Quantum Algorithms Resolve a New Type of Problem

In 1994, a mathematician discovered how to make a quantum machine do things that no other classical computer does. This work showed that a machine that was based on quantum mechanics might in principle break down large numbers into their prime factors. This is a feat so problematic for classical computers that it has become the basis of much of the internet security.

There was surge of optimism. Researchers thought that we might able to create quantum algorithms that solve many different problems.

However, progress has stalled. Ryan O’Donnell, Carnegie Mellon University said that it was a slowing down. People were amazed and thought it was amazing. But scientists discovered that they could achieve dramatic speedups for a narrow set of problems contain by standard set known as NP. This means that they could efficiently verify solutions like factoring.

This was true for almost three decades. In April, researchers discovered a completely new type of problematic that quantum computing device could solve exponentially quicker than the classical. This involves computing the inputs to complex mathematical process based only on its messy outputs. It is not clear if the problem is unique or a part of a larger set of problems.

A computer scientist Vinod Vaikuntanathan (Massachusetts Institute of Technology) said that there is an excitement. “Many people are wondering what else might be out there.”

This result enabled Simon to see the potential for quantum computers to have dramatic superiority. His paper was submitted to a leading conference, but it was not accepted. Peter Shor, is junior member on the conference’s program committee was fascinated by the paper. Shor was in New Jersey at Bell Laboratories at the time. Shor discovered that Simon’s algorithm could be used to calculate Oracle’s period, if one exists.

Shor’s result was notable. His quantum algorithm could reduce huge numbers into their prime factors in a matter of seconds, something no other classical algorithm can does. Researchers discovered many other effective quantum algorithms over the years. While some of these algorithms, such as Shor’s, provided an exponential advantage, no one was able to prove a quantum advantage over any NP problems that wasn’t periodic.

Two computer scientists, Scott Aaronson of University of Texas, Austin & Andris Ambainis of University of Latvia, noticed this dearth of progress. The existence of nonrandom structures, like periodicity, is a proofs of quantum advantage.

Their conjecture placed a limit on quantum computers’ capabilities. It only stated that there was no significant speedup for a particular types of unstructured NP problems — those with either yes & no answers. The conjecture did not apply to problems that required more precise, quantitative answers. This is what we call a search problem.

This in mind, Takashi Yamakawa, NTT Social Informatics Laboratories, and Mark Zhandry, NTT Research, and Princeton University, decided to test a particular search problem that Oded Regev introduced in 2005.

Imagine a collection of weather vanes that point in the same direction. Give each one a measured shove, and then let the wind blow their way. Regev wanted to find out, based upon their final directions, where each of them pointed in the beginning. This problem is known as “learning with errors” because the shove, the wind and other factors act as a source for random error in the original direction. It is difficult to solve both for classical and quantum algorithms.

Yamakawa & Zhandry modified the setup. They altered the strength of the starting shoves to make them more predictable. They also made the wind random, so it could be decided by an oracle. This allowed for greater randomness in some cases and complete dormancy in others.

These modifications allowed researchers to discover that a quantum algorithm can efficiently find the initial direction. They also found that classical algorithms must be slower than an exponential factor. They then modified their algorithm to solve the real-world problem. This replaces Shor’s oracle with a mathematical equation.

Computer scientists continue to work on this problem. Vaikuntanathan says it is similar to another problem that can arise when data compression is used: Two bits can accidentally be compressed into the same location, overwriting them. There is some similarity between the problem of anticipating collisions so you can avoid them and the one that arises when trying to predict them. He said, “That’s basically a class problem which looks like this.” These problems may be solved quantumly.

It was hoped that an unstructured problem such as the one presented might be solved even on quantum computers in their early stages, thus allowing for testing. Because they are random, unstructured problems may require less programming time or be more sensitive to noise. The new problem is too complex for quantum computers to solve. It’s a strange problem. Aaronson said that he had never thought of defining it. It has many very attractive features, but it is hard to see in retrospect.

This is the first instance of dramatic quantum advantage in an unstructured NP issue. Are there other problems that quantum technology could make possible? There is now more reason to think so.

O’Donnell stated that “it’s somewhat overturned [our beliefs] about what problems quantum computers could solve.”