By the early eighties, Fredkin, Feynman, Minsky and others were exploring the notion that the laws of physics could be simulated with computers. Feynman's particular contribution was to bring quantum mechanics into the discussion and his ideas played a key role in the development of quantum computation. It was shown in 1995 by Barenco et al that all quantum computation processes could be written in terms of local operations and CNOT gates. We show how one of the most important of all physical systems, the quantized bosonic oscillator, can be rewritten in precisely those terms and therefore described as a quantum computational process, exactly in line with Feynman's ideas. We discuss single particle excitations and coherent states.
By the early eighties, Fredkin, Feynman, Minsky and others were exploring the notion that the laws of physics could be simulated with computers. Feynman's particular contribution was to bring quantum mechanics into the discussion and his ideas played a key role in the development of quantum computation. It was shown in 1995 by Barenco et al that all quantum computation processes could be written in terms of local operations and CNOT gates. We show how one of the most important of all physical systems, the quantized bosonic oscillator, can be rewritten in precisely those terms and therefore described as a quantum computational process, exactly in line with Feynman's ideas. We discuss single particle excitations and coherent states.
Quantum simulation involves using a quantum system to study another quantum system. The proposal to simulate quantum physics using quantum mechanical devices, as outlined by Richard Feynman in 1982, stems from classical computers' challenges in simulating quantum systems. The pursuit of simulating complex quantum systems has led to the intersection of quantum mechanics and quantum computing. This offers new opportunities to understand and manipulate the quantum world. Central to describing quantum dynamics is the Hamiltonian operator H^, which governs the time evolution of physical systems and serves as the mathematical foundation for quantum simulations. This thesis explores the methodologies and advancements in simulating Hamiltonian operators on quantum computers, focusing on decomposing these operators into quantum gates and the efficiency gains over classical computational approaches. Initially, the thesis explains the fundamental theory and basics of quantum computing. Further it dives into the foundations of the Hamiltonian operator within quantum mechanics, highlighting its role in describing the dynamics of quantum systems. The complexity and high-dimensional nature of Hamiltonian matrices necessitate using quantum computers to simulate quantum phenomena accurately and efficiently. The process of simulating the Hamiltonian operator on a quantum computer involves a detailed examination of the strategies for decomposing the operator into a sequence of quantum gates. This decomposition is crucial for translating the abstract mathematical description of quantum systems into the physical operations of a quantum computer. By utilizing the principles of quantum mechanics, such as superposition and entanglement, quantum gates manipulate qubits to mimic the evolution of quantum states as the Hamiltonian operator dictates. Latter portions of the thesis are dedicated to comparing the efficiency of quantum simulations against traditional classical simulations and inspecting efficient quantum simulations. Quantum computers offer a polynomial or an exponential speedup for certain classes of problems, such as those involving highly entangled quantum states or requiring the simulation of complex quantum dynamics. This advantage is due to the quantum computer's ability to naturally represent and process quantum information - a task for which classical computers require exponential resources as the system size increases. The efficiency of simulating the Hamiltonian operator on a quantum computer is further enhanced through optimization techniques and algorithmic improvements. These include error mitigation strategies and the development of quantum algorithms tailored to specific types of Hamiltonian operators. The thesis presents a comprehensive analysis of these methods, demonstrating how they make quantum simulations more practical and accessible for scientific research and technological development. In conclusion this thesis is a literature review that examines the transformative potential of quantum simulations. By efficiently simulating the Hamiltonian operator on quantum computers researchers can explore quantum phenomena with precision, paving the way for breakthroughs in quantum physics, materials science, and beyond. The advancements in quantum computing technology and algorithmic approaches discussed here highlight the current capabilities and future explorations in the quantum realm.
Following an introduction on interferometric physical computers, the discussion turns to Feynman’s contribution to quantum computing and qbits. The ψ + , ψ − , ψ + , and ψ − quantum entanglement states are expressed in terms of the σ x , σ y , σ z , and I matrices and the discussion continues, on quantitative terms, Pauli gates, the Hadamard gate, and the CNOT gate. The equivalence between σ x and the polarization rotational matrix R is noted. Interaction between the ψ + , ψ − , ψ + , and ψ − states, R and the Hadamard gate give rise to whole families of quantum mathematical operations. Physical representations for R and the Hadamard gate are explicitly identified.
