[math-fun] 3x+1 problem
I seem to have deleted a recent message which gave the bound to which the 3x+1 problem has been checked (something like 10^17) and an address of a Jeff Lagarias web page which gave info on the problem. Could someone kindlily resend to me? Thanks! R.
Does anyone know any more details about this announcement? X-URL: http://story.news.yahoo.com/news?tmpl=story&u=/nf/20031002/bs_nf/22407 Researchers Create Super-Fast Quantum Computer Simulator Thu Oct 2, 3:03 PM ET Mike Martin , sci.NewsFactor.com Using a classical computer to manipulate Shor's algorithm -- a centerpiece of quantum-computer science -- Japanese researchers hope they can shore up the shaky marriage of quantum mechanics and computing that scientists hope will someday produce offspring of super-fast HAL-9000 style machines. "The proposed system will be a powerful tool for the development of quantum algorithms," claim University of Tokyo scientists Minoru Fujishima, Kento Inai, Tetsuro Kitasho and Koichiro Hoh. Shoring up Shor Shor's algorithm uses the probabilities of quantum mechanics to factor large numbers in fewer steps than possible with more conventional means. Factoring a number means breaking it into a product of its prime factors. Fifteen, for instance, breaks into the product of the prime numbers 3 and 5. Using conventional factoring algorithms, the time it takes to factor a number increases exponentially -- 2^N -- with the size of the number, where the exponent N is the number of digits. To factor the 5-digit number 65,448, for instance, it might take 2^5, or 32 minutes, using conventional computer algorithms. In 1994, AT & T researcher Peter Shor created an algorithm based on quantum probabilities wherein the time required to factor a number grows only as a polynomial function -- N^2 -- of the number's size. N again is the number of digits. In theory, factoring 65,448 with Shor's algorithm would take seven minutes less than with the conventional method -- 5^2, or 25 minutes. Simulate To Emulate Aircraft engineers design jumbo jets using ground-based simulators to emulate real flying. Likewise, computer engineers hope to design quantum-computer software and hardware by simulating quantum computation on classical computers. "In order to develop software efficiently, a system which can emulate large-scale problems at high speed is required," Fujishima told NewsFactor. Running algorithms designed for quantum computers on a classic Compaq or Dell (Nasdaq: DELL - news) isn't easy. Shor's algorithm, for instance, only produces a correct result that is "highly probable," so it must be run repeatedly to increase this probability. The number of simulated "quantum bits," or qubits, required to process these re-runs quickly grows cumbersome in a classical simulation of the algorithm. Now, however, "something very important has happened," said Texas A&M electrical engineering professor Laszlo Kish. "Fujishima and coworkers have found the way to avoid the complexity required to run Shor's quantum algorithm on a classical computer using CMOS technology." Fujishima and his team invented a tool called a "quantum index processor" (QIP) to increase the speed at which classical computers run quantum algorithms, such as Shor's. "We found that the computation time of the QIP is 10^26 times faster than that of the conventional emulator," Fujishima explained. "Our emulator, named a "quantum index processor," emulates the procedure of quantum computing at high speed by using the latest CMOS LSI," a type of semiconductor. Closer to Real "The speed of this computer emulator is 10^26 times (100 trillion trillion times) faster than our workstations," Kish told NewsFactor -- faster, in fact, than any other quantum simulator built to date, and more closely approaching the speed of a real quantum computer. If we are ever to realize the quantum future of computers, "we need classical (CMOS) special-purpose computing approaches, perhaps inspired by quantum algorithms like Fujishima's present method," added Kish, who recently showed that quantum computers may never become reality without some serious retooling. "We should stop wasting money for 'quantum dreams,'" he says.
On Thu, 2 Oct 2003, Henry Baker wrote:
Does anyone know any more details about this announcement?
You mean, besides the fact that the author's understanding of even the basics of the factoring problem is appalling? I factored his "32-minute" number in about 30 seconds in my head! David Moulton
X-URL: http://story.news.yahoo.com/news?tmpl=story&u=/nf/20031002/bs_nf/22407
Researchers Create Super-Fast Quantum Computer Simulator Thu Oct 2, 3:03 PM ET
Mike Martin , sci.NewsFactor.com
...
