reduction allows to generate the scattering matrices of new quantum graphs from assembling of simpler graphs. The reduction of quantum channels is also defined. The implementation of the quantum gates corresponding to the reduced unitary operator is investigated, although no explicit construction is presented. The situation is different for the reduction of quantum channels for which explicit implementations are given.
Abstract Given a unitary operator in a finite dimensional complex vector space, its unitary reduction to a subspace is defined. The application to quantum graphs is discussed. It is shown how the reduction allows to generate the scattering matrices of new quantum graphs from assembling of simpler graphs. The reduction of quantum channels is also defined. The implementation of the quantum gates corresponding to the reduced unitary operator is investigated, although no explicit construction is presented. The situation is different for the reduction of quantum channels for which explicit implementations are given.
Maps of the form Phi(X) =sum_{i=1}^s A_iXA^*, where A_1, . . . ,A_s are fixed complex n by n matrices and X is any complex n by n matrix are used in quantum information theory as representations of quantum channels. This article deals with computable conditions for the existence of decoherence--free subspaces for Phi. Since the definition of decoherence-free subspace for quantum channels relies only on the matrices A1, . . . ,As, the term of common reducing unitary subspace is used instead of the original one. Among the main results of the paper, there are computable conditions for the existence of common eigenvectors. These are related to common reducing unitary subspaces of dimension one. The new results on common eigenvectors provide new effective condition for the existence of common invariant subspaces of arbitrary dimensions.
Pure states of three-qubit quantum systems are represented by eight-dimensional complex vectors, whereas their unitary transformations are described by 8×8 complex matrices. Motivated by this correspondence, we extend our algebraic model for two-qubit systems to eight-dimensional spaces endowed with commutative and associative multiplication. To each eight-dimensional number we assign a matrix and prove that the resulting correspondence is a homomorphic bijection. We derive conditions under which the associated matrix is unitary and hence represents a three-qubit quantum gate. This leads to a specific algebraic notion of a unitary state: a normalized state whose multiplication matrix is unitary. We construct 16,384 basis spaces; within each fixed space, multiplication generates a commuting family of quantum gates. For any two unitary states in the class considered, we give an explicit transition-gate construction within one of these algebraic families. For real components, the unitary states in a representative basis space form a union of thirty-two four-dimensional manifolds in R8. The construction therefore yields new structured classes of three-qubit unitary operators and invariant state sets, while its extension to larger multiqubit systems remains open.
Pure states of two-qubit quantum systems are described by a four-dimensional vector of complex numbers, and unitary operators transferring a two-qubit quantum system from one state to another have the form of a 4 × 4 matrix with complex elements. This fact brings to mind the idea of studying the spaces of four-dimensional numbers with complex components. Moreover, the results obtained by the authors for four-dimensional numbers with real components inspire some optimism. In this paper we construct four-dimensional spaces of complex numbers by analogy with four-dimensional spaces of real numbers. Each four-dimensional number is mapped to a matrix formed from its components and it is proved that the constructed mapping is a bijection and a homomorphism. In the space of fourdimensional numbers, of the eight basis elements, half are real and half are imaginary. The presence of such symmetry distinguish these spaces from the space of quaternions, in which one basis element is real and the rest are imaginary. The symmetry of the basis numbers makes these spaces a natural generalization of one-dimensional and two-dimensional (complex) algebra. The conditions under which the corresponding matrices are gates for two-qubit quantum systems are defined. The notion of unitary state of a two-qubit quantum system is introduced, to which various gates from commutative groups of gates correspond. It is shown that any gate of a unitary state transforms a unitary state into a unitary state and a non-unitary state into a non-unitary state. Almost all gates used in the construction of quantum circuits, in particular H, SWAP, CX, CY, CZ, have the same properties. The problem of searching for a gate that transfers a quantum system from one unitary state to another unitary state has been solved. Thus, with the help of fourdimensional spaces of complex numbers it was possible to construct whole classes of two-qubit gates, which opens new possibilities for the construction of quantum algorithms. The results obtained have important theoretical and practical implications for quantum computing.
A new approach to quantum walks is presented. Considering a quantum system undergoing some unitary discrete-time evolution in a directed graph G, we think of the vertices of G as sites that are occupied by the quantum system, whose internal state is described by density operators. To formulate the unitary evolution, we define reflections in the tensor product of an internal Hilbert space and a spatial Hilbert space. We then construct unitary channels that govern the evolution of the system in the graph. The discrete dynamics of the system (called quantum walks) is obtained by iterating the unitary channel on the density operator of the quantum system. It turns out that in this framework, the action of the unitary channel on a density operator is described by the usual matrix multiplication.
A special class of quantum channels, named subspace local (SL), are defined and investigated. The proposed definition of subspace locality of quantum channels is an attempt to answer the question of what kind of restriction should be put on a channel, if it is to act `locally' with respect to two `locations', when these naturally correspond to a separation of the total Hilbert space in an orthogonal sum of subspaces, rather than a tensor product decomposition. It is shown that the set of SL channels decomposes into four disjoint families of channels. Explicit expressions to generate all channels in each family is presented. It is shown that one of these four families, the local subspace preserving (LSP) channels, is precisely the intersection between the set of subspace preserving channels and the SL channels. For a subclass of the LSP channels, a special type of unitary representation using ancilla systems is presented.
In this paper, we consider a class of Unitary Quantum Walks on arbitrary graphs, parameterized by a family of scattering matrices. These Scattering Quantum Walks model the discrete dynamics of a system on the graph’s edges, where a scattering process at the vertices is governed by the scattering matrices assigned to each vertex. We show the Scattering Quantum Walks encompass several known Quantum Walks. We further introduce two classes of Scattering Open Quantum Walks on arbitrary graphs, also parameterized by scattering matrices: one defined on the edges, the other on the vertices of the graph. We show these walks give rise to proper quantum channels and describe their main spectral and dynamical properties, relating them to naturally associated classical Markov chains.