One of the emerging technologies is quantum computing and its simulation. Quantum mechanics deals with the particles at atomic or subatomic state. Electrons, photons are the examples of it. Thus, quantum physics plays a very important role in quantum computing. Using simple principals of superposition and entanglement quantum computers can do wonders of fast computation which a classical computer cannot. In 1980 Feynman proposed ideal model for computer-based quantum mechanical system and its simulation. This has revolutionised computer difficulties. Since 1982 in a journey of quantum computing, there were many ups and downs. Numerous advances have been made to uncover quantum mechanics. Nowadays there has been a commercial interest because of the performance of quantum computers over classical computers.
There were few basic aspects of computers and computation which intrigued Richard P. Feynman, and made the subject sufficiently interesting that he would spend time teaching it. Feynman himself did the lecture on quantum computation. Charles Bennett has already prepared the way by talking about reversible computation, and Norman Margolus on the billiard ball computer, so Feynman developed quantum computation in ways which made its connection to classical reversible computers obvious. His lecture was chiefly developing the idea that an entirely reversible quantum system, without damping processes, could perform universal computation. The insight that quantum computers were really different came only later, and to others, not to Feynman. Feynman only emphasized that the physical scale and speed of computers were not limited by the classical world, since conceptually they could be built of reversible components at the atomic level.
Key to solving many societally relevant problems in materials science, quantum chemistry, and nuclear physics is understanding their fundamental microscopic quantum behavior. All of these problems can be phrased as fermionic (e.g., electrons, quarks) and bosonic particles (e.g., lattice vibrations, photons) interacting with each other. However, solving these systems of strongly interacting fermions and bosons is out of reach for classical computers and also non-error-corrected qubit-only quantum computers. In this paper, we introduce a hybrid qubit and bosonic-oscillator quantum-simulation framework, which is so efficient that it could simulate some of these strongly correlated problems even without error correction. Specifically, we leverage the available universal bosonic gate set with O ( 1 ) operations per gate as a function of the bosonic cutoff. This enables our framework to provide exact decompositions of higher-order fermion-boson interactions, including notoriously challenging terms such as ( 2 + 1 ) D gauge theories. We show how to compile dynamics, ancilla-free partial error detection, nonlocal observable measurements, and ground-state energy estimation using both qubit-boson variational quantum eigensolvers and quantum signal processing. We introduce a “post-qubit quantum advantage” experiment which is out of reach for both near-term qubit-only quantum computers and classical computers. We justify this proposal by estimating bosonic gate fidelities through high–boson-occupation Lindblad simulation and by comparing to qubit-only algorithms. For the latter, we develop a new qubit-based algorithm for simulating a beam splitter with a boson number cutoff N max , which scales as O ˜ ( N max 1 + o ( 1 ) ) , asymptotically outperforming previously known qubit methods. However, to reach the same circuit fidelity as qubit-boson hardware with 99.9% per-gate fidelity, a qubit quantum computer using the best-known method would still need fidelities of at least 99.9999%, likely requiring substantial error correction. Our framework can be used in superconducting hardware, which we focus on, as well as trapped-ion and neutral-atom hardware. This work establishes digital hybrid qubit-boson quantum simulation as a promising and advantageous method for solving the many challenging, strongly correlated, fermion-boson models of relevance to science and technology.
Why quantum computing? As we all know very well, conventional (electronic) computers have enormous power which, moreover, is constantly increasing. With their help gigantic quantities of data can be handled and evaluated. This is of crucial importance, for instance, in modern high energy physics experiments. On the other hand, computers can be used to model very complex systems with the aim of predicting, for example, future developments in our environment, notably climatic changes. Thanks to their incredible speed computers are, of course, destined to treat mathematical problems for which specific algorithms exist very efficiently. To mention a simple example, the digits of π are readily evaluated up to a very high order. Last, but not least, their potential to solve differential equations numerically, even when they are nonlinear, makes computers indispensable in modern research. So, one might ask, is there any reason to look for fundamentally new ways of computing as they are seemingly offered by quantum processes? What advantages can be expected when the laws of quantum mechanics, rather than those of classical physics, are exploited for computation? Two answers were given: first, the simulation of complex quantum systems on classical computers meets serious difficulties that might be overcome with the help of quantum computers. This idea was suggested in 1982 by R. Feynman, and in the 1990s it was indeed shown by several groups of researchers that quantum mechanical systems for which no efficient simulation on a classical computer is known can be efficiently simulated on a quantum computer.