Using conventional factoring algorithms, the time it takes to factor a number increases exponentially -- 2^N -- with the size of the number, where the exponent N is the number of digits. To factor the 5-digit number 65,448, for instance, it might take 2^5, or 32 minutes, using conventional computer algorithms.
In 1994, AT & T researcher Peter Shor created an algorithm based on quantum probabilities wherein the time required to factor a number grows only as a polynomial function -- N^2 -- of the number's size. N again is the number of digits.
In theory, factoring 65,448 with Shor's algorithm would take seven minutes less than with the conventional method -- 5^2, or 25 minutes.
I wouldn't dismiss this so quickly. It sounds like the researchers have developed a fast way to "emulate" quantum computer algorithms on conventional systems. Sounds intriguing. Does anyone know more details? DHB -----Original Message----- On Thu, 2 Oct 2003, Henry Baker wrote:
Does anyone know any more details about this announcement?
You mean, besides the fact that the author's understanding of even the basics of the factoring problem is appalling? I factored his "32-minute" number in about 30 seconds in my head! David Moulton
X-URL: http://story.news.yahoo.com/news?tmpl=story&u=/nf/20031002/bs_nf/22407
Quoting "David P. Moulton" <moulton@idaccr.org>:
On Thu, 2 Oct 2003, Henry Baker wrote:
Does anyone know any more details about this announcement?
You mean, besides the fact that the author's understanding of even the basics of the factoring problem is appalling? I factored his "32-minute" number in about 30 seconds in my head!
David Moulton
Not so long ago the astounding news was that someone had used a five bit quantum computer to factor 15!! Much longer ago I recall when people were demonstrating a little plastic cube which had a !transistor! embedded in it along with a little speaker and a theromocouple. It squealed when you held it between your fingers. Totally, absolutely amazing! Don't forget Moore's Law. - hvm ------------------------------------------------- Obtén tu correo en www.correo.unam.mx UNAMonos Comunicándonos
On Fri, 3 Oct 2003 mcintosh@servidor.unam.mx wrote:
Quoting "David P. Moulton" <moulton@idaccr.org>:
On Thu, 2 Oct 2003, Henry Baker wrote:
Does anyone know any more details about this announcement?
You mean, besides the fact that the author's understanding of even the basics of the factoring problem is appalling? I factored his "32-minute" number in about 30 seconds in my head!
David Moulton
Not so long ago the astounding news was that someone had used a five bit quantum computer to factor 15!! Much longer ago I recall when people were demonstrating a little plastic cube which had a !transistor! embedded in it along with a little speaker and a theromocouple. It squealed when you held it between your fingers. Totally, absolutely amazing!
Don't forget Moore's Law.
- hvm
I think some people were missing the point of my comment. I'm not saying anything bad about the actual research, which may be quite interesting; I'm saying that the writer was botching things to an astounding degree. And remember, the "32-minute" number was supposed to take 32 minutes to factor "using conventional computer algorithms". Even naive trial division with such a number will take a tiny fraction of a second. Richard Schroeppel says
I'm inclined to give the science writer the benefit of the doubt on explaining factoring time. It's so oversimplified it's wrong, but for a target audience that needs an explanation of what factoring is at all, What can you do? Any article with the subexpression O(e^(1.923*cbrt(logN (loglog N)^2))) isn't going to be read by this audience.
"It's so oversimplified it's wrong" is quite an understatement! I would complain less if the author had used "microseconds", say, rather than "minutes" in his estimates. And I would complain less if he had used a number to be factored that didn't have so many small factors, or even was odd! And while he shouldn't necessarily give the correct running times, he states the ones he uses, which are totally ludicrous, with such authority and definitiveness. While it's good that a general audience can read a story about factoring and quantum computers, I think it's a mistake to give such an incorrect picture of the truth that it might interfere with other understanding that could result from reading some other article. David Moulton
participants (5)
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David H Bailey -
David P. Moulton -
Henry Baker -
mcintosh@servidor.unam.mx -
Richard Guy