A new quantum algorithm for classical mechanics with an exponential speedup

One of the most important insights you take away from a physics undergrad is that you can model much of physical phenomena as a harmonic oscillator. The reason for this is quite simple<p>1. Every closed system has a fixed total energy, so many systems just settle into an oscillating state, where kinetic energy converts into potential and back.<p>2. Most real world systems are approximately closed, so they leak energy till they have low total energy (this also follows from the second law).<p>3. An oscillating system with low total energy can have its potential energy accurately approximated with a quadratic function. Or in other words a harmonic oscillator.<p>So, while I can&#x27;t say if there are many interesting&#x2F;useful coupled classical oscillator systems that need an exponential speedup for us to study, it is nevertheless exciting to hear that such systems do admit a quantum speedup.

Very interesting indeed:<p>&gt; discovery of a new quantum algorithm that offers an exponential advantage for simulating coupled classical harmonic oscillators.<p>&gt; To enable the simulation of a large number of coupled harmonic oscillators, we came up with a mapping that encodes the positions and velocities of all masses and springs into the quantum wavefunction of a system of qubits. Since the number of parameters describing the wavefunction of a system of qubits grows exponentially with the number of qubits, we can encode the information of N balls into a quantum mechanical system of only about log(N) qubits.

The approach maps the classical mechanics of coupled harmonic oscillators to a particular quantum system. I&#x27;m curious if there is some (unrelated) classical system whose quantization is that same quantum system. In other words, does this quantum system have a natural interpretation as the quantum mechanical version of some classical system? If so, it&#x27;s presumably very different from (e.g., a lot smaller than?) the oscillator system which motivated the study of the quantum system.

The thing that stands out to me is the described proof of BQP-completeness where they say they prove any quantum system can be similarities as balls and springs, but later they say you may need an exponential number of springs for a classical simulation. That sounds like the BQP reduction would be exponential, guess I&#x27;ll have to read the paper to see what I&#x27;m missing.

I&#x27;m skeptical. If you have an exponential speedup for simulations of coupled oscillators, you can rig a system of coupled oscillators into a general purpose computer [0] and therefore have an exponential speedup for any computation. That seems too good to be true.<p>[0] <a href="https:&#x2F;&#x2F;www.zyvex.com&#x2F;nanotech&#x2F;mechano.html" rel="nofollow noreferrer">https:&#x2F;&#x2F;www.zyvex.com&#x2F;nanotech&#x2F;mechano.html</a>

This could be pretty exciting. For all the talk about quantum advantage, the number of quantum algorithms that have exponential speedup is super small; as the blog mentions it&#x27;s essentially Shor&#x27;s prime factoring and quantum simulations.

&gt; Further, we use this mapping to prove that any problem efficiently solvable by a quantum algorithm can be recast as a problem involving a network of coupled oscillators, albeit exponentially many of them.<p>Is this a new result, giving that quantum field theory is described in terms of quantum harmonic oscillators?

Need to take a closer look at the paper, but a classical approach to this problems is to map it to an eigenvalue problem, so could we say that they have found a quantum speed up for solving eigenvalue problems?

I suspect an analog computer would work just as well for modeling coupled harmonic oscillators.

Is this similar to Ed Gerck&#x27;s &quot;finding&quot; of a QC algorithm that can break RSA-2048?

Usually these types of articles are total nonsense, but this is legit.<p>A very cool result!<p>It&#x27;d be interesting to see how many other systems can be approximated by the system they&#x27;ve solved for (without incurring an exponential penalty in the translation).

I&#x27;ll read it when I&#x27;m home. But I want to say that the fact that this is from google &quot;quantum AI&quot; makes me doubt the legitimacy. They are really ruining their reputation with all the absurd quantum stuff they have been publishing, e.g: their wormhole stuff and a lot of quantum neural networks bs.