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The following question is related to this one: Will deep learning neural networks run on quantum computers?. I found it complementary and necessary because the previous answers are not completely related with my concerns.

Primarily, my question is related to hardware-based neural networks. As proposed by Precioso et al. in 'Training and operation of an integrated neuromorphic network based on metal-oxide memristors' the use of solid-state or even molecular memristors can be a hardware proposal to implement efficient and non-volatile neural networks.

Thus, the question is: Is there a quantum logic operation which can be used to improve the algorithms present in neural networks? Can quantum logic add any possible strategy in order to increase the efficiency of such networks?

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Again, this is still an open question.

There are two lines of work that come to mind when you talk of "hardware-based neural networks" which try/claim to use photonics as a mean to speed-up processing, and make direct reference to speeding up machine learning tasks. Shen et al. 2016 (1610.02365) propose a method to implement "fully-optical neural networks" which they claim to provide advantages in terms of computational speed and power efficiency. A similar in principle (but not in method) idea is the one pursued by the LightOn startup (see the papers referenced in their website). Very roughly speaking, the idea is here to exploit the natural dynamics of complex media, and therefore the natural dynamics of scattered light, to again get better processing speed and power consumption performances (note that I'm just stating the claims made in the papers here, as the details and validity of performances and advantages can be hard to judge in this case).

Note that, in both cases, only classical light is used. In other words, there is arguably nothing "quantum" about these works, so you may not consider them totally relevant here. However, both platforms could naturally be used (and have been used for different experiments) with single photons, and therefore can be used for processing of quantum information. However, building a "quantum neural network" is not just a matter of building a neural network-like evolution for a quantum system, as that would likely provide no advantage at all.

Talking instead of "quantum logic operation which can be used to improve the algorithms present in neural networks", this is the idea behind a good chunk of research being done on quantum (assisted) machine learning, some references for which can be found in the question you linked as well as in this other one. A notable example is HHL09, which provides a way to speed-up the problem of inverting a linear system, which is an important part of many machine-learning algorithms.

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