e-book Problems in quantum mechanics

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By now, most people have heard that quantum computing is a revolutionary technology that leverages the bizarre characteristics of quantum mechanics to solve certain problems faster than regular computers can. Those problems range from the worlds of mathematics to retail business, and physics to finance.

Gödel and Turing enter quantum physics

If we get quantum technology right, the benefits should lift the entire economy and enhance U. The promise of quantum computing was first recognized in the s yet remains unfulfilled.

This loss of coherence called decoherence , caused by vibrations, temperature fluctuations, electromagnetic waves and other interactions with the outside environment, ultimately destroys the exotic quantum properties of the computer. Given the current pervasiveness of decoherence and other errors, contemporary quantum computers are unlikely to return correct answers for programs of even modest execution time.

While competing technologies and competing architectures are attacking these problems, no existing hardware platform can maintain coherence and provide the robust error correction required for large-scale computation. A breakthrough is probably several years away. The billion-dollar question in the meantime is, how do we get useful results out of a computer that becomes unusably unreliable before completing a typical computation?

Answers are coming from intense investigation across a number of fronts, with researchers in industry, academia and the national laboratories pursuing a variety of methods for reducing errors. One approach is to guess what an error-free computation would look like based on the results of computations with various noise levels.

A completely different approach, hybrid quantum-classical algorithms, runs only the most performance-critical sections of a program on a quantum computer, with the bulk of the program running on a more robust classical computer. While classical computers are also affected by various sources of errors, these errors can be corrected with a modest amount of extra storage and logic. That is just a tiny fraction of the number of classical bits your device has available to it, typically hundreds of billions. The trouble is, quantum mechanics challenges our intuition. So we struggle to figure out the best algorithms for performing meaningful tasks.

Problems in Quantum Mechanics

To help overcome these problems, our team at Los Alamos National Laboratory is developing a method to invent and optimize algorithms that perform useful tasks on noisy quantum computers. Algorithms are the lists of operations that tell a computer to do something, analogous to a cooking recipe. Compared to classical algorithms, the quantum kind are best kept as short as possible and, we have found, best tailored to the particular defects and noise regime of a given hardware device. That enables the algorithm to execute more processing steps within the constrained time frame before decoherence reduces the likelihood of a correct result to nearly zero.

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In our interdisciplinary work on quantum computing at Los Alamos, funded by the Laboratory Directed Research and Development program, we are pursuing a key step in getting algorithms to run effectively. The main idea is to reduce the number of gates in an attempt to finish execution before decoherence and other sources of errors have a chance to unacceptably reduce the likelihood of success.

Unless I am much mistaken, it would exceed the force of human wit to consider so many causes of motion at the same time, and to define the motions by exact laws which would allow of an easy calculation. Since essentially all relevant physical systems are composed by a collection of interacting particles, the many-body problem is extremely important.

Home: Quantum Mechanics

In the present article, I will focus on the quantum many-body problem which has been my main topic of research since The complexity of quantum many-body systems was identified by physicists already in the s. Around that time, the great physicist Paul Dirac envisioned two major problems in quantum mechanics.

The second problem was precisely the quantum many-body problem. Luckily, the quantum states of many physical systems can be described using much less information than the maximum capacity of their Hilbert spaces. Simply put, a quantum wave function describes mathematically the state of a quantum system. The first quantum system to receive an exact mathematical treatment was the hydrogen atom. The latter is the sum of two terms:. The eigenvalues and the corresponding eigenstates are. For concreteness, let us consider the following example: the quantum harmonic oscillator.

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The QHO is the quantum-mechanical counterpart of the classical harmonic oscillator see the figure below , which is a system that experiences a force when displaced from its initial that restores it to its equilibrium position. The animation below compares the classical and quantum conceptions of a simple harmonic oscillator. While a simple oscillating mass in a well-defined trajectory represents the classical system blocks A and B in the figure above , the corresponding quantum system is represented by a complex wave function. Though it is intuitive to think of spin as a rotation of a particle around its own axis this picture is not quite correct since then the particle would rotate at a faster than light speed which would violate fundamental physical principles.

Problems on Quantum Mechanics from Previous Gate Exams

If fact spins are quantum mechanical objects without classical counterpart. Quantum spin systems are closely associated with the phenomena of magnetism. Magnets are made of atoms, which are often small magnets. When these atomic magnets become parallelly oriented they give origin to the macroscopic effect we are familiar with. I will now provide a quick summary of the basic components of machine learning algorithms in a way that will be helpful for the reader to understand their connections with quantum systems.

Machine learning approaches have two basic components Carleo, :. Artificial neural networks are usually non-linear multi-dimensional nested functions. Their internal workings are only heuristically understood and investigating their structure does not generate insights regarding the function being it approximates.

Restricted Boltzmann Machines are generative stochastic neural networks. They have many applications including:. They are different from other more popular neural networks which estimate a value based on inputs while RBMs estimate probability densities of the inputs they estimate many points instead of a single value.

RBMs have the following properties:. The energy functional to be minimized is given by:. The joint probability distribution of both visible and hidden units reads:. Tracing out the hidden units, we obtain the marginal probability of a visible input vector:.

Since, as noted before, hidden visible unit activations are mutually independent given the visible hidden unit activations one can write:.

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Finally, the activation probabilities read:. The training steps are the following :. The following analysis is heavily based on this excellent tutorial. The three figures below show how a RBM processes inputs. They learn to reconstruct the data performing a long succession of passes forward and backward ones between its two layers. In the backward pass, as shown in the diagram below, the activation functions of the nodes in the hidden layer become the new inputs.

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The product of these inputs and the respective weights are summed and the new biases b from the visible layer are added at each input node. Naturally, the reconstructions and the original inputs are very different at first since the values of w are randomly initialized. However, as the error is repeatedly backpropagated against the w s, it is gradually minimized. Joining both conditional distributions, the joint probability distribution of x and a is obtained i.

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  • The question is how to reformulate the time-independent Schrodinger equation, which is an eigenvalue problem, as a machine learning problem. As it turns out, the answer has been known for quite some time, and it is based on the so-called variational method , an alternative formulation of the wave equation that can be used to obtain the energies of a quantum system.

    Using this method we can write the optimization problem as follows:.