Utvecklandet av kvantdatorn är ett ytterst lovande projekt som kombinerar teoretisk och experimental kvantfysik, matematik, teori om kvantinformation och datalogi. Under första steget i utvecklandet av kvantdatorn låg huvudintresset på att skapa några algoritmer med framtida tillämpningar, klargöra grundläggande frågor och utveckla en experimentell teknologi för en leksakskvantdator som verkar på några kvantbitar. Då dominerade förväntningarna om snabba framsteg bland kvantforskare. Men det verkar som om dessa stora förväntningar inte har besannats helt. Många grundläggande och tekniska problem som dekoherens hos kvantbitarna och instabilitet i kvantstrukturen skapar redan vid ett litet antal register tvivel om en snabb utveckling av kvantdatorer som verkligen fungerar. Trots detta kan man inte förneka att stora framsteg gjorts inom kvantteknologin. Det råder givetvis ett stort gap mellan skapandet av en leksakskvantdator med 10-15 kvantregister och att t.ex. tillgodose de tekniska förutsättningarna för det projekt på 100 kvantregister som aviserades för några år sen i USA. Det är också uppenbart att svårigheterna ökar ickelinjärt med ökningen av antalet register. Därför är simulering av kvantdatorer i klassiska datorer en viktig del av kvantdatorprojektet. Självklart kan man inte förvänta sig att en kvantalgoritm skall lösa ett NP-problem i polynomisk tid i en klassisk dator. Detta är heller inte syftet med klassisk simulering. Den klassiska simuleringen av kvantdatorer kommer att täcka en del av gapet mellan den teoretiskt matematiska formuleringen av kvantmekaniken och ett förverkligande av en kvantdator. Ett av de viktigaste problemen i vetenskapen om kvantdatorn är att utveckla ett nytt symboliskt språk för kvantdatorerna och att anpassa redan existerande symboliska språk för klassiska datorer till kvantalgoritmer. Denna avhandling ägnas åt en anpassning av det symboliska språket Mathematica till kända kvantalgoritmer och motsvarande simulering i klassiska datorer. Konkret kommer vi att representera Simons algoritm, Deutsch-Joszas algoritm, Grovers algoritm, Shors algoritm och kvantfelrättande koder i det symboliska språket Mathematica. Vi använder samma stomme i alla dessa algoritmer. Denna stomme representerar de karaktäristiska egenskaperna i det symboliska språkets framställning av kvantdatorn och det är enkelt att inkludera denna stomme i framtida algoritmer. / Quantum computing is an extremely promising project combining theoretical and experimental quantum physics, mathematics, quantum information theory and computer science. At the first stage of development of quantum computing the main attention was paid to creating a few algorithms which might have applications in the future, clarifying fundamental questions and developing experimental technologies for toy quantum computers operating with a few quantum bits. At that time expectations of quick progress in the quantum computing project dominated in the quantum community. However, it seems that such high expectations were not totally justified. Numerous fundamental and technological problems such as the decoherence of quantum bits and the instability of quantum structures even with a small number of registers led to doubts about a quick development of really working quantum computers. Although it can not be denied that great progress had been made in quantum technologies, it is clear that there is still a huge gap between the creation of toy quantum computers with 10-15 quantum registers and, e.g., satisfying the technical conditions of the project of 100 quantum registers announced a few years ago in the USA. It is also evident that difficulties increase nonlinearly with an increasing number of registers. Therefore the simulation of quantum computations on classical computers became an important part of the quantum computing project. Of course, it can not be expected that quantum algorithms would help to solve NP problems for polynomial time on classical computers. However, this is not at all the aim of classical simulation. Classical simulation of quantum computations will cover part of the gap between the theoretical mathematical formulation of quantum mechanics and the realization of quantum computers. One of the most important problems in "quantum computer science" is the development of new symbolic languages for quantum computing and the adaptation of existing symbolic languages for classical computing to quantum algorithms. The present thesis is devoted to the adaptation of the Mathematica symbolic language to known quantum algorithms and corresponding simulation on the classical computer. Concretely we shall represent in the Mathematica symbolic language Simon's algorithm, the Deutsch-Josza algorithm, Grover's algorithm, Shor's algorithm and quantum error-correcting codes. We shall see that the same framework can be used for all these algorithms. This framework will contain the characteristic property of the symbolic language representation of quantum computing and it will be a straightforward matter to include this framework in future algorithms.

