The design and understanding of quantum-dot cellular automata (QCA) networks has been largely influenced by limitations in the approximation methods used in common design tools. In some cases, such limitations have led to unrealistic selections of clock zones which are not feasible for nanoscale QCA implementations given current fabrication constraints on clocking electrodes. A better understanding of the behaviour of larger QCA networks of perhaps tens to hundreds of QCA devices is needed. One approach is by investigating the low energy spectrum; however, diagonalization of the system Hamiltonian even in the 2-state approximation is impractical beyond 20 or so devices. In this work, we present a methodology for understanding the spectrum of the full network in terms of contributions from components of the network. We show that important features of the low energy spectrum can be attributed to specific critical components, and present one scheme for decomposing the network into these components. In addition, we address the question of computing the low energy spectrum of large QCA networks. A method based on basis reduction which naturally emerges from the component decomposition is successfully applied to a 49 cell XOR gate with results compared against a density matrix renormalization group implementation.
The design and understanding of quantum-dot cellular automata (QCA) networks has been largely influenced by limitations in the approximation methods used in common design tools. In some cases, such limitations have led to unrealistic selections of clock zones which are not feasible for nanoscale QCA implementations given current fabrication constraints on clocking electrodes. A better understanding of the behaviour of larger QCA networks of perhaps tens to hundreds of QCA devices is needed. One approach is by investigating the low energy spectrum; however, diagonalization of the system Hamiltonian even in the 2-state approximation is impractical beyond 20 or so devices. In this work, we present a methodology for understanding the spectrum of the full network in terms of contributions from components of the network. We show that important features of the low energy spectrum can be attributed to specific critical components, and present one scheme for decomposing the network into these components. In addition, we address the question of computing the low energy spectrum of large QCA networks. A method based on basis reduction which naturally emerges from the component decomposition is successfully applied to a 49 cell XOR gate with results compared against a density matrix renormalization group implementation.
Understanding the dynamic behaviour of nanoscale quantum-dot cellular automata (QCA) networks involves the simulation of large numbers of QCA devices, with the complexity of the full quantum treatment exponential in the network size. Previous attempts to limit this complexity introduce simplifying assumptions with known inaccuracies. In this thesis we investigate an alternate approach, extracting performance metrics through analysing the low energy eigenspectrum of a clocked network. We make two major contributions. In the first part of this thesis, we study the use of silicon dangling bonds (SiDBs) as a platform for combinatorial logic, and ultimately nanoscale QCA. We present models for understanding the preferred configurations and dynamics of charges in these structures. We consider the clocking of SiDB-based QCA wires, and reveal a complicated trajectory of charge states that serve as a challenge for QCA operation. By studying SiDB-based QCA from the framework of the familiar 3-state model, in which these preferred charge states translate to eigenstates of a system Hamiltonian, we determine conditions for which SiDB-QCA wires can cor- rectly operate when clocked. These conditions are potentially impractical unless net-neutral SiDB arrangements can be achieved. The remaining bulk of the thesis revolves around the link between QCA clocking and quantum annealing. We first investigate the adiabaticity of simple 2-state QCA networks under zone clocking. We present upper bounds on the clocking frequency beyond which adiabaticity falls below a 99% threshold, and demonstrate how we can efficiently estimate clocking performance using only a few of the energy eigenstates. Due to a natural mapping between QCA cells and superconduct- ing flux qubits, the potential for investigating performance using a physical quantum annealer is explored. Methods for embedding QCA networks onto the annealer are discussed and a selection of annealing results are analyzed. Finally, we establish a method for decomposing the system Hamiltonian into contributions from given components, and a means to identify meaningful components which critically affect clocking performance. This framework reveals a heuristic algorithm for approximating the low energy eigenspectra of large QCA networks, enabling future investigations into the performance of networks well beyond previous size limitations.
