Sharif Digital Repository

Matrix Product States (MPS) is an analytical formalism to construct solvable many-body quan-tum models, which is based on the concepts of quantum information and computation theory. In this dissertation, We briefly review the history of matrix product states. We follow these states from quantum Markovian states(QMS) and valence bond solid states(VBS) to finitely correlated states(FCS) and matrix product states(MPS). We then present our method of study-ing and working with MP states and we introduce and study in detail the 1D and quasi-1D solvable spin models that we have been able to successfully construct in this frame work and calculate the corresponding correlations and quantities. These... 

One of the problems that we are deal with in quantum information is secure transition of information. In Quantum State Sharing, we can encode our data on the quantum state and share the quantum state among some people such that none of the can be able to retrieve the state without collaboration with others. In this thesis, we will be concerned with reviewing of classical and quantum secret sharing and finally introduce another method in order to share a two qubit secret between N parties such any of the members can retrieve the state only with collaboration with other parties. It will be shown that using only Bell Pairs as recourse and measurement basis makes this model more efficient than... 

Quantum random walk is a computational model in quantum computation which is as powerful as other models like quantum circuit model. One dimensional random walks can be implemented in the laboratory by using a two-level quantum coin (e.g. the two states of a photon). For implementing higher dimensional random walks, one should simulate quantum coins with higher number of levels. This is difficult to implement experimentally. Various proposals try to bypass this problem, like the proposal of alternate walks in [C. DiFranco et al., Phys. Rev. Lett. 106, 080502(2011)]. Here we suggest an alternate solution: We use the bi-partite structure of some lattices to effectively act as a two-level... 

One of the problems facing quantum computation is errors due to interaction with the environment which destroy coherence of quantum states. Most schemes to design a quantum computer therefore focus on finding ways to minimize the interactions of the qubits with the environment. Constructing such systems with large numbers of qubits which are infallible is a hard task and far from being achieved in the near future. There is another quantum computational model which is called topological quantum computation, proposing a different solution. Qubits of this model are quasiparticles of a 2-dimensional topologically ordered system that are called Anyons. In this model gates are nonabelian... 

Entanglement plays a major role in quantum communication as the main resource of the network and makes it possible to perform protocols such as quantum teleportation and quantum cryptography which are not achievable in classical networks. Yet, being a very fragile resource against noise, which inevitably is more destructive as the distance increases and as time passes, makes it a very hard to share entangled pairs between two points at a long distance. This discouraging result for linear networks, has prompted investigation of entanglement percolation and entanglement distribution in regular one- and two-dimensional networks with various (i.e rectangular, triangular and hexagonal)... 

Quantum mechanics provides a deep understanding of atoms and their interaction with light. As long as we consider only microscopic systems on the scale of an atomic radius, objections to quantum properties such as quantum superposition are nevertheless rare, mainly because of the overwhelming experimental evidence. When it comes to macroscopic systems, many things are not clear anymore. For example,everyday objects of macroscopic size do not exist in superposition of their different states. The reason is that a quantum system interacts with its environment locally,which destroys non-local quantum correlation within the system, larger objects interact with the environment more intensively and... 

Quantum mechanics, because of its special structure, on one hand offers us possibilities for doing un-classical tasks and on the other hand imposes limitations on performing some quantum tasks. We know these limitations as no-go theorems in Quantum mechanics. Studying these no-go theorems and properties which can be used as Quantum resources, has a wide variety of applications in Quantum information and Quantum computation, also they can lead us to a deeper understanding of the Quantum mechanics theory.One of these non-classical properties is Superposition. Quantum superposition is both a source for other Quantum properties like entanglement, and a Quantum resource for Quantum tasks. But... 

Two qubit density matrices which are of X-shape, are a natural generalization of Bell Diagonal States recently simulated on the IBM quantum device. We propose a quantum circuit for simulation of a general X-state on the same quantum device and study its properties for several values of the extended parameter space. We also show by specific measurements, that their X-shape is robust against noisy quantum gates. To further physically motivate this study, we invoke the two-spin Heisenberg XYZ system and show that for a wide class of initial states, it leads to dynamical density matrices which are X-states. Due to the symmetries of this Hamiltonian, we show that by only two qubits, one can... 

One of the main challenges in the communication and storage of information is its evolution due to the environmental noises. There are various techniques in coding theory to decrease the probability of changing the message using redundancy. Among all of the possible codings, the maximum rate that can be achieved for reliable communication through a channel is called the capacity of that channel. In the classical information domain, this quantity has been investigated in the Shannon's second theorem and is equivalent to a finite convex optimization problem. This problem is much more complicated when dealing with quantum information and thus studying special common channels has been doubled in... 

One of the most popular topics in quantum information is the transfer of quantum states by the evolution of spin chains. To transfer quantum information one must transfer an arbitrary quantum state from one point to another in a spin network. The common idea for this is to use the coupling strengths of adjacent spins and apply a specific local magnetic field to the spin chains. In many of these chains, the state transfer is not done complete and the fidelity between the final state and the initial state is less than one. Recently, chains with a special Hamiltonian have been designed by which a desired state can be placed on one side of the chain and after a specific time, the state can be... 

