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Why exactly is machine learning on quantum computers different than classical machine learning? Is there a specific difference that allows quantum machine learning to outperform classical machine learning?

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Potentially, the same advantage that quantum computing can provide over classical computing. By "quantum machine learning", in the way you seem to be using the term here, people usually refer to quantum algorithms developed to solve tasks usually handled by machine learning, that is, very roughly speaking, pattern recognition tasks (though in the quantum case algorithms aimed at performing linear algebra operations are also included in the category).

Just as there are tasks for which quantum algorithms are known to, or promise to provide enhancement over classical computers, think Grover's or Shor's, one can hope pattern recognition tasks can be similarly sped up. That's pretty much the point of quantum machine learning, also often referred to as "quantum-enhanced machine learning" in this context, to distinguish it from applications of classical machine learning to quantum information tasks, which is an entirely different field of study.

See also this question and links therein.

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  • $\begingroup$ Thank you very much for sharing. So essentially, let's assume that I run the exact same ML algorithm on a quantum computer and on a classical computer. Will the main benefit of running on the quantum computer be computational time? Also would this be a "quantum-enhanced machine learning algorithm"? $\endgroup$ – Rob James Sep 2 at 1:47
  • $\begingroup$ @RobJames sure, quantum computers can run classical algorithms, but there is no gain in doing that. That's because quantum algorithms work in fundamentally different ways than their classical counterparts. See e.g. quantumcomputing.stackexchange.com/a/6630/55. The advantage of quantum algorithms that provide computational speed-up over their classical counterparts is indeed that they provide computational speed-up over their classical counterparts. It is not known what tasks can be speed-up by quantum algorithms though. $\endgroup$ – glS Sep 2 at 7:58
  • $\begingroup$ That makes sense. So for example, with an algorithm like QSVM (Quantum Support Vector Machines), there will be a significant computational speed-up over classical SVMs? Would QSVM be considered a "quantum-enhanced ML algorithm"? $\endgroup$ – Rob James Sep 2 at 12:44
  • $\begingroup$ yes, that would be one example. The field is still relatively new though, so the advantage of most algorithms comes with caveats, and not everyone agrees on whether these would be actually usable in practice. Stating the exact conditions under which a given algorithm could provide an actual practical speed-up is usually nontrivial. $\endgroup$ – glS Sep 2 at 14:44
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As so often, and especially in young research areas, the answer depends quite a lot on how you break down the question. Let me try a few examples:

Does quantum mechanics change what is theoretically learnable?

A beautiful paper is this reference which states a few complex results in rather clear words. Again, it depends very much on what you define as "learning". Overall, exponential speedups in the number of data samples seem to not be possible in many settings, but exponential time complexity speedups very well possible.

What asymptotic computational speedups can quantum computing provide for machine learning?

Probably the most studied approach here is to outsource linear algebra subroutines such as matrix inversion or singular value decomposition to quantum computers. These subroutines appear for example in convex optimisation used in linear regression or kernel methods.

Quantum computing research is traditionally very focused on exponential speedups, which have been claimed in many quantum machine learning papers. But they rely on lots of assumptions about how you load your data into a quantum computer, and how to process the results. The assumptions require deep technical knowledge to grasp, and it is not always clear how good classical methods are in this case. For example, the quantum algorithm may require a sparse data matrix for an exponential speedup over the vanilla classical method, but under this assumption there is another classical method that is much faster too. Some quantum algorithms have since been "de-quantised", which is a euphemism for "found to not really provide an exponential speedup if the same assumptions are imposed on classical algorithms".

Standard quantum algorithms can often give you a quadratic speedup for sampling and unstructured search problems. But classical methods are quite fast at heuristic sampling in the first place (think of contrastive divergence), and search problems so vast that a quadratic speedup does not make them tractable either.

Can quantum computations give rise to machine learning models that generalise well?

Most of the work in near-term quantum machine learning, that is QML using small and noisy devices that are the current "prototypes" of quantum computers, are interested in what models quantum computers very naturally give rise to. Do they look like neural networks? Or like anything else? Are they good generalisers? Can they be trained efficiently?

Of course, speedups are important here to - if we find a quantum model that is powerful, but easy to classically simulate, one does not need the quantum computer in the first place (still, a quantum computer may just be the fastest hardware in absolute terms to process those models, and therefore still advantageous). But much more important is to show that the quantum model generalises well.
This type of research, much driven by "variational" or "trainable" or "parametrised" quantum circuits which are optimised with the usual classical techniques of deep learning, has only few answers yet to the question of quantum advantages. There are interesting clues though - quantum models of this type are mathematically speaking linear algebra computations applied to data mapped into the very large Hilbert spaces of quantum systems. They are also modular and trainable like neural networks.

If one accepts that quantum computers are strictly speaking more powerful than classical computers, the quantum model could in principle express a larger class of functions. But how to utilise this potential advantage in a concrete quantum algorithm design is very hard to point out and not "automatically given". For example, one can show that certain ways to encode data into a quantum computer gives quantum models only access to very trivial function classes, and they are unlikely to learn anything interesting.

One reason why this is an extremely challenging (but very interesting) research area is this: if the goal is to build powerful generalisers, but the theoretical foundations of generalisation are poorly understood even in classical machine learning, and our current devices are too small and noisy to run meaningful empirical benchmarks, how can one actually show that the quantum model has an advantage? In other words, a lot of work is needed to even find a satisfying investigation framework for the question "are quantum models better machine learning models?".

So, overall I'd say like every decent research field, the art is to reformulate the question until we can answer it - at which stage the answer is usually hard to understand for non-experts.

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  • $\begingroup$ Thanks so much for typing all that up. So, given a Quantum Support Vector Machine model and a classical Support Vector Machine model, do you suggest that there could be a quadratic speedup? Does that mean that there are certain problems that would take an unreasonable large amount of time classically but could be approached through QML algorithms? $\endgroup$ – Rob James Sep 11 at 1:21
  • $\begingroup$ What I meant to say is that whether a Quantum Support Vector Machine is faster than a classical SVM depends on a) what you understand a QSVM to be (there are tons of different ideas), and b) what you are trying to learn/ what data you have. But yes, in principle a quantum computer could implement something very similar to an SVM, but invert the kernel Gram matrix in linear time. And yes, most people believe the second question can be answered positive, but it is very hard to find problems useful for a broad range of applications and prove that the quantum methods are better. $\endgroup$ – Maria Schuld Sep 14 at 5:55
  • $\begingroup$ Thank you very much for the clarification. I realize it would be difficult to theoretically prove whether a certain problem is more efficiently computed using QML, but is there an experimental way to prove the time difference? For example, could I simply run an algorithm on a classical and quantum computer to make a judgment on efficiency? $\endgroup$ – Rob James Sep 15 at 2:40
  • $\begingroup$ If you want to compare absolute runtime (instead of counting the asymptotic growth of gate numbers with the input size), you can of course do that with experiments. One needs to be careful with comparisons though: to get the same precision of results to a classical computer, we need error correction, which will - once available - blow up qubit numbers and absolute runtime... Also, experiments today are necessarily small, and you may not see the advantages of the QC yet. It's tough :) $\endgroup$ – Maria Schuld Sep 17 at 8:33

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