Abstract The semiclassical approximation of quantum computing and quasi-qubits (s-bits) have been obtained by us as a result of our work over the past few years. This work can be conventionally divided into two parts. The first part, let’s call it the programming model, contains a computer model of quasi-qubits and quantum computing. The second part, let’s call it the microelectronic model, describes the microelectronic realization of qubits in the semiclassical approximation (quasi-qubits) and exists in the form of block diagrams, which are supposed to be easy to manufacture. How did we get the semiclassical approximation? The difficulty in solving such a problem was that microparticles in quantum mechanics are described in an infinite-dimensional Hilbert space. Classical models are much poorer in the number of variables; therefore, it is impossible to describe quantum mechanical objects by classical methods due to the small number of available parameters.
In this section we introduce the framework of quantum mechanics as it pertains to the types of systems we will consider for quantum computing. Here we also introduce the notion of a quantum bit or ‘qubit’, which is a fundamental concept for quantum computing. At the beginning of the twentieth century, it was believed by most that the laws of Newton and Maxwell were the correct laws of physics. By the 1930s, however, it had become apparent that these classical theories faced serious problems in trying to account for the observed results of certain experiments. As a result, a new mathematical framework for physics called quantum mechanics was formulated, and new theories of physics called quantum physics were developed in this framework. Quantum physics includes the physical theories of quantum electrodynamics and quantum field theory, but we do not need to know these physical theories in order to learn about quantum information. Quantum information is the result of reformulating information theory in this quantum framework. We saw in Section 1.6 an example of a two-state quantum system: a photon that is constrained to follow one of two distinguishable paths. We identified the two distinguishable paths with the 2-dimensional basis vectors and then noted that a general ‘path state’ of the photon can be described by a complex vector with |α0|2 +|α1|2 = 1. This simple example captures the essence of the first postulate, which tells us how physical states are represented in quantum mechanics. Depending on the degree of freedom (i.e. the type of state) of the system being considered, H may be infinite-dimensional. For example, if the state refers to the position of a particle that is free to occupy any point in some region of space, the associated Hilbert space is usually taken to be a continuous (and thus infinite-dimensional) space. It is worth noting that in practice, with finite resources, we cannot distinguish a continuous state space from one with a discrete state space having a sufficiently small minimum spacing between adjacent locations. For describing realistic models of quantum computation, we will typically only be interested in degrees of freedom for which the state is described by a vector in a finite-dimensional (complex) Hilbert space.
The advent of computers has significantly diminished the need for manual computation. As technology has evolved, it has led to the creation of classical computers with high performance, throughput, and memory capacity. Classical computers store data in binary format, using bits that can either be 0 or 1, typically represented by low and high voltage levels, respectively. While effective, this method sometimes demands considerable resources for certain computations. In contrast, quantum computers represent the cutting edge of computing technology. They utilize quantum bits or qubits, which exploit quantum mechanical phenomena like superposition and entanglement, allowing for more efficient problem-solving. This paper presents a simplified overview of the core concepts of quantum computing.
We propose a divide-and-conquer method for the quantum-classical hybrid algorithm to solve larger problems with small-scale quantum computers.Specifically, we concatenate a variational quantum eigensolver (VQE) with a reduction in the system dimension, where the interactions between divided subsystems are taken as an effective Hamiltonian expanded by the reduced basis.Then the effective Hamiltonian is further solved by the VQE, which we call deep VQE.Deep VQE allows us to apply quantum-classical hybrid algorithms on small-scale quantum computers to large systems with strong intrasubsystem interactions and weak intersubsystem interactions, or strongly correlated spin models on large regular lattices.As proof-of-principle numerical demonstrations, we use the proposed method for quasi-one-dimensional models, including one-dimensionally coupled 12-qubit Heisenberg antiferromagnetic models on kagome lattices as well as two-dimensional Heisenberg antiferromagnetic models on square lattices.The largest problem size of 64 qubits is solved by simulating 20-qubit quantum computers with a reasonably good accuracy approximately a few %.The proposed scheme enables us to handle the problems of > 1000 qubits by concatenating VQEs with a few tens of qubits.While it is unclear how accurate ground-state energy can be obtained for such a large system, our numerical results on a 64-qubit system suggest that deep VQE provides a good approximation (discrepancy within a few percent) and has room for further improvement.Therefore, deep VQE provides us a promising pathway to solve practically important problems on noisy intermediate-scale quantum computers.
In recent years, we have observed a new revolution in computer science, driven by the increasing availability of computational power and data. This historic event has seen the capillary diffusion of what is commonly known as artificial intelligence (AI). In particular, we have seen a rapid improvement of large language models that now can interact with people on a level that appeared impossible just a few years ago. Despite this newly found focus on AI and computer science, the technical difficulties that limit the increase in computational power are increasing, and the prediction described by Moore's law no longer holds. For this reason physicists, computer scientists, mathematicians, and engineers are studying another computational paradigm called quantum computing. Similar to the rise of large language models, quantum computing has gained significant attention in the last ten years due to the development of initial quantum hardware and its rapid advancements. The quantum computing paradigm offers a fundamentally different set of tools with respect to what is called classical computation. Quantum properties and effects like superposition, entanglement, and interference give us the ability to solve problems that we could not approach classically. In this thesis, we explore the intersection between artificial intelligence and quantum computing from an experimental point of view. We researched quantum AI models, utilizing both the classical and quantum paradigms using variational architectures. This thesis is organized as follows: part 1 contains an introduction to quantum computation, starting from the postulate of quantum mechanics, through an introduction to QML and finally the state of the art. Part 2 contains the description of the research we performed. It's divided into two main chapters containing fundamentally different approaches to quantum computing. Each chapter contains section illustrating our research on that approach.
Part I: Semiclassical Gravity Efficiently Solves
NP-Complete Problems
Assuming the gravitational field is classical and that it couples to quantum fields via the semiclassical Einstein field equations (EFEs), we show that the weak-field dynamics of a massive and non-relativistic qubit can in principle be used to solve an
NP-complete problem in polynomial time. We attribute this vast computational power to the non-linear qubit dynamics afforded by the semiclassical EFEs. Consequently, the above two assumptions entail a violation of the Physical Extended Church--Turing Thesis, which we regard as evidence for the quantization of gravity.
Part II: Spacetime Quasicrystals
We generalize self-similar quasicrystals, such as the Penrose tilings, from Euclidean space to Minkowski spacetime. We construct the first examples of Lorentzian quasicrystals and identify their key novel features. We then explore possible connections to fundamental physics. First, we propose a speculative scenario in which our
(3+1)D universe is embedded in a particularly symmetric
(9+1)D torus
T9,1, previously identified as the most symmetric toroidal compactification of the superstring. We suggest that this construction may shed light on the seesaw relation
MPlMvac≈MEW2 between the Planck, vacuum-energy, and electroweak scales. Second, we replace the random causal sets of causal set theory with ordered, highly symmetric quasi-crystalline causal sets. We find suggestive evidence that our quasicrystals outperform Poisson random sets in the number-volume correspondence in both
(1+1)D and
(3+1)D, where such behaviour has been conjectured to be impossible. Finally, extending hyperuniformity from space to spacetime, we show that our
(1+1)D quasicrystal is the first demonstrated example of a spacetime-hyperuniform structure.