Quantum Algorithms Conquer a New Kind of Problem | Quanta Magazine

Quantum Algorithms Conquer a New Kind of Problem | Quanta Magazine

In 1994, a mathematician figured out how make a quantum computer do something that no ordinary classical computer could. The work revealed that, in principle, a machine based on the rules of quantum mechanics could efficiently break a large number into its prime factors — a task so difficult for a classical computer that it forms the basis for much of today’s internet security.

A surge of optimism followed. Perhaps, researchers thought, we’ll be able to invent quantum algorithms that can solve a huge range of different problems.

But progress stalled. “It’s been a bit of a bummer trajectory,” said Ryan O’Donnell of Carnegie Mellon University. “People were like, ‘This is amazing, I’m sure we’re going to get all sorts of other amazing algorithms.’ Nope.” Scientists discovered dramatic speedups only for a single, narrow class of problems from within a standard set called NP, meaning they had efficiently verifiable solutions — such as factoring.

That was the case for nearly three decades. Then in April, researchers invented a fundamentally new kind of problem that a quantum computer should be able to solve exponentially faster than a classical one. It involves calculating the inputs to a complicated mathematical process, based solely on its jumbled outputs. Whether the problem stands alone or is the first in a new frontier of many others has yet to be determined.

“There is a sense of excitement,” said Vinod Vaikuntanathan, a computer scientist at the Massachusetts Institute of Technology. “A lot of people are thinking about what else is out there.”

Computer scientists try to understand what quantum computers do better by studying mathematical models that represent them. Often, they imagine a model of a quantum or classical computer paired with an idealized calculating machine called an oracle. Oracles are like simple mathematical functions or computer programs, taking in an input and spitting out a predetermined output. They might have a random behavior, outputting “yes” if the input falls within a certain random range (say, 12 to 67) and “no” if it doesn’t. Or they might be periodic, so that an input between 1 to 10 returns “yes,” 11 to 20 yields “no,” 21 to 30 produces “yes” again, and so on.

Let’s say you have one of these periodic oracles and you don’t know the period. All you can do is feed it numbers and see what it outputs. With those constraints, how fast could a computer find the period? In 1993, Daniel Simon, then at the University of Montreal, found that a quantum algorithm could calculate the answer to a closely related problem exponentially faster than any classical algorithm.

The result enabled Simon to determine one of the first hints of where dramatic superiority for quantum computers could be expected. But when he submitted his paper to a leading conference, it was rejected. The paper did, however, interest a junior member of the conference’s program committee — Peter Shor, who at the time was at Bell Laboratories in New Jersey. Shor went on to find that he could adapt Simon’s algorithm to calculate the period of an oracle, if it had one. Then he realized he could adapt the algorithm once again, to solve an equation that behaves similarly to a periodic oracle: the equation that describes factoring, which is periodic.

Shor’s result was historic. The quantum algorithm he discovered could rapidly reduce gigantic numbers into their constituent prime factors, something that no known classical algorithm can do. In the years that followed, researchers discovered other efficient quantum algorithms. Some of them, like Shor’s algorithm, even provided exponential advantage, but no one could prove a dramatic quantum advantage on any NP problem that wasn’t periodic.

This dearth of progress led two computer scientists, Scott Aaronson of the University of Texas, Austin, and Andris Ambainis of the University of Latvia, to make an observation. Proofs of quantum advantage always seemed dependent on oracles that had some kind of nonrandom structure, such as periodicity. In 2009, they conjectured that there couldn’t be dramatic speedups on NP problems that were random, or unstructured. No one could find an exception.

Their conjecture put a bound on the powers of quantum computers. But it said only that there were no dramatic speedups for a specific type of unstructured NP problem — those with yes or no answers. If a problem involved figuring out more specific, quantitative answers, which is known as a search problem, the conjecture didn’t apply.

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