杏吧原创

WE HAVE a hardware problem. Every computer ever built is just a snazzier version of a blueprint drawn up by the mathematician Alan Turing in the 1930s. He created a hypothetical 鈥渦niversal Turing machine鈥�, a stripped-down processor which, he proved, could do anything that any digital computer would ever be able to do.

By going back to the very roots of what maths can and cannot prove, Turing was also able to show that there are some things that none of our computers could ever do鈥攑roblems no program could ever solve. This result places limits on all of science. When physicists use computers to investigate the workings of the Universe, there are certain things they just won鈥檛 be able to find out.

But who says we鈥檙e stuck with ordinary computers? Cristian Calude and Boris Pavlov of the University of Auckland in New Zealand have proved that Turing didn鈥檛 have all the answers. They have found a trick that could sidestep Turing鈥檚 computational barrier and give access to otherwise forbidden knowledge.

While some things are uncomputable by Turing machines, that doesn鈥檛 mean they鈥檙e totally off-limits. If digital processing can鈥檛 give you the answer, what about doing the number crunching with a totally different physical mechanism? One possibility, Calude says, is a quantum computer.

These machines鈥攊f anyone ever manages to build one鈥攑erform computations using particles such as electrons and photons, which can exist in different quantum states. The states act just like digital 1s and 0s, with the crucial difference that the strange laws of quantum mechanics allow these particles to exist in a 鈥渟uperposition鈥� of states. That means they can be in two or more states at once鈥攕uch as spinning clockwise and anticlockwise simultaneously, or having several different energies at once. Quantum computers can exploit this counterintuitive property to carry out thousands of separate computations simultaneously.

It鈥檚 miles faster than classical computing, but by itself that鈥檚 not enough to get you past the Turing barrier. After all, as the Caltech physicist and Nobel laureate Richard Feynman said 20 years ago, if you鈥檙e going nowhere, then simply doing things more quickly will just get you nowhere faster. Calude has something more subtle in mind.

His suggestion is to think bigger: why not create a superposition of every conceivable state at once? Something like a hydrogen atom has infinitely many possible energy levels. While the levels start out well-spaced, they get closer as the energies grow higher, until they become almost indistinguishable. In a paper to be published in the inaugural edition of MIT鈥檚 new journal Quantum Information Processing, Calude and Pavlov have shown that a superposition of an infinite number of energy states would allow a quantum computer to do things no classical computer can ever manage鈥攁lmost like running 鈥渇orever鈥� in a finite time.

This leap means that a quantum computer can overcome Turing鈥檚 most famous barrier to computing power: the 鈥渉alting problem鈥�. Given any computer program and an input, can a Turing machine tell in advance whether that program will eventually halt or grind away forever? Turing himself proved the answer is no鈥攖he program might stop after a couple of days, or a billion years, but the only way to find out is to run it and wait. That might seem no more than a curiosity, but getting round this barrier would be a gigantic breakthrough for science, solving many important questions in maths and physics.

Stop and search

Goldbach鈥檚 conjecture, for example, states that every even number is the sum of two primes, and it can be recast as a halting problem. All you do is write a computer program that searches for a counterexample鈥攖hat is, for an even number that is not the sum of two primes. If it finds one, it stops, proving Goldbach鈥檚 conjecture wrong. If it goes on forever, the conjecture is, apparently, right. A host of other mathematical questions can be written as halting problems鈥攊ncluding the Riemann hypothesis, an unproven assumption upon which a great deal of physics and maths rests (New 杏吧原创, 11 November 2000, p 32).

So how would a quantum processor overcome Turing鈥檚 barrier and see whether these programs halt? Any given program investigating another can do only one thing: run the program under test and then ask the question, 鈥淒oes it halt within x amount of time?鈥� Because the answer can only be yes or no, the processor might report that the program under test doesn鈥檛 stop, whereas, in fact, it just hadn鈥檛 stopped within that time. However long a time interval you allow, if the answer is always no you can still never believe it.

But if you assign different investigating questions to an infinite superposition of quantum states, something remarkable happens. While the program under test runs as normal, the questions in the infinite superposition ask every possible 鈥渄oes it stop鈥� question at once. Calude has used this to uncover a subtle signal in the program under test. This signal is invisible to any classical analysis but shows up under the infinite superposition, and gives a clue as to whether or not the program will halt.

