New quantum algorithms finally crack nonlinear equations

The article seem to imply that quantum computers are a solution to the problem that non-linear differential equations are chaotic (as in: tiny changes in initial conditions lead to vastly different solutions).<p>My impression of quantum computers are they will allow us to get to solutions to algorithmic problems <i>faster</i>.<p>I fail to see how they could possibly work around the chaos inherent to some diffeqs (e.g. N-body problems).<p>[edit]:and after reading the article more carefully, I&#x27;m still not sure how QC has <i>any</i> effect on the chaotic nature of non-linear diffeqs . All I see in here is that they&#x27;re trying to map non-linear diffeqs to linear systems via approximations so they can run it on a QC.

&gt; The new approaches disguise that nonlinearity as a more digestible set of linear approximations, though their exact methods vary considerably.<p>The idea of reducing a nonlinear problem to a sequence of linear problems is the bread and butter of nonlinear PDE, both theoretical and numerical. The trick is always in the reduction and I guess here you want the reduction to have a nice &quot;quantum&quot; solution.<p>I&#x27;m not qualified to say what either of these papers have to do with &quot;chaos&quot;. You&#x27;ll notice that this word only appears in one of the papers and then only once in the intro.

What is the mesmerizing image at the top of the article ?

I find it fascinating that we&#x27;re using an overall linear model (Quantum Physics) of our non-linear reality, particularly the bits that are less linear (Bose Einstein condensates) to approximate linear solutions to non-linear models (flow dynamics, etc) of said non-linear reality.

&gt;&quot;So the trick is finding a way to mathematically<p><i>convert a nonlinear system into a linear one</i>.<p>“We want to have some linear system because that’s what our toolbox has in it,” Childs said. The groups did this in two different ways.<p>Childs’ team used<p><i>Carleman linearization</i><p>, an out-of-fashion mathematical technique from the 1930s, to transform nonlinear problems into an array of linear equations.&quot;<p>[...]<p>&quot;It modeled any nonlinear problem as a Bose-Einstein condensate. This is a state of matter where interactions within an ultracold group of particles cause each individual particle to behave identically. Since the<p><i>particles are all interconnected</i><p>(PDS: You mean like a <i>WAVE</i> ??? &lt;g&gt;)<p>,<p><i>each particle’s behavior influences the rest</i><p>(PDS: You mean like a <i>WAVE</i> ??? &lt;g&gt;)<p>, feeding back to that particle in a loop characteristic of nonlinearity.&quot;<p>[...]<p>&gt;“Give me your favorite nonlinear differential equation, then I’ll build you a Bose-Einstein condensate that will simulate it,” said Tobias Osborne&quot;<p>If you can get a Bose-Einstein condensate for a given nonlinear differential equation -- <i>perhaps the reverse is true as well</i> -- perhaps, <i>for a given Bose-Einstein condensate, you can get back a nonlinear differential equation...</i><p>If that&#x27;s true, and if it&#x27;s also true that the nonlinear differential equation can be turned back into a linear differential equation (via Carleman linearization), and if so, then you can possess a linear differential equation representing your Bose-Einstein condensate, er, wave, er, fluid equation, er, linear differential equation, er, Bose-Einstein condensate... &lt;g&gt;<p>Perhaps all of these things -- are just different ways of viewing, different VIEWS -- of the same underlying physical phenomena...<p>To quote a famous Musician(!):<p>&quot;We always didn&#x27;t feel the same, <i>we just saw it from a different point of VIEW...</i>&quot;<p>-Bob Dylan, &quot;Tangled Up In Blue&quot;