The interest in the Landau-Streater channel has been mainly due to its abstract mathematical properties. We show that in three dimensions and with a slight modification, this channel can be realized as a rotation of qutrit states in random directions by random angles. Therefore and because of the potential use of qutrits in quan- tum processing tasks and their realization in many different platforms, the modified Landau-Streater channel can be used as a very simple and realistic noise model, in the same way that the depolarizing channel is for qubits. We will make a detailed study of this channel and derive its various properties. In particular, we will use the recently proposed flag... 

In this thesis, we have studied some exact solutions for low dimensional strongly correlated systems, especially spin chains. In the first chapter using simple AKLT model and introducing Valence Bond Solid States we have prepared good area to study Finitely Correlated States. In addition to presenting relation between these concepts, we have expressed Matrix Product States as a simpler formalism to deal with Finitely Correlated States. In the second chapter we have discussed the Optimum Ground State concept and we have studied a spin-32 with nearest neighbor interaction and a spin-1 model with next nearest neighbor interaction
as example  

One of the serious challenges we are facing in making quantum computers is the quantum error correction. Whereas system interact with the environment, quantum states are easily inf luenced and are suf fering f rom noise and error. T his makes the act of quantum computing be disrupted.T heory of quantum error correction has emerged for facing such a serious challenge. In this theory, many ways that are combined with several branches of mathematics were invited that are essentially dif ferent to classical error correction. one of these is stabilizer codes that show a rich structure and simple f rom classical error correction codes and in addition many quantum error correction codes can be... 

In this thesis, we investigate the problem of quantum estimation in open quantum sys- tems. To this end, after a review of classical and quantum estimation theory, and the role of different properties of the quantum systems in enhancing estimation of the closed systems, we will study different characteristics of the open systems. Non-Markovianity of a dynamics is a quantum property which is a special feature of open systems dynamics. It is expected that this property plays a role in the precession of the estimation. Since there is no universal measure for distinguishing Markovian and non-Markovian dynamics, we first use quantum discord between the system and the environment as a measure for... 

antum computers are necessary to simulate quantum systems. e fragility of qubits in presence of decoherence and external noise is the biggest obstacle in realizing a quantum computer. To overcome su problems, topological quantum computation has been introduced by Kitaev that combines the main quantum feature of the quantum world, namely, superposition of states, with the robustness of classical bits whi is the result of a macroscopic number of very small entities, comprising ea bit. In this way, topological features whi are robust against local perturbations are used for storing information. Kitaev’s model is a famous many-body model, whi has topologically degenerate ground states,... 

one of the important challenges for implementation of Quantum Computation is fabrication of Quantum Wires that transport the Quantum Information from one part of the Quantum Hardware to the other parts. Heisenberg Spin Chains are promising candidate for this. In article [28] Gaussian wave packets on the spin chains are used to encode the logical qubits. In this model, to implement unitary operation on these qubits, extra interaction are designed in some parts of the spin chains. As Gaussian wave packets move along this interactions, the desired unitary operators act on them. In fact, this is an static model without external control to perform Quantum Computation. In spite of many interesting... 

One of the most important problems of quantum information theory is to try to determine the channel capacity of noisy quantum channels. Alice would like to send Bob information over many uses of a noisy quantum communication link. As the line is usually noisy, this cannot be done perfectly, and so they must use suitable coding methods to minimize the error. Different coding methods yield different rates of information transition and different errors, the channel capacity is defined as the maximum rate at which information may be transferred with vanishing error in the limit of a large number of channel uses. There are a variety of different capacities, depending upon whether Alice and Bob... 

Providing an Algorithm whi? can solve the sear? problem using minimum resources(su? as time, memory and so on)has been ?allenged by computer science experts for many years. By discovering the quantum computer it can be asked that does the quantum algorithm provide a be?er solution to sear? problem? ?e answer is yes. In this thesis the grover’s algorithm, the quantum sear? algorithm using the quantum walk, probabilistic quantum sear? algorithm and finally the two-qubit grover sear? algorithm implementation in physical world will be discussed  

Since the inception of the fields of quantum information and computation, the task of coherently transferring quantum states, through long and short distances has been of utmost importance. While photons are the ideal carriers of quantum information over long distances, it has become evident that the best possible method for transferring quantum information over short distances, i.e. through regular arrays of qubits, is to exploit the natural dynamics of the many body system. This idea was first introduced in the work of Bose [2] who showed that the natural dynamics of a Heisenberg ferromagnetic chain can achieve high-fidelity transfer of spin states over distances as long as 80 lattice... 

Ising model. It means that the partition function of each lattice model is equal to the partition function of 2D Ising model. This completeness is proved by using concepts and techniques from quantum information theory and is based on the universality of cluster states. We have now proved this important result, by using independent and general concepts and methods which are accessible to a wide audience of researchers across many disciplines. Furthermore, our method has the advantage of eing algorithmic in nature so that, by following a set of simple graphical transformations, one is able to transform any discrete lattice model to an Ising model defined on a (polynomially) larger 2D...