What level of understanding should you have of Quantum Physics to write a hard science fiction novel? And how should you learn Quantum Physics in order to be able to write a hard science fiction novel? What's the least time-consuming way of doing so?
To echo the comment from F1Krazy, you only need knowledge of quantum physics if your story depends upon quantum physics. You could write about information technology, orbital mechanics, synthetic biology, or a dozen other topics without ever mentioning (or understanding) quantum mechanics.
While there are many different ways to write fiction, it would seem to me that the more successful stories start out as stories rather than accumulated research notes. As often as not, the story makes up technology that might possibly exist in order to make the story work. Two common examples are faster-than-light travel and time travel. Most physicists will argue that both phenomena are not possible, given our current knowledge of the universe. But there are a lot of good and even great stories that depend upon those two actions being possible. The story does some hand waving to make worm holes or black holes or additional dimensions of space and time or something that sounds-like-science and that allows the story to proceed.
If you accept that the story dictates the technological/science needs, then write the story. Determine the minimum amount of hand-waving that you need to make the story work. Search the web for information on that topic. See if you can find a scientist who is prepared to chat about the topic. Go to the library. Go to Worldbuilding.Stackexchange.com, demonstrate that you have done your homework, and ask a specific question. In any case, be prepared to bend the story at least to some extent to fit better with the science/technology.
Finally, read widely, both fiction and non-fiction. Try to understand what works to keep you interested. Then write, knowing that most of what you write will not be published. Write for practice, to polish skills, to try out different approaches. Then read some more. Rinse and repeat.
I cannot do better than the answer by JoeStonecash in addressing the big-picture question, and I encourage you to take that advice and in particular to not worry about quantum physics until it somehow becomes critical to your story. At the same time, I wholeheartedly agree with the comment by Patrick Stevens that "the subject is unique among the sciences in how close to total nonsense most people's understanding of it is." So I wanted to also address the part of your question that is specific to quantum physics, where I think the main thing is to know and avoid the major misrepresentations that are out there, at least if you're going for the hard-science angle.
As a disclaimer, I am not a physicist, but I work with physicists and have tried to limit myself to things I am confident in. Like JoeStonecash mentioned, you should confirm anything written below with other sources before using it in your writing.
The most misunderstood part of quantum physics, in my experience reading science fiction, is what quantum physics says happens when something is "observed". Let's start by setting the record straight: The terms "observation" and "measurement" just mean that a system interacts with some other sufficiently large system. These interactions cause "decoherence", and the smaller system rapidly evolves to a state where we can reasonably describe the result of the measurement (and only that result) with classical terms and ordinary grade-school probability. We say that the system's wave-function has "collapsed". But the nature of the system is still quantum and, generally speaking, any unmeasured aspects of the system will still defy classical description, even in the moment of observation.
Now to address some associated misconceptions in science fiction:
Although observation is usually ridiculously fast, that does not make it literally instantaneous. With an extremely careful experiment, you can cut off the interaction with the larger system before the decoherence gets far enough along for a classical description to make much sense.
An "observer" does not have to meet any mysterious criteria like being conscious or being human, and the physics does not change depending on whether it is a person doing the looking. For an interaction to behave like an observation, the only thing necessary is that the "observer" is big enough. (And remember, that's "big" by quantum physicists' standards, so not very.)
Systems do not alternate between being wholly quantum at normal times and wholly classical during measurement. As mentioned above, a measurement can usually only make one aspect of a system look classical. For example, imagine a lone particle. If you measure its position, decoherence will put it in a state consistent with having a definite location, but that same decoherence will ruin your ability to even apply the classical concept of momentum. Likewise, if you instead choose to measure momentum, it is nonsense to talk about the particle's classical position during the measurement. The problem isn't that the position and momentum are somehow hidden from us, nor that either possible measurement would somehow disturb the "true" values, it's that the idea that particles are billiard balls with classical positions and momenta is too naïve to ever match reality. At best, we can make one concept or the other apply at a time, but when we're not forcing reality's hand, it's likely that neither applies. (This example is the oft-misunderstood, could-be-better-named Heisenberg Uncertainty Principle. There are lots of other uncertainty principles, each giving a limit on the simultaneous applicability of classical ideas.)
Relatedly, a "superposition" of two classical states X and Y does not mean "X and Y are simultaneously true". Nor does it mean "either X or Y is true, but we don't know which". What it means is that neither X nor Y is an accurate description of the reality, so if we take a measurement with X and Y as possible outcomes, the ensuing decoherence could put the system in either state.
Since "superposition" does not mean "or" or "and", quantum computation is not about trying a bunch of classical computations simultaneously. Rather, we start by planning a measurement that will give us one of several possible answers. The physics tells us that, for each answer, the likelihood of decohering to that answer depends on a certain amplitude. So we design a process that tugs correct answers' amplitudes consistently in one direction but wrong answers' amplitudes in opposing directions. The wrong answers' tugs cancel each other out, and so the associated amplitudes end up small and unlikely play into the final measurement. As you can imagine, taking a problem and finding a process that works to cancel out every wrong answer's amplitude is incredibly difficult (except in certain special cases)—it's not a magic bullet.
