A Quantum Linear Equation Solver

joe writes "Aram Harrow and colleagues have just published on the arXiv a quantum algorithm for solving systems of linear equations (paper, PDF). Until now, the only quantum algorithms of practical consequence have been Shor's algorithm for prime factoring, and Feynman-inspired quantum simulation algorithms. All other algorithms either solve problems with no known practical applications, or produce only a polynomial speedup versus classical algorithms. Harrow et. al.'s algorithm provides an exponential speedup over the best-known classical algorithms. Since solving linear equations is such a common task in computational science and engineering, this algorithm makes many more important problems that currently use thousands of hours of CPU time on supercomputers amenable to significant quantum speedup. Now we just need a large-scale quantum computer. Hurry up, guys!"

Comments

Looks like a big deal to me.

[jonniesmokes]

Finally a cool article on /. This is extremely cool! There are a lot of problems in the real world that have extremely large sparse matrices that need to be inverted. Fluid dynamics and solutions to Maxwells equations come to mind. But I am sure there are other applications in relativity and plasma physics. Estimating a solution to a linear dynamic system of say 2^128 degrees of freedom in only 128 cycles would change a lot of things.

And... Yes, we are working very hard on building the computers.

Avinatan Hassidim and Seth Lloyd

[aram.harrow]

are my coauthors, and the ordering of author names was alphabetical, and doesn't reflect our level of contributions (which were more or less equal).

So please cite this as "Harrow, Hassidim and Lloyd" and not "Harrow and coauthors."

That said, we're pleased about that attention. :)

In response to one question: the matrix inversion problem for the parameters we consider is (almost certainly) not NP-complete, but instead is BQP-complete, meaning that it is as difficult as any other problem that quantum computers can solve. We plan to update our arXiv paper shortly with an explanation of this point.

Re:Polynomial Speedup

[blueg3]

To rephrase the other response, log vs. polynomial is the same as polynomial vs. exponential. Moving from polynomial time to logarithmic time is an exponential speedup (just as is moving from exponential time to polynomial time is).

O(n^3) vs. O(log n)
->
O(exp(n^3)) vs. O(n)

Re:Comparing Apples to Apples

[aram.harrow]

This is a great question. Here's how I like to think about it: If A is a stochastic matrix and b is a probability distribution that we can sample from, then given a few assumptions, we can sample from Ab efficiently. This is more or less the idea behind so-called Monte Carlo simulations, which are a tremendously useful tool in classical computing. However, we don't know how to get sampling access to A^{-1} b. Our algorithm gives us something like sampling access to A^{-1} b. Not exactly, because we're talking about quantum amplitudes, rather than probabilities. But more importantly, taking the inverse can make a sparse matrix dense, and (as we often see for problems admitting quantum speedups) sampling-based approaches to computing it fail because the samples have alternating signs.