We analyze the basic aspects of the quantum information theory, namely the quantum version
of the theory that is behind all the classical implementations like computers and communications.
We focus on quantum computation, defining its fundamental unit: the qubit,
namely a quantum two-level system that will be described in detail in this thesis. We then
concentrate on the possible transformations gates that can be applied to qubits, both unitary
and non-unitary ones.
The challenges in the construction of qubits are impressive: in particular, it is difficult to
isolate the system from the environment. This interaction is modeled by non-unitary transformations.
The characteristic times connected to these processes are the relaxation time, the
time in which a state decays in another state, and the decoherence time, the time in which
quantum coherence is lost. These times are fundamental in quantum computing since they
quantify how many operations can be performed on qubits and still obtain reliable results.
We measure these characteristic times on the IBM quantum processor ’ibmq_16_melbourne’
performing three different experiments: one for the relaxation time and two for the decoherence
time, namely Ramsey and Echo experiments.
Finally, we implement a quantum algorithm, an algorithm that uses the resources of quantum
mechanics, like entanglement. We focus on finding a protocol to define the quantum
version of the elementary cellular automata, the quantum elementary cellular automata, and
run it on the IBM processor. They are dynamical systems defined on a lattice in which the
evolution is defined using simple local update rules.
Comparing the physical results with theoretical expectations and noise-affected simulations
we show the performance of the IBM processor.
We analyze the basic aspects of the quantum information theory, namely the quantum version
of the theory that is behind all the classical implementations like computers and communications.
We focus on quantum computation, defining its fundamental unit: the qubit,
namely a quantum two-level system that will be described in detail in this thesis. We then
concentrate on the possible transformations gates that can be applied to qubits, both unitary
and non-unitary ones.
The challenges in the construction of qubits are impressive: in particular, it is difficult to
isolate the system from the environment. This interaction is modeled by non-unitary transformations.
The characteristic times connected to these processes are the relaxation time, the
time in which a state decays in another state, and the decoherence time, the time in which
quantum coherence is lost. These times are fundamental in quantum computing since they
quantify how many operations can be performed on qubits and still obtain reliable results.
We measure these characteristic times on the IBM quantum processor ’ibmq_16_melbourne’
performing three different experiments: one for the relaxation time and two for the decoherence
time, namely Ramsey and Echo experiments.
Finally, we implement a quantum algorithm, an algorithm that uses the resources of quantum
mechanics, like entanglement. We focus on finding a protocol to define the quantum
version of the elementary cellular automata, the quantum elementary cellular automata, and
run it on the IBM processor. They are dynamical systems defined on a lattice in which the
evolution is defined using simple local update rules.
Comparing the physical results with theoretical expectations and noise-affected simulations
we show the performance of the IBM processor.
In quantum computers a unitary transformation is applied to a given initial state of a set of qubits through several quantum gates. Tests were conducted using Grover's algorithm and implemented in a variety of different ways on the publically available IBM quantum computers. Essentially, quantum computers manipulate quantum states instead of classical bits. Quantum information can be represented by the elementary units called quantum bits abbreviated as qubits or qbits. This chapter discusses the fundamental properties of quantum systems that are relevant to information processing. Quantum computation needs something much more than that quantum parallelism to be useful. This is achieved in Deutsch's algorithm which combines quantum parallelism with interference. One can simulate a classical circuit using a quantum circuit. That is, it is possible to perform classical computations on quantum computers. But the advantage of quantum computing is that powerful functions may be computed by making use of quantum parallelism, the fundamental feature of many quantum algorithms.
