It is shown in the paper that the unitary quantum dynamics in quantum mechanics is the universal quantum driving force to speed up a quantum computation. This assertion supports strongly in theory that the unitary quantum dynamics is the fundamental and universal principle in nature. On the other hand, the symmetric structure of Hilbert space of a composite quantum system is the quantum-computing resource that is not owned by classical computation. A new quantum-computing speedup theory is set up on the basis of the unitary quantum dynamics. Both the unitary quantum dynamics and the symmetric structure and property of the Hilbert space of the quantum system are mainly responsible for an exponential quantum-computing speedup for a general efficient quantum algorithm. The inherent importance for the unitary quantum dynamics to speed up a quantum computation lies in the unique ability of the unitary quantum dynamics to build the effective interaction between the symmetric structure of the Hilbert space of the quantum system and the mathematical symmetric structure of a problem to be solved on the quantum system. This unique ability could result in an essential difference of computational power between quantum and classical computations by combining the symmetric structure and property of the Hilbert space. The new quantum-computing speedup theory also provides reasonable mechanisms for exponential quantum-computing speedup for the existing efficient quantum algorithms based on the quantum parallel principle. These existing quantum algorithms including the hidden-subgroup-problem quantum algorithms and conventional quantum search algorithms have the common character that the symmetric structure of the Hilbert space does not have any effective effect on these quantum algorithms. This could be the main reason why these quantum algorithms are quite special and considered to be semiclassical.
It is shown in the paper that the unitary quantum dynamics in quantum mechanics is the universal quantum driving force to speed up a quantum computation. This assertion supports strongly in theory that the unitary quantum dynamics is the fundamental and universal principle in nature. On the other hand, the symmetric structure of Hilbert space of a composite quantum system is the quantum-computing resource that is not owned by classical computation. A new quantum-computing speedup theory is set up on the basis of the unitary quantum dynamics. Both the unitary quantum dynamics and the symmetric structure and property of the Hilbert space of the quantum system are mainly responsible for an exponential quantum-computing speedup for a general efficient quantum algorithm. The inherent importance for the unitary quantum dynamics to speed up a quantum computation lies in the unique ability of the unitary quantum dynamics to build the effective interaction between the symmetric structure of the Hilbert space of the quantum system and the mathematical symmetric structure of a problem to be solved on the quantum system. This unique ability could result in an essential difference of computational power between quantum and classical computations by combining the symmetric structure and property of the Hilbert space. The new quantum-computing speedup theory also provides reasonable mechanisms for exponential quantum-computing speedup for the existing efficient quantum algorithms based on the quantum parallel principle. These existing quantum algorithms including the hidden-subgroup-problem quantum algorithms and conventional quantum search algorithms have the common character that the symmetric structure of the Hilbert space does not have any effective effect on these quantum algorithms. This could be the main reason why these quantum algorithms are quite special and considered to be semiclassical.
The quantum-computing speedup theory considers the symmetric structures and properties of quantum systems as the fundamental Quantum-Computing-Speedup (QCS) resources which are responsible for exponentially speeding up quantum computing and simulating. At present a large and important problem is how to make use of the fundamental QCS resources to speed up essentially quantum computing and simulating. Here the author makes a great effort toward solving this important problem. The theoretical research work in this paper is mainly divided into the two Parts I and II. The Part I investigates mainly the multiple-quantum operator algebra spaces. And the relationships are analyzed among the multiple-quantum operator algebra spaces, quantum simulating for the unitary time-evolutional processes, and the fundamental QCS resources which exist in the different kinds of basic quantum spaces: the multiple-quantum operator algebra spaces, the density operator spaces, and the Hilbert spaces. It concludes that the multiple-quantum operator algebra space must be positioned as the central place where the fundamental QCS resources are exploited to speed up quantum computing and simulating. The Part II investigates mainly the subspace-selective unitary manipulation based on the Hilbert-space symmetric structures. Recognize that the multiple-quantum operator algebra space is the central place. Then those fundamental QCS resources original from the Hilbert space (a quantum-state space) must be explicitly taken into account in the multiple-quantum operator algebra space (a linear operator space). This is an important problem. The subspace-selective unitary manipulation is able to solve this problem. It aims to harness the fundamental QCS resources original from the Hilbert space to speed up quantum computing and simulating in the multiple-quantum operator algebra space.
