Archive for the ‘mathematics’ category: Page 14

Dec 19, 2021

Beyond Qubits: Unlocking the Third State in Quantum Processors

Posted by in categories: energy, mathematics, quantum physics

By Alex Hill, Senior Quantum Systems Engineer

Qubits are the basic building block of a quantum processor, and are so named because they represent a continuum of complex superpositions of two basic quantum states. The power of qubits comes in part from their ability to encode significantly more information than a classical bit — an infinite set of states between 0 and 1. In mathematical terms, quantum gates that manipulate the state of individual qubits are unitary operators drawn from SU.

Rigetti’s superconducting quantum processors are based on the transmon design [1]. Each physical qubit is an anharmonic oscillator, meaning that the energy gaps between subsequent qubit energy states decrease as the qubit climbs higher up the state ladder. We typically only address the first two states, 0 and 1 (in the literature, sometimes referred to as g(round) and e(xcited)); however, the design of our qubits supports even higher states. The simple structure of the transmon energy levels gives superconducting qubits the unique ability to address many of these states in a single circuit.

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Dec 19, 2021

Cosmologists Close in on Logical Laws for the Big Bang

Posted by in categories: cosmology, mathematics, physics

Physicists are translating commonsense principles into strict mathematical constraints for how our universe must have behaved at the beginning of time.

Dec 17, 2021

Mathematician Hurls Structure and Disorder Into Century-Old Problem

Posted by in categories: computing, mathematics

The mathematician Ben Green of the University of Oxford has made a major stride toward understanding a nearly 100-year-old combinatorics problem, showing that a well-known recent conjecture is “not only wrong but spectacularly wrong,” as Andrew Granville of the University of Montreal put it. The new paper shows how to create much longer disordered strings of colored beads than mathematicians had thought possible, extending a line of work from the 1940s that has found applications in many areas of computer science.

The conjecture, formulated about 17 years ago by Ron Graham, one of the leading discrete mathematicians of the past half-century, concerns how many red and blue beads you can string together without creating any long sequences of evenly spaced beads of a single color. (You get to decide what “long” means for each color.)

This problem is one of the oldest in Ramsey theory, which asks how large various mathematical objects can grow before pockets of order must emerge. The bead-stringing question is easy to state but deceptively difficult: For long strings there are just too many bead arrangements to try one by one.

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Dec 16, 2021

Building the Mathematical Library of the Future

Posted by in categories: futurism, mathematics

A small community of mathematicians is using a software program called Lean to build a new digital repository. They hope it represents the future of their field.

Dec 14, 2021

From flashing fireflies to cheering crowds: Physicists unlock secret to synchronisation

Posted by in categories: computing, information science, mathematics, physics

Physicists from Trinity have unlocked the secret that explains how large groups of individual “oscillators”—from flashing fireflies to cheering crowds, and from ticking clocks to clicking metronomes—tend to synchronize when in each other’s company.

Their work, just published in the journal Physical Review Research, provides a mathematical basis for a phenomenon that has perplexed millions—their newly developed equations help explain how individual randomness seen in the and in electrical and computer systems can give rise to synchronization.

We have long known that when one clock runs slightly faster than another, physically connecting them can make them tick in time. But making a large assembly of clocks synchronize in this way was thought to be much more difficult—or even impossible, if there are too many of them.

Dec 11, 2021

Machine learning speeds up vehicle routing

Posted by in categories: information science, mathematics, robotics/AI, transportation

Strategy accelerates the best algorithmic solvers for large sets of cities.

Waiting for a holiday package to be delivered? There’s a tricky math problem that needs to be solved before the delivery truck pulls up to your door, and MIT researchers have a strategy that could speed up the solution.

The approach applies to vehicle routing problems such as last-mile delivery, where the goal is to deliver goods from a central depot to multiple cities while keeping travel costs down. While there are algorithms designed to solve this problem for a few hundred cities, these solutions become too slow when applied to a larger set of cities.

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Dec 8, 2021

What Are The Milestones Of Robotaxi Service?

Posted by in categories: mathematics, robotics/AI, transportation

More than a score of companies are pushing to be early winners in the race for self-driving taxis — robotaxis — with the potential that brings to capture the entire value chain of car transport from your riders. They are all at different stages, and they almost all want to convince the public and investors that they are far along.

To really know how far along a project is, you need the chance to look inside it. To see the data only insiders see on just how well their vehicle is performing, as well as what it can and can’t do. Most teams want to keep those inside details secret, though in time they will need to reveal them to convince the public, and eventually regulators that they are ready to deploy.

Because they keep them secret, those of us looking in from the outside can only scrape for clues. The biggest clues come when they reach certain milestones, and when they take risks which tell us their own internal math has said it’s OK to take that risk. Most teams announce successes and release videos of drives, but these offer us only limited information because they can be cherry picked. The best indicators are what they do, not what they say.

Dec 7, 2021

DeepMind’s AI Helped Crack Two Mathematical Puzzles That Stumped Humans for Decades

Posted by in categories: biological, information science, mathematics, robotics/AI, time travel

Working with two teams of mathematicians, DeepMind engineered an algorithm that can look across different mathematical fields and spot connections that previously escaped the human mind. The AI doesn’t do all the work—when fed sufficient data, it finds patterns. These patterns are then passed on to human mathematicians to guide their intuition and creativity towards new laws of nature.

“I was not expecting to have some of my preconceptions turned on their head,” said Dr. Marc Lackenby at the University of Oxford, one of the scientists collaborating with DeepMind, to Nature, where the study was published.

The AI comes just a few months after DeepMind’s previous triumph in solving a 50-year-old challenge in biology. This is different. For the first time, machine learning is aiming at the core of mathematics—a science for spotting patterns that eventually leads to formally-proven ideas, or theorems, about how our world works. It also emphasized collaboration between machine and man in bridging observations to working theorems.

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Dec 5, 2021

DeepMind’s AI helps untangle the mathematics of knots

Posted by in categories: mathematics, robotics/AI

Computer simulations and visualizations of knots and other objects have long helped mathematicians to look for patterns and develop their intuition, says Jeffrey Weeks, a mathematician based in Canton, New York, who has pioneered some of those techniques since the 1980s. But, he adds, “Getting the computer to seek out patterns takes the research process to a qualitatively different level.”

The authors say the approach, described in a paper in the 2 December issue of Nature1, could benefit other areas of maths that involve large data sets.

Dec 4, 2021

AI Is Discovering Patterns in Pure Mathematics That Have Never Been Seen Before

Posted by in categories: biological, mathematics, robotics/AI

We can add suggesting and proving mathematical theorems to the long list of what artificial intelligence is capable of: Mathematicians and AI experts have teamed up to demonstrate how machine learning can open up new avenues to explore in the field.

While mathematicians have been using computers to discover patterns for decades, the increasing power of machine learning means that these networks can work through huge swathes of data and identify patterns that haven’t been spotted before.

In a newly published study, a research team used artificial intelligence systems developed by DeepMind, the same company that has been deploying AI to solve tricky biology problems and improve the accuracy of weather forecasts, to unknot some long-standing math problems.

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