Quantum mechanics has proven its position as the primary theory for accurately describing microscopic domains. Numerous fields in physics utilize quantum mechanics to great effects, with one prominent application being quantum computing. Quantum computing extends the field of quantum mechanics to information theory, striving to perform computational tasks utilizing quantum phenomena with notably greater efficiency compared to classical alternatives. In this thesis, I will go through the basics required in the field of quantum computing. To understand quantum computing demands adapting a new type of matrix mechanical view. This involves considering Hilbert spaces, which are linear vector space having a well defined inner product. Hilbert spaces contain state vectors describing quantum states, as well as operators for altering and measuring these states. The dynamics of quantum systems are governed by three quantum postulates. These specify Hilbert spaces and state vectors as the fundamental components of quantum systems, Hermitian operators as providing the physical manifestation of these systems, and finally, the Schrödinger's equation to describe the time evolution of such systems. These notions are required in producing quantum computing. Quantum computing uses qubits as the computational basis, analogous to classical bits but with the added capability of quantum superposition. Qubits are represented as two-state vectors, and can be physically realized with any quantum-behaving binary-valued system, like the electron spins. Qubits can be combined into higher dimensional states with the use of tensor products, causing the system's Hilbert space to expand as 2^n with n being the number of qubits, allowing for exponential computational speedup. Computational logic can be constructed by acting on the qubits using unitary operators, known as quantum gates. The combinations of qubits and quantum gates can be visualized as quantum circuits, depicted using specialized circuit diagram notations. An essential part of quantum computing, and the phenomena enabling exponential computational speedup, is quantum entanglement. Entanglement arises in interacting qubits, intertwining them so that the individual states become correlated. Entanglement can be characterized by being unable to reduce a combined state into its pure constituents. The amount of entanglement in a system can be measured using von Neumann entropy, analogous to Shannon entropy in classical information theory. Entanglement serves various purposes in quantum computing and is a defining feature distinguishing it from the classical domain. In this thesis, the entanglement properties of a periodic four-qubit circuit with alternating quantum gates were simulated. The simulation was performed computationally, and measured entanglement entropy as a function of the amount of circuit periods. The results demonstrated different characteristics of entanglement depending on the parameters used for the quantum gates, with the entanglement either being periodic, non-occurring or chaotic in nature.
The last two decades have seen an enormous increase in the computational power of digital computers. This was due to the rapid technical development in manufacturing processes and controlling semiconducting structures on submicron scale. Concurrently, the electric circuits have encountered the first signs of the realm of quantum mechanics. Those effects may induce noise and thus they are typically considered harmful. However, the manipulation of the coherent quantum states might turn out be the basis of powerful computers – quantum computers. There, the computation is encoded into the unitary time evolution of a quantum mechanical state vector. Eventually, quantum mechanics could enable one, for example, to read secret electronic messages which are encrypted by the widely employed RSA cryptosystem – a task which is extremely laborious for the current digital computers. This thesis presents a theoretical study of the coherent manipulations of pure quantum states in a quantum register, that is, quantum algorithms. An implementation of a quantum algorithm involves the initialization of the input state and its manipulation with quantum gates followed by the measurements. The physical implementation of each gate requires that it is decomposed into low-level gates whose physical realizations are explicitly known. Here, the problem is examined from two directions. Firstly, the numerical optimization scheme for controlling time-evolution of a closed quantum system is discussed. This yields a method for implementing quantum gates acting on up to three quantum bits, qubits. The approach is independent of the physical realization of the quantum computer, but it is considered explicitly for a proposed inductively coupled Josephson charge qubit register. Secondly, the techniques of numerical matrix computation are utilized to find a general method for decomposing an arbitrary n-qubit gate into a sequence of elementary gates, which act on one or two qubits. The results of this thesis help to improve the implementation of quantum algorithms. The quantum circuit construction developed in the thesis is the first one to achieve the asymptotically minimal complexity in the number of elementary gates. In context of acceleration of quantum algorithms we present a gate-level study of Shor's algorithm and show how to accelerate the algorithm by merging several elementary gates into multiqubit gates. Finally, the requirements set by the resulting gate array are compared to the properties of superconducting qubits. This allows us to discuss the feasibility of the Josephson charge qubit register, for instance, as hardware for breaking the RSA cryptosystem.