A unitary dynamical state-locking (UNIDYSLOCK) process is a unitary process that can transform two orthogonal states to two non-orthogonal states. Its inverse process, i.e., the QUANSDAM process could realize an exponential QC speedup in solving an unstructured search problem in the quantum-computing (QC) speedup theory (X. Miao, arXiv:1105.3573 [quant-ph] (2011)). In this paper the principle of how a UNIDYSLOCK process works is described on the basis of the QC speedup theory. A UNIDYSLOCK process does not exist at all in pure quantum mechanics. That it can change the quantum-state difference is original from the fundamental interaction between the quantum physical laws (i.e., the unitary quantum dynamics and the Hilbert-space symmetric structure) and the mathematical-logical principle of the search problem. It is characteristic for the QC speedup theory. Even if a usual quantum computational (Qc) process that is reversible or unitary contained a UNIDYSLOCK (or QUANSDAM) process, the contribution of the UNIDYSLOCK (or QUANSDAM) process to the QC speedup of the usual Qc process would be secondary or negligible. Whether or not a UNIDYSLOCK (or QUANSDAM) process can make an essential contribution to a QC speedup can distinguish the QC speedup theory from any usual Qc theory. In the paper the QC speedup mechanism of the HSSS quantum search process is also studied. The search process is essentially different from a usual quantum search algorithm. It works in both the usual Hilbert space and the math Hilbert space. Due to the math Hilbert space a mathematical-parallel operation is characteristic for the QC speedup theory, while a quantum-parallel operation is characteristic for the usual Qc theory. The fundamental interaction mentioned above and the fundamental quantum-computing resource play the main role in speeding up the search process.
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 Physical Impossibility of Quantum Computational Speedup: Two Independent Arguments from Quantum-Geometry Dynamics Quantum computing promises computational advantages that are, in the most celebrated cases, exponential. Shor's algorithm factors integers exponentially faster than the best known classical algorithm; Grover's algorithm searches an unstructured database with a quadratic speedup. These results are mathematically rigorous within their assumed framework. This paper asks not whether the mathematics is correct but whether the physical assumptions underlying it are. I argue that Quantum-Geometry Dynamics (QGD) — an axiomatic framework for physics derived from the discreteness of space and the kinetic nature of matter — establishes two independent foundational preclusions of quantum computational speedup. These are not engineering obstacles or decoherence problems that better technology can overcome. They are consequences of the physical structure of discrete space, and no improvement in qubit isolation, error correction, or fabrication technique can supply a physical resource that does not exist. The first preclusion concerns physical states. In QGD, matter is composed of preons⁺ that propagate through quantum-geometrical space by preonic leaps, each of which is a definite transition from one spatial unit to the next in the direction of the momentum vector. A preon(+) always occupies a definite position. There is no ontological superposition. What quantum mechanics represents as a superposed state is, in QGD, an epistemic description of our ignorance of the system's actual preonic microstate. Since ontological superposition is the physical resource on which quantum parallelism depends, its absence means that no physical system can instantiate the state required for exponential speedup. The second preclusion concerns physical processes. Causal succession in discrete space is categorically and structurally irreversible: every preonic leap has a definite direction, and no physical process can reverse the causal order of preonic states. This is not thermodynamic irreversibility arising from large numbers but a property of every individual preonic transition. Since the quantum circuit model is built on reversible unitary gates, and reversible computation has no physical instantiation in a discrete-space framework, the operational basis of quantum computing is equally precluded. I show how both arguments bite at specific steps in Shor's algorithm and Grover's algorithm. In Shor's algorithm, the first preclusion destroys the state preparation step and the second destroys the quantum Fourier transform. In Grover's algorithm, the first preclusion destroys the uniform superposition and the second destroys the diffusion operator and oracle. Either preclusion alone is sufficient; together they attack quantum speedup at both ends — what goes in and how it is processed. The black hole information paradox, correctly understood within QGD, provides independent confirmation: the paradox dissolves precisely because physical states are always definite and physical evolution is categorically irreversible — the same two features that preclude quantum speedup. The Uniqueness Theorem for QGD further strengthens the claim: if the two-axiom discrete-space structure is the unique viable foundational description of physical reality, these preclusions are not features of one possible framework among many but consequences of the physical structure of the universe. This paper supersedes the earlier paper Quantum Computing Under Quantum-Geometry Dynamics (Zenodo, DOI: 10.5281/zenodo.19522137) and should be read in its place. This paper is part of the Minimal Physically Derivable Theory (MPDT) programme.
Introduction Discussions of quantum-computational algorithms in the literature refer to various features of quantum mechanics as the source of the exponential speed-up relative to classical algorithms: superposition and entanglement, the fact that the state space of n bits is a space of 2 n states while the state space of n qubits is a space of 2 n dimensions, the possibility of computing all values of a function in a single computational step by “quantum parallelism,” or the possibility of an efficient implementation of the discrete quantum Fourier transform. Here I propose a different answer to the question posed in the title, in terms of the difference between classical logic and quantum logic, i.e., the difference between the Boolean classical event structure and the non-Boolean quantum event structure. In a nutshell, the ultimate source of the speed-up is the difference between a classical disjunction, which is true (or false) in virtue of the truth values of the disjuncts, and a quantum disjunction, which can be true (or false) even if none of the disjuncts is either true or false. In the following, I will discuss the information-processing in Deutsch's XOR algorithm (the first genuinely quantum algorithm) and related period-finding quantum algorithms (Simon's algorithm and Shor's factorization algorithm). It is well known that these algorithms can be formulated as solutions to a hidden-subgroup problem. Here the salient features of the information-processing are presented from the perspective of the way in which the algorithms exploit the non-Boolean logic represented by the projective geometry (the subspace structure) of Hilbert space.
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.