The d -level or qudit one-way quantum computer (d1WQC) is described using the valence bond solid formalism and the generalized Pauli group. This formalism provides a transparent means of deriving measurement patterns for the implementation of quantum gates in the computational model. We introduce a new universal set of qudit gates and use it to give a constructive proof of the universality of d1WQC. We characterize the set of gates that can be performed in one parallel time step in this model.
The d -level or qudit one-way quantum computer (d1WQC) is described using the valence bond solid formalism and the generalized Pauli group. This formalism provides a transparent means of deriving measurement patterns for the implementation of quantum gates in the computational model. We introduce a new universal set of qudit gates and use it to give a constructive proof of the universality of d1WQC. We characterize the set of gates that can be performed in one parallel time step in this model.
We propose a way of universal quantum computation by doing joint measurements on distributed singlets. We show how these joint measurements become local measurements when the singlets are interpreted as the virtual components of a large valence-bond state. This proves the equivalence of the cluster-state-based quantum computational model and the teleportation-based model, and we discuss several features and possible extensions. We show that all stabilizer states have a very simple interpretation in terms of valence-bond solids, which allows to understand their entanglement properties in a transparent way.
The concept of a qudit (a d-level system) cluster state is proposed by generalizing the qubit cluster state [Phys. Rev. Lett. 86, 910 (2001)] to higher-dimensional Hilbert space according to the finite-dimensional representations of quantum plane algebra. We demonstrate their quantum correlations and prove a theorem which guarantees the availability of the qudit cluster states in quantum computation. We explicitly construct the network to show the universality of the one-way computer based on the defined qudit cluster states and single-qudit measurement. A protocol of implementing one-way quantum computer is suggested using the high-dimensional ``Ising'' model which can be found in many magnetic systems.
Quantum computing with qudits, quantum systems with d>2 levels, offers a powerful extension beyond qubits, expanding the computational possibilities of quantum systems, allowing the simplification of the implementation of several algorithms and, possibly, providing a foundation for optimised error correction. In this work, we propose a novel formulation of qudit gates that is universally applicable for any number of levels d, without restrictions on the dimensionality. By extending the mathematical framework of quantum gates to arbitrary dimensions, we derive explicit gate operations that form a universal set for quantum computation on qudits of any size. We demonstrate the validity of our approach through the implementation of the Quantum Fourier Transform (QFT) for arbitrary d, verifying both the correctness and utility of our generalized gates. This novel methodology broadens the design space for quantum algorithms and fault-tolerant architectures, paving the way for advancements in qudit-based quantum computing.
Quantum computers have the potential to solve problems that are believed to be classically intractable. However, building such a device is proving to be very challenging. In this thesis, two physically promising settings for quantum computation are investigated: the one-way quantum computer and ancilla-based quantum gates. The majority of both the theoretical and experimental focus in the field of quantum computation has been on computation using 2-level quantum systems, known as qubits. In contrast to this, in this thesis I consider the relatively less well-understood setting of quantum computation using continuous variables or d-level quantum systems, called qudits. I develop a simple notation that encompasses each different encoding, and is applicable to a `general quantum variable'. These ideas are then used to investigate computational depth (a proxy for time) in quantum circuits and one-way quantum computations in this general quantum variable setting. In doing so, the parallelism inherent in the one-way quantum computer is made precise. In the second half of this thesis, a range of techniques are proposed for implementing entangling gates on a well-isolated computational register via interactions with `ancillary' systems. In particular, ancilla-based quantum gates for general quantum variables are investigated - including the interesting case of hybrid quantum computation, whereby more than one encoding is used in tandem. The methods proposed herein each have their own unique advantages, such as: reducing gate-counts in certain circuits, allowing for inherently parallel computation, or minimising the physical requirements for universal quantum computation. In particular, the final gate techniques that are proposed in this thesis may implement any quantum computation using only a single fixed ancilla-register interaction gate and ancillas prepared in simple states. This then allows the computational register to consist of well-isolated `memory' quantum variables and the ancillas need only be optimised for a single high-quality fixed interaction gate. Hence, this provides a simple and highly promising setting for physically implementing a quantum computer.
