No distributed quantum advantage for approximate graph coloring | Hacker News Reader

No distributed quantum advantage for approximate graph coloring

arxiv.org

71 pointsnabla916 comments

At first, the results claimed here seem to clash with the fact that we know, for instance, that it is NP-hard to 5-color a graph which is promised to be 3-colorable [1]. Of course, the devil is in the details - on page 11, where they describe the "LOCAL" computation model they work in, they explain:

"In each round the processors may perform unbounded computations on their respective local state variables and subsequently exchange of messages of arbitrary size along the links given by the underlying input graph."

Given this setup, it should be possible to solve any problem in O(n) rounds, and indeed they point this out in the introduction:

"If the chromatic number of G is χ, in this setting it is trivial to find a χ-coloring in T = O(n) rounds, as in O(n) rounds all nodes can learn the full topology of their own connected component and they can locally find an optimal coloring by brute force without any further communication."

While this paper is very interesting from a theoretical perspective, it is saying more about the LOCAL model of distributed computing than it is about quantum computing. Nobody should come away from this with the conclusion that we have somehow proved that quantum algorithms are useless for real-world problems.

[1] https://arxiv.org/abs/1811.00970

The assumption that output distributions don't violate the no-signaling from the future principle may be a useful abstraction, but it seems a little too neat for real-world quantum interactions. It's a model, not physical reality. Also, the necessity of knowing specific graph parameters in advance could limit the usability of these algorithms in unpredictable distributed systems.

It seems from the paper that their algorithm only needs to know the size of the input graph and nothing else.

The lower bound is graph-theoretical and based on graph-parameters which are not considered by the algorithm.

It is not known if there is there any graph problem that someone cares about for which there exist algorithm with distributed quantum advantage?

For many graph problems it's possible show to show that there is no quantum advantage by by defining a model so that it can do anything except violating causality.

This paper shows that approximate graph coloring has not quantum advantage.

n00b question: what are some important problems for which quantum computation has shown to have an advantage

you aren't the first asking, so someone made a website listing stuff

https://quantumalgorithmzoo.org/

factoring large integers into primes (shor's algorithm). that and grover's search promise to provide ways to crack a lot of encryption.

also quantum computing should improve both classical and quantum physical simulations.

The other side of Shor's algorithm is that you can use it to solve discrete logarithm (which additionally breaks elliptic curve crypto; factoring breaks RSA).

I don't agree that Grover's algorithm is as fundamental in breaking encryption, since it "only" yields a quadratic speed-up. You'd have similar effective security as in the pre-quantum world by doubling the key length; you could do this by, say, switching to AES-256 instead of AES-128. Meanwhile, switching from RSA-2048 to RSA-4096 only helps if it means the problem size now exceeds the size of the biggest quantum computers.

I think there is promise with tons of optimization problems (machine learning, approximating differential equations). I hope these would be significantly faster in practice.

It would be cool if we had quantum accelerated game physics engines. Super realistic games that are fast.

Simulations could scale better, so we could make them a lot more useful.

Unless you are simulating quantum effects, these sorts of simulations are unlikely to be faster on a quantum computer. Also, attempts at using the Ising model for general-purpose optimization have found themselves somewhat limited in application.

For people doing drug discovery and quantum physics/chemistry research, they will see an exponential speedup from quantum computing, but I think you're overstating how applicable this technology is.

Good news for cryptographers in case we need an alternative to LWE and coding-based PQC.

I'm not sure this implies much about PQC. The model here is LOCAL distributed computing which is pretty far from what you're concerned about in PQC.

Why are they trying to colour graphs with quantum computers? Just reading the title

I suspect the point is that all NP-hard problems are equivalent to each other, often one can be transform one NP-hard problem to another. This shows that quantum will not save us. This is bad but also good for encryption. NP remains an important topic post-quantum that cannot be hand-waved away.

NP hard problems aren’t all equivalent, that’s NP complete.

Thanks!

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