More precisely, it gives a measure of the proportion of misleading answers to the 鈥渄oes it stop鈥� question. Although a non-halting program gives you an infinite number of worthless 鈥渘o鈥� answers, Calude and Pavlov have shown that, as the quantum program runs, it can measure the error introduced by these worthless answers. And the longer you run the quantum algorithm, the less significant the error becomes. 鈥淭his is our key result,鈥� Calude says. 鈥淚t gives insight into the behaviour of non-halting problems no other mathematical result has been able to give.鈥�

Calude and Pavlov鈥檚 algorithm doesn鈥檛 provide a definite 鈥測es鈥� or 鈥渘o鈥� to the question of whether a program halts, but it does give an answer accompanied by a percentage certainty. If you want to get a more accurate result, you just run the quantum program for longer.

Calude is extremely proud of this result: he believes it could be implemented on a real-life quantum computer, laying much that is 鈥渦nknowable鈥� open to attack. 鈥淯sing infinite superpositions is rather theoretical, but not necessarily non-practical or non-testable,鈥� he says.

He is bound to face a great deal of scepticism, of course. Most quantum computing researchers still think Feynman was right; quantum computers remain bound by Turing鈥檚 barrier. 鈥淚f you look at the theory of quantum mechanics, everything in there is computable,鈥� says Richard Jozsa, a quantum algorithms researcher at Bristol University. And by computable, he means that a Turing machine could eventually do it. Jozsa does admit that there鈥檚 more to the quantum world than we know about just yet, so we can鈥檛 rule out finding ways past the Turing barrier in future. But he says he hasn鈥檛 seen any way to do it so far.

Proof positive

Calude counters that such doubts are only a matter of opinion, and don鈥檛 prove that quantum theory isn鈥檛 up to the challenge. He thinks Feynman鈥檚 20-year-old pronouncement has closed people鈥檚 minds. 鈥淧eople were brainwashed by Feynman into thinking it was impossible,鈥� he says. Calude says his paper offers a proof that quantum computers can鈥攊n theory at least鈥攂reak the Turing barrier.

In practice, however, this may prove enormously difficult. Gregory Chaitin, a mathematician based at IBM鈥檚 Yorktown Heights laboratory, thinks the signal that hints in advance whether the program will ever halt will be too small for measurements to pick up. 鈥淚 think they involve being able to measure real numbers with infinite precision, which I don鈥檛 think is possible,鈥� he says.

But Calude believes that such problems are often not insurmountable鈥攁nd he has reason to be bullish. Last year he managed to do another calculation that Chaitin had thought beyond all hope (see 鈥淎 glimpse of the impossible鈥�). Similarly, Calude is confident that reading the hidden quantum signal鈥攁nd thus breaking the Turing barrier鈥攚ill be possible. And he鈥檚 not alone. In the past couple of years, several more researchers have begun to ask whether Turing only had half the picture. Tien Kieu of Swinburne University of Technology in Australia, for example, has also come up with a way for quantum computers to surpass Turing鈥檚 barrier. Like Calude鈥檚 method, it exploits the possibility of infinite computations by encoding the problem in the energy states of, say, an atom or molecule. Others have suggested that black holes or DNA might provide a way to peek into the unknowable (see 鈥淭o infinity and beyond鈥�). 鈥淚t seems the problem鈥檚 ripe for solving,鈥� Calude says.

Testing time

But it鈥檚 not yet clear whether the theory will translate into hard knowledge. Although Calude is convinced his maths is right, only an experiment can reveal whether his idea is practical. And Kieu believes that the Universe might not be around for long enough to complete the execution of such a quantum algorithm.

Even so, Calude thinks that some fundamental aspect of maths and physics will have to change. 鈥淎lready we鈥檝e realised that classically uncomputable is not the same as quantum-mechanically uncomputable,鈥� he says. And that alone might be enough to challenge the nature of mathematical thought about, for instance, what constitutes proof of a theorem. 鈥淏ecause of these new computational models, the idea of 鈥榩roof鈥� might鈥攁nd I personally believe that it will鈥攃hange,鈥� he says. 鈥淎nd if the nature of proof changes, the idea of mathematical knowledge will also change.鈥�

This could mark the return to a more positive mathematical era. In 1930, the German mathematician David Hilbert showed his colleagues that maths would meet all the challenges that faced it. Then, in the following year, Kurt G枚del deflated the optimism with proofs that there were things in the mathematical universe that might be true, but were unprovable. It was G枚del鈥檚 work that enabled Turing to show that his computation machine would not be able to answer certain questions鈥攕uch as the halting problem.

But, thanks to Calude, that definition of 鈥渦ncomputable鈥� will have to go. Who knows what other 鈥渋nsurmountable鈥� barriers will crumble to the ground? An audacious assault on the limits of knowledge could reveal and unravel more than we ever thought possible. All we need is a machine that can break the laws of logic.