A distributed quantum computer holds the potential to emulate a larger quantumcomputer by being partitioned it into smaller modules where local operations (LO)can be applied, and classical communication (CC) can be utilized between thesemodules. Finding algorithms under LOCC restrictions is crucial for leveraging thecapabilities of distributed quantum computing, This thesis explores probabilisticexact LOCC supermaps, that maps 2-qubit bipartite unitary operations to its inver-sion and complex conjugation. Presented are LOCC unitary inversion and complexconjugation supermaps that use 3 calls of the operation, achieving success proba-bilities of 3/128 and 3/8, respectively. These supermaps are discovered through anexamination of the Kraus Cirac decomposition and its interaction with single qubitunitary inversion supermaps. These results can be used for time reversal of as welland noise reduction in closed distributed quantum systems / En distribuerad kvantdator har potentialen att emulera en större kvantdator genom att delas upp i mindre moduler, där lokala operations (LO) kan appliceras och klassisk kommunikation (CC) användas. För att effektivt kunna använda algoritmer på en distribuerad kvantdator måste de anpassas för LOCC restriktioner. Denna avhandling studerar probabilistiskt exakta LOCC superavbildningar, somavbildar 2-qubits bipartita unitära operationer till deras invers och komplexkonjugat. I avhandlingen presenters en LOCC unitär inversion- samt en komplexkonjugatsuperavbildning vilka använder 3 anrop av operationen och lyckas med sannolikhet 3/128 respektive 3/8. Dessa superavbildningar hittades genom att studera Kraus Cirac-uppdelningen och dess interaktion med 1-qubits inversionsuperavbildningar. Förhoppningsvis kan dessa resultat användas till att invertera tiden samt brusreducering på distribuerade kvantsystem.

This thesis focuses on two aspects of Quantum Information Processing (QIP) and contains experimental implementation by Nuclear Magnetic Resonance (NMR) spectroscopy. The two aspects are: (i) development of novel methodologies for improved or fault tolerant QIP using longer lived states and geometric phases and (ii) implementation of certain quantum algorithms and theorems by NMR. In the first chapter a general introduction to Quantum Information Processing and its implementation using NMR as well as a description of NMR Hamiltonians and NMR relaxation using Redfield theory and magnetization modes are given. The second chapter contains a study of relaxation of Pseudo Pure States (PPS). PPS are specially prepared initial states from where computation begins. These states, being non-equilibrium states, relax with time and hence introduce error in computation. In this chapter we have studied the role of Cross-Correlations in relaxation of PPS. The third and fourth chapters, respectively report observation of cyclic and non-cyclic geometric phases. When the state of a qubit is subjected to evolution either adiabatically or non-adiabatically along the surface of the Bloch sphere, the qubit sometimes gain a phase factor apart from the dynamic phase. This is known as the Geometric phase, as it depends only on the geometry of the path of evolution. Geometric phase is used in Fault tolerant QIP. In these two chapters we have demonstrated how geometric phases of a qubit can be measured using NMR. The fifth and sixth chapters contain the implementations of “No Deletion” and “No Cloning” (quantum triplicator for partially known states) theorems. No Cloning and No Deletion theorems are closely related. The former states that an unknown quantum states can not be copied perfectly while the later states that an unknown state can not be deleted perfectly either. In these two chapters we have discussed about experimental implementation of the two theorems. The last chapter contains implementation of “Deutsch-Jozsa” algorithm in strongly dipolar coupled spin systems. Dipolar couplings being larger than the scalar couplings provide better opportunity for scaling up to larger number of qubits. However, strongly coupled systems offer few experimental challenges as well. This chapter demonstrates how a strongly coupled system can be used in NMR QIP.

Integrated quantum photonics show monolithic waveguide chips to be a promising platform for realizing the next generation of quantum optical circuits. This work proposes the implementation of quantum page Rank algorithm on a photonic waveguide lattice. Our contributions are as follows: Continuous-time quantum stochastic walk(QSW)-an alternate paradigm of quantum computing, is a hybrid quantum walk that incorporates both unitary and non-unitary effects. We propose the use of QSW which necessitates the hopping of the quantum crawler on a directed graph, for the quantum page Rank problem. We propose the implementation of quantum page Rank on a photonic waveguide lattice, where we allow the density matrix to evolve according to the Lindblad-Kossakowski master equation, the diagonal of which gives the quantum page Rank. We have also shown the use of the metric of positional Kolmogorov Complexity as an efficient tool for determining whether or not the quantum channel has been compromised. We appositionally encode multi-photon decoy pulses within the stream of single photon pulses. This positional encoding is chosen in such a way as to have low Kolmogorov complexity. The PNS attack on the multi-photon decoy pulses causes a dip in the ratio of the transmittance of the decoy pulses to the signal pulses in the conventional analysis.