"Entanglement" is not a concept that's useful for communication. Where it comes in is that, while we can't normally predict how decoherence will go, we can give probabilities, and the probabilities for systems can be correlated—we call such systems "entangled". For example, you and I could set up two particles so that we know that we'll get the same result if we measure their spins. Then, even if neither particle could be accurately called spin-up or spin-down prior to measurement (remember, classical descriptions don't normally make sense except during a corresponding observation), I can trust that my future measurement will play out the same way yours will. So we do have access to a shared secret. But there's nothing I can really do to send you information: you have no way of knowing if or when I've made my observation.
If you can avoid all of these errors, readers like me will be ecstatic. Even if some of the other details are off, I would be delighted to read a work that gets this stuff right.
I would argue in a slightly different vein than the other (good) answers. Technically, you don't need to know how quantum physics works at all, to write hard science fiction that uses it. The only thing you need to know is what it can do. Or rather what people that understand it believe it will be able to do in the future.
There is of course the engineering-porn side of hard sci-fi with its endless infodumps, but in general, those do not advance the story and are somewhat rightfully considered a bad habit. There are also stories that hinge on how a very specific concept in engineering or physics is implemented precisely, but there are many times as much that don't.
For most people, technology is just a tool. To answer your question, I need to know that the words I am typing right now will appear on this page where you can read them after I press the big button at the bottom. I do not need to be able to program, or to know how a computer works, or the internet, or how to set up a server, I only need to know what actions I can take and what effects they have. And even though in fact I actually know all this, I rarely feel the need to tell people about it if they don't explicitly ask or it somehow helps me to make a point.
The same is true for the characters in your novel. To them, their quantum-whatever-device might be an everyday object, so they are used to it and don't reflect on its inner workings. Or some Q'ish science boffin might hand them this brand new device, telling them that they shouldn't even try to understand how it works, it's way above their paygrade, but in recognition of that it's been made foolproof. Just press that button and things happen.
What makes your story hard science fiction is that the existence of such a device in the future is plausible. Leave it to the reader to figure out the details if they want to. In fact, you can get away with more this way than if you explain everything. It is much harder to prove that something cannot be done than that something cannot be done in this very specific way. Equally, any plotholes of the "why didn't they construct device X using the same technology as in device Y" are less likely to occur if you don't know the details of Y that might prevent this.
So just grab some popular science magazine or read around on Wikipedia about proposed technologies related to your plot and figure out what proponents say that they can do and what critics say their problems are. Then find some place in between and use them for your story.
To take your quantum physics example, if you read around a bit in science news, you will find people talking about quantum computers and quantum encryption. Of the former people apparently think that they might be able to crack any classical encryption, but right now they are big experimental devices requiring clean rooms and special cooling, and so on. Of the latter, they think that it is uninterceptable, but you need a line of sight or a direct fiber connection to exchange "entangled particles" or whatever that is (I am being deliberately obtuse here).
This is already enough material to spawn a mid-21st-century hard sci-fi spy story: Agent 008 obtains an encrypted storage device with important information from the last remnants of resistance in now fully occupied Hong Kong and somehow has to get it back home for decryption by the agencies quantum computers, all without it being intercepted. His quantum encrypted satellite line is way too slow for this amount of data, all fast modes of communication are watched and the borders are closed. How can he get the storage device out of the country without causing an international incident? How long does he have until his opponents notice what a treasure he is carrying?
Good questions, and great answers. I would like to add, because I love a good science fiction novel, you should find a balance between writing about science fiction and reality. Is it safe to say you're trying to supplement quantum physics for fantasy and magic? Because science fiction is still fiction, it doesn't discount the fact that things could happen within the laws of logic and reason. If the requirement is that one needs an advanced degree in astro-physics to understand what's going on then you maybe find yourself explaining to the reader too much about science and bore him or her. Conversely if your reader is expected to know the scientific fundamentals prior to giving your novel a read, then there is no problem. What I'm saying is that you should try to avoid it getting too geeky or nerdy. So what is good plain old science fiction? I say the more quantum physics the better! This will separate you from other science fiction by really challenging the mind of the reader and contest the limits of what's truly possible.
Is the story like Frank Herbert's Dune trilogy, or is it something like the movie Total Recall. Or, is your story so original that it has never been done before. These are only a couple of my favorites, and I would like to list more to make it easier to depict what it is exactly you're asking.
There is no "should" about that and if there were, it would still have almost nothing to do with writing.
Broadly, "writing" is a process for which "quantum physics" or any other topic provides some content.
Even if you plan to write an SF novel aimed largely at a readership of quantum physicists, simply read what your search engine lists as the three or four most publicly accessible texts.
Failing that, make it up. Who but a quantum physicist will challenge you?