Molecular quantum-dot cellular automata (QCA) may provide low-power, high-speed computational hardware for processing classical information. Simulation and modeling play an important role in the design of QCA circuits because fully coherent models of QCA scale exponentially with the number of devices, and such models are severely limited in size. For larger circuits, approximations become necessary. In the era of fault-tolerant quantum computation, however, it may become possible to model large QCA circuits without such limitations. This work explores the use of the noisy-intermediate scale quantum (NISQ) variational quantum eigensolver (VQE) method for estimating the ground state of QCA circuits. This is relevant because the computational result of a QCA calculation is encoded in the circuit’s ground state. In this study, VQE is used to model logic circuits, including binary wires, inverters, and majority gates. VQE models are performed by ideal simulators, noisy simulators, and actual quantum hardware. This study demonstrates that VQE may indeed be used to model molecular QCA circuits. It is observed that using modern NISQ hardware, results are still quite sensitive to noise, so measures should be taken to minimize noise. These include simplifying the ansatz circuit whenever possible and using low-noise hardware.
Simulations of quantum-dot cellular automata (QCA) on classical computers are highly limited due to the exponential growth in resources required for the numerical simulation of quantum mechanics involving networks of finite state nodes. Recent advancements in computing based on networks of flux-qubits, and in particular the platform technology developed by D-Wave Systems Inc., have made it possible to explore QCA networks that are intractable on classical machines. However, the embedding of such networks onto the available processor architecture is a key challenge in setting up such simulations. In this work, two approaches to embedding QCA circuits are characterized: a dense placement algorithm that uses a routing method based on negotiated congestion; and a heuristic method implemented in D-Wave's SAPI package. Both embedding methods are characterized using a set of basic QCA benchmark circuits of various sizes and complexities. When including diagonal interactions only in the case of an inverter, both methods were able to embed a 4-bit 2-1 multiplexer circuit containing 192 non-driver QCA cells onto the 512 qubit D-Wave Vesuvius chip architecture. Including diagonal interactions for all cells, both methods successfully embedded a serial adder circuit containing 126 non-driver cells.
The design of arithmetic and logical unit is a crucial block schematic of quantum computing, which has gained popularity in recent years. While division is a successive subtraction, a logical multiplier employs the technique of successive addition, which are the essential blocks of the quantum computing. The key components including comparator design, adders, and subtractors are considered with the help of Quantum-Dot Cellular Automata (QCA) in this work. Such essential blocks of the Arithmetic Logic Unit (ALU) are designed, implemented, and tested in the simulation using QCADesigner version 2.0.3. The results are found to be appealing with the standard truth tables of the above logical circuits. Hence, the designs could be utilized in the practical applications of Internet of Things, Block chains and Cryptographic solutions catering to the challenges of industry, healthcare and finance sectors. Addressing challenges such as a reduced number of qubits and complexity reduction could make way for more robust and practical ALU designs in QCA-based quantum systems. In the proposed novel work, the number of qubits required has been reduced to 24, offering a notable reduction compared to several recent approaches. Hence, the novelty of the work lies in the architectural-level reorganization and logic-level minimization strategy, which collectively enable a significant reduction in design complexity (up to 93%) without introducing additional fabrication or simulation overhead, optimizing clock zone assignments to enhance timing efficiency.
Over an arbitrary commutative ring R, we develop a theory of quantum cellular automata. We then use algebraic K-theory to construct a space Q(X) of quantum cellular automata (QCA) on a given metric space X. In most cases of interest, π0Q(X) classifies QCA up to quantum circuits and stabilization. Notably, the QCA spaces are related by homotopy equivalences Q(∗)≃ΩnQ(Zn) for all n, which shows that the classification of QCA on Euclidean lattices is given by an Ω-spectrum indexed by the dimension n. As a corollary, we also obtain a non-connective delooping of the K-theory of Azumaya R-algebras, which may be of independent interest. We also include a section leading to the Ω-spectrum for QCA over C∗-algebras with unitary circuits.
Advancements in computing based on qubit networks, and in particular the flux-qubit processor architecture developed by D-Wave System's Inc., have enabled the physical simulation of quantum-dot cellular automata (QCA) networks beyond the limit of classical methods. However, the embedding of QCA networks onto the available processor architecture is a key challenge in preparing such simulations. In this work, two approaches to embedding QCA circuits are characterized: a dense placement algorithm that uses a routing method based on negotiated congestion; and a heuristic method implemented in D-Wave's Solver API package. A set of benchmark QCA networks is used to characterise the algorithms and a stochastic circuit generator is employed to investigate the performance for different processor sizes and active flux-qubit yields.