A quantum computer is similar to an ordinary computer, except that instead of bits it uses two-level quantum systems (qubits), and the usual logical operations on bits are replaced with noiseless quantum dynamics in the form of unitary gates or Hamiltonian interactions. Many sets of resources that are universal for quantum computation (that is, capable of implementing any quantum computation) have been found, but general conditions for universality are still only partially understood. Improving our understanding of these conditions contributes to two broad goals. On the one hand, understanding what class of physical systems is suitable for quantum computation allows us choose the systems that are most easily controlled or built in the laboratory. On the other hand, identifying the unifying characteristics of such a class provides insight into the fundamental physics responsible for the much greater computational power of quantum computers. The original results presented in this thesis are a contribution to our understanding of the physical requirements for universal quantum computation. These results lead to the identification of a new, large class of universal dynamics. More specifically, the main problem that is addressed is the following: consider a physical system that is made up of a finite number of qubits that are subject to a natural, fixed interaction described by a two-body Hamiltonian. Suppose we have the ability to control the system only by applying fast one-qubit operations. Under what conditions is this system capable of universal quantum computation? The first original result of the thesis is to prove that universal quantum computation is possible under these conditions given the single extra constraint that the Hamiltonian must entangle, or connect, all of the qubits. In other words, since a two-body Hamiltonian is expressible as a sum of coupling terms each of which acts nontrivially on a pair of qubits, consider the graph whose vertices correspond to qubits and whose edges connect qubits that are coupled by a term in the Hamiltonian. Then this Hamiltonian, together with fast one-qubit unitaries, is universal for quantum computation precisely when the corresponding graph is connected. This result suggests many potential generalizations, three of which are explored in this thesis. The first generalization is to show that the same result holds for a two-body Hamiltonian that acts on D-dimensional quantum systems (qudits), given fast one-qudit unitaries. A second extension is to the case of a many-body Hamiltonian, that is, a Hamiltonian that has coupling terms that act on more than two qubits. Again, we find that under mild restrictions, such Hamiltonians are universal provided they are connected and that arbitrary one-qubit unitaries are available. We also discover and explore some surprising structure in many-body Hamiltonians that is associated with the parity of the couplings. A third related problem that is addressed is the question of what two-qubit unitaries are universal for quantum computation. This problem was originally solved by J.-L. and R. Brylinski [BB02], who gave a technical proof of the universality of any two-qudit gate that is not the swap gate or a product of one-qudit unitaries, given one-qudit unitaries as a free resource. We develop an elementary and constructive proof for the case of qubits, and prove that our construction is very close to optimal in the number of uses of the two-qubit gate. Given a qualitative understanding of what dynamical resources are universal for quantum computation, it is natural to consider developing a quantitative measure of the usefulness of quantum dynamics for quantum information processing. This is done in the final chapter of the thesis, in which the beginnings of a quantitative theory of quantum dynamics are developed.
In this book chapter, we provide a tutorial introduction to one-way quantum computation and many of the techniques one can use to understand it. The techniques which are described include the stabilizer formalism and the logical Heisenberg picture. We highlight ways in which it is useful to understand one-way computation beyond simple equivalence with the quantum circuit model. We briefly review current proposals of implementations and experimental progress and summarize some recent related theoretical developments. Although the chapter is primarily didactic in focus, we include a number of new methods and observations. These include: a simpler and more compact formulation of one-way quantum computation in the stabilizer formalism; A new way of implementing unitaries diagonal in the computational basis; New results on the family of operations which may be implemented in a single round of measurements; A method for constructing compact one-way patterns by decomposing unitaries in terms of diagonal unitaries and Clifford group transformations.