This paper proposes a novel approach to quantum algorithm design by leveraging the principles of cellular automata (CA). The core idea is to utilize CA as a hardware abstraction layer for quantum computation, mimicking the evolution rules of qubits. This allows for a parallelized design and optimization of quantum algorithms. Specifically, quantum gates are mapped to CA rules, enabling the simulation of qubit superposition and entanglement through cellular interactions and information transfer. This work explores the theoretical framework and outlines a methodology for designing quantum algorithms using CA, focusing on the potential for enhanced computational efficiency and scalability. The design methodology includes defining the CA grid, encoding quantum states into cellular configurations, translating quantum gates into CA rule updates, and analyzing the resulting algorithm execution. The presented framework offers a fundamentally different perspective on quantum algorithm development, potentially bridging the gap between classical algorithmic design and the unique characteristics of quantum systems. Further research will focus on exploring specific quantum algorithms and evaluating the performance of the CA-based approach.
Quantum computation is a part of quantum information theory. By using quantum mechanical phenomena it is possible to speed up the solving of some problems with quantum algorithms over classical probabilistic algorithms. In some cases quantum algorithms can utilise quantum parallelism in order to speed up problem solving. In this thesis experiments on IBM quantum computers are compared with ideal error free simulations. In the first part quantum teleportation is shown in practice, similarly the performance of Deutsch–Jozsa, Bernstein–Vazirani, Simon’s and Grover’s search algorithms are shown in action on IBM quantum computers and each is compared against an ideal simulation. The last experiment shows how the binary success probability of an unsharp measurement using Lüders instrument can be improved by adding sequential measurements by using an ideal simulation and a simulation using simple noise model.
Complexity is an intuitively recognized feature of nature, but where does it come from? Is complexity only apparent at the classical level, or can it be found at the underlying quantum level as well? We approach these questions by exploring models consistent with basic quantum theory but which also hold promise for exhibiting complex behaviors – a set of models we call the quantum elementary cellular automata (QECA). Previously, various quantum cellular automata models have been studied for both their quantum information theoretic properties and their ability to simulate key physics equations like the Schrodinger and Dirac equations. We take the unique approach of analyzing QECA as complexity-generating systems. Doing so requires a more precise notion of what we mean by complex. This is done by proposing nine axes of complexity along which the complexity of any system may be quantified. When applied to QECA, we find evidence of complexity using three of these axes, namely diversity, persistent dynamical macrostates, and connectivity. The studies presented are numerical simulations done without approximation using highly optimized exact diagonalization code which supports a Hilbert space of up to 2^27 dimensions. The code is written entirely in the high-level open source programming language Python, making it easily expandable to future projects requiring exact simulation of quantum systems. A careful description of an algorithm critical to our method as well as our use of high performance computing resources on a cluster supercomputer is given. Powerful quantifiers of entanglement and connectivity such as von Neumann entropy and complex network measures computed on quantum mutual information adjacency matrices provide analysis tools for the simulations. Each network measure is defined then tested on well-characterized entangled states from quantum information theory, like the GHZ and W states and singlet state arrays. The network measures known as network density, clustering coefficient, and disparity are specifically considered. We find these network measures offer unique information regarding the structure of two point correlations in the states produced by QECA dynamics, as compared to each other and as quantified by a principal component analysis. Using such measures, we address the complexity of QECA models at three levels of specificity. First, a broad analysis of tens of thousands of simulations gives an overview of the variety of dynamics available to the models. We quantify the diversity of our simulations as the density of simulations which appear, on average, unlike typical entangled quantum states. Second, a more selective analysis identifies QECA by their complexity dynamics, in the frequency domain. A few QECA which exhibit persistent dynamical macrostates in the form of highly structured entanglement are also described. Entanglement dynamics are quantified by the distribution of changes in bond entropy (the von Neumann entropy of all bipartitionings of the QECA system). Finally, we take a detailed…