Calude has little respect for 鈥渦nbreakable鈥� barriers. Last year, New 杏吧原创 published a story about Omega, a bizarre number linked to Turing鈥檚 proof that there are things computers can鈥檛 do (10 March 2001, p 28). There was thought to be no way to even begin calculating the random sequence of digits that make up Omega. But we鈥檙e now able to publish the first 64 digits (below).

Contrary to all expectations, calculating these bits wasn鈥檛 that hard. The first n digits of Omega represent the probability that a program that鈥檚 less than n bits long (when translated into binary from whatever programming language is used) will halt. Calude, together with Michael Dinneen and Chi-Kou Shu, also at Auckland University, ran all possible programs that are 1 bit long, then 2 bits, then 3 and so on up to 84 bits, to see whether they halted. They made their task more manageable by weeding out all the programs that duplicated each other. This reduced the size of the computational task by a factor of more than 10¹⁷, but they still had to work with around 3 gigabytes of compressed data.

That didn鈥檛 give the first 84 digits exactly, however. Calude couldn鈥檛 ignore the potential that longer programs might affect Omega鈥檚 first few digits: it鈥檚 conceivable that a large set of very long programs could contribute to the values of the first bits of Omega.

The breakthrough Calude鈥檚 team made was to discover a structure that all halting programs longer than 84 bits must have. They found that, in the finite set of programs the team had computed, these programs all have to start with a particular sequence of bits. This limits the contribution that the remaining infinite set of halting programs can make to the first bits of Omega. 鈥淚n our case, this set cannot influence the first 69 bits of Omega,鈥� says Calude. 鈥淗owever, for technical reasons, only the first 64 are exact.鈥�

Calude may well have prised open the door to the Universe鈥檚 deepest secrets. 鈥淥mega鈥檚 first few thousand digits contain the answers to more mathematical questions than can be written down in the entire Universe,鈥� says IBM鈥檚 Charles Bennett, a pioneer in the field of quantum information. John Casti of the Santa Fe Institute echoes this. Omega鈥檚 digits encode the 鈥渟ecret of the Universe鈥�,鈥� he says. 鈥淎lmost every unsolved problem in mathematics and many in physics and elsewhere could be settled by knowing enough digits of Omega.鈥�

But Calude doesn鈥檛 think their method for unravelling Omega will take them much further. In fact, there鈥檚 not much hope of significant progress at all, according to Omega鈥檚 discoverer, Gregory Chaitin of IBM鈥檚 Yorktown Heights laboratory. He says the fact that Calude could calculate these 64 digits simply shows that there鈥檚 no classically uncomputable mathematical problem that can be tackled by a program just 64 bits long. Finding more of Omega鈥檚 digits would take us closer to finding the threshold between computable and uncomputable, though, so it鈥檚 worth pursuing, Calude says. But until we have a very different kind of computer, the secrets of the Universe will stay hidden.

Quantum computers aren鈥檛 the only potential way to access the inaccessible. Two Hungarian researchers, Gabor Etesi of Kyoto University in Japan and Istvan Nemeti of the Mathematical Institute of the Hungarian Academy in Budapest, have proposed exploiting an exotic possibility first imagined more than 50 years ago. The German mathematician Hermann Weyl suggested that a Turing machine undergoing acceleration close to the speed of light would suddenly have time on its side. That鈥檚 because, in Einstein鈥檚 theory of relativity, something that鈥檚 accelerating experiences a 鈥渄ilation鈥� or slowing of time relative to observers moving under a different accelerating force. Etesi and Nemeti have shown that a pair of machines, each operating in a different region of a rotating black hole could carry out an infinite number of calculations in a finite time and go beyond the Turing barrier. 鈥淥ur computer is not only a consistent thought experiment, but also a realistic idea,鈥� Gabor says. But, he adds, you won鈥檛 see one any time soon. It would involve using a rotating black hole to provide the acceleration鈥攁nd no one鈥檚 even found one yet.

Coming back down to Earth, DNA computers might provide another way forward. Instead of lines of programming code, DNA computers could use chemical sequences that sort through all possible solutions to a problem. Once the computation is complete, you just fish the answer out of the test tube. Although this process is largely just a 鈥渨et鈥� version of the established speed-up algorithms for quantum computers, Calude believes there鈥檚 another form of DNA computing that might beat the Turing barrier: 鈥渕embrane computing鈥�, invented by George Paun of the Romanian Academy鈥檚 Institute of Mathematics in Bucharest. This proposes using cell-like arrangements of processing units, separated by membranes that can dissolve or divide. The physical arrangement of the computer is then constantly changing in a way that allows particularly powerful processing.