We briefly look at the background of the research field of "quanta and information" from the epistemological point of view. Then quantum computation is very briefly explained and in particular quantum brachistochrone as my own research. The optimal quantum control to drive a transition from a given state to a target state in the shortest time naturally leads to the two state vector formulation of quantum mechanics advocated by Aharonov et al.
We briefly look at the background of the research field of "quanta and information" from the epistemological point of view. Then quantum computation is very briefly explained and in particular quantum brachistochrone as my own research. The optimal quantum control to drive a transition from a given state to a target state in the shortest time naturally leads to the two state vector formulation of quantum mechanics advocated by Aharonov et al.
Minimum-time quantum control protocols can be obtained from the quantum brachistochrone formalism [Carlini, Hosoya, Koike, and Okudaira, Phys. Rev. Lett. 96, 06053, (2006)]. We point out that the original treatment implicitly applied the variational calculus with fixed boundary conditions. We argue that the genuine quantum brachistochrone problem involves a variational problem with a movable endpoint, contrary to the classical brachistochrone problem. This formulation not only simplifies the derivation of the quantum brachistochrone equation but introduces an additional constraint at the endpoint due to the boundary effect. We present the general solution to the full quantum brachistochrone equation and discuss its main features. Using it, we prove that the speed of evolution under constraints is reduced with respect to the unrestricted case. In addition, we find that solving the quantum brachistochrone equation is closely connected to solving the dynamics of the Lagrange multipliers, which is in general governed by nonlinear differential equations. Their numerical integration allows generating time-extremal trajectories. Furthermore, when the restricted operators form a closed subalgebra, the Lagrange multipliers become constant and the optimal Hamiltonian takes a concise form. The new class of analytically solvable models for the quantum brachistochrone problem opens up the possibility of applying it to many-body quantum systems, exploring notions related to geometry such as quantum speed limits, and advancing significantly the quantum state and gate preparation for quantum information processing.
The quantum brachistochrone problem -- to find the time-optimal transition between given initial and final quantum states -- is investigated in this bachelor's thesis. First the quantum equivalent of distance (the Fubini-Study metric) is formulated, using geometry of spheres. Together with constraints, it is used to create the quantum action. Functional derivatives are then taken of said action to find equations of motion (eoms). For the unconstrained case (excluding finite energy), these eoms are solved in closed form, whilst general to be solved formulae are obtained for the constrained case. The latter is used to explicitly solve an example quantum brachistochrone problem: a spin-1/2 particle in a controllable magnetic field constrained to an x-y plane. Finally, the link with quantum computing is illustrated through the time-optimization of unitary transformations that the quantum brachistochrone yields us.
Quantum Information is the science that aims to use the unusual behavior of the microscopic world, governed by the laws of Quantum Mechanics, in order to improve the way in which we compute or communicate information. Though the first ideas in this direction come from the early 80’s, it is in the last decade when Quantum Information has suffered an spectacular development. It is impossible to resume in a paper like this one the importance and complexity of the field. Therefore, I will limit to briefly explain some of the initial ideas (considered classical by now), and to briefly suggest some of the modern lines of research. By the nature of this exposition, I have decided to avoid rigor and to concentrate more in ideas and intuitions. Anyhow, I have tried to provide with enough references, in such a way that an interested reader could find there proper theorems and proofs.
Quantum brachistochrone (QB) is a quantum analogue of classical brachistochrone (shortest path). It is a solution to the following problem: How can we perform a desired quantum operation (or obtain a desired final quantum state) most quickly, by a time-dependent Hamiltonian subject to given constraints? Finding QB is a fundamental problem in quantum mechanics in its own right. Moreover, it will be useful in the study of quantum information and quantum engineering, such as quantum speed limits and implementations of quantum computers. A general framework for finding QBs, called QB formalism, has been developed. It is based on Pontryagin's maximum principle. We review the basics of the QB formalism, give simple examples, and briefly discuss some related studies. This article is part of the theme issue 'Shortcuts to adiabaticity: theoretical, experimental and interdisciplinary perspectives'.
Information is physical: the laws which govern its encoding, processing and communication are bound by those of its unavoidably physical embodiment. In today's informatics, information obeys the laws of Newton's and Maxwell's classical physics: this assertion holds all the way from commercial computers down to (up to?) their most abstract models, e.g. Turing machines and lambda-calculus. Today's computation and communication are classical. Research in quantum information processing and communication was born some twenty five years ago, as a child of two major scientific achievements of the 20th century, namely quantum physics and information sciences. The driving force of this interdisciplinary research is that of looking for the consequences of having information encoding, computation and communication based upon the laws of quantum physics, i.e. the ultimate knowledge that we have, today, of the foreign world of elementary particles, as described by quantum mechanics. Breakthroughs in cryptography, communications, information theory, algorithmics and, more recently, in abstract computational models, programming languages and semantics frameworks, have shown that this transplantation of information from classical to quantum has far reaching consequences, both quantitative and qualitative, and opens new avenues for research within the foundations of computer science.From a computer scientist's point of view, which is my point of view, I will explain the basics, survey the main achievements, and outline the current hot topics and major challenges of this promising and stimulating research. I will not assume any prior knowledge in quantum mechanics from the audience.