We wrote a book intended to give an intuition for how quantum algorithms work for an audience who do not necessarily want to dive into the detailed linear algebra:
I really like a cloud chamber that I can put together in the pub whilst potentially drunk. A pint glass and some dry ice does the job (dry ice being obtainable from a fire extinguisher - not that I'd recommend you grab one off the pub wall without replacing it refilled).
Since a lot of general intros to QM are being suggested (rather than technical/historical texts), then here's another recommendation which is not technical nor historical, but comes as close as you can get to a proper understanding of QM without mathematical detail:
Not to be too contrary, but I disagree with this sentiment. First let me say I'm assuming that the OP wants to involve a historical narrative to gain a deeper conceptual understanding of quantum mechanics.
Under that assumption, the interesting thing about QM is that there is very little inherent conceptual justification for the technical descriptions we have of quantum systems other than that 'they work'. Or more precisely, understanding why QM systems are mathematically described in the way they are is still an active area of research.
Therefore, often the closest you can come to a deeper understanding is by tracing how someone stumbled into, or was forced into, using a particular mathematical approach. At least then you are left at the same precipice of knowledge over which experts have been dangling for nearly 100 years now.
I hope that one day we can give deeper explanations for why we have Hilbert spaces and what-not, but until then, learning the history is the closest you can get to understanding why things ended up the way they are on a technical level. And assuming a decent mathematical background, there's no reason you can't also learn the technical detail itself for the first time right along with that historical narrative.
There are some translation/copy-writing issues but don't let that put you off. It has a solid coverage of quantization of Hamiltonian mechanics in both matrix mechanics and wave mechanics, and then the unification of the two - all following the historical narrative.
Building a good-enough optical table relatively cheaply isn't too hard. Use cinder blocks for legs and then alternately layer plywood and carpet, with a layer of spaced out wheelbarrow inner-tubes midway through. You can even add a final layer of sheet steel if you're using magnetically mounted optical components. You can test the stability by reflecting a laser-pen off a dish of water and watching the reflection on a piece of paper/wall. Most important is to be away from traffic etc...
We wrote our O'Reilly book, "Programming Quantum Computers", precisely to fit your use case. It assumes no advanced mathematics, doesn't shy away from really delving into how algorithms work, and also has an online simulator to let you experiment with actual code. I am of course biased, but I would say that it's the resource out there requiring the least mathematics needed to get some practical knowledge and a chance to experiment in code:
Shameless plug dealt with, the text I'd next point you to for being grounded more in the Real world rather than Hilbert space is Mermin's. Modulo his insistence on using the term QBit rather than qubit, it's a great pedagogical work by someone with a very deep understanding of quantum mechanics. It's also in hardcover, which also helps lend it more weight:
As others have recommended, anything by Scott Aaronson is gold. Computational complexity is his passion, and although I think his work is very thorough and accessible, I would suggest it's a little less hands-on. However for very, very deep insights there's nowhere better to go. Alongside his book, Aaronson's blog at https://www.scottaaronson.com/blog/ is also revered by both QC enthusiasts and professionals alike and a great place to follow debate on the latest developments in the field.
The most mathematically demanding text (or most thorough, depending on how you look at it) I'd consider is Nielsen and Chuang (a.k.a "Mike 'n' Ike", a.k.a "The bible"). It's slightly out of date in some more recent concepts regarding the implementation of quantum computing (not wrong, just a tad incomplete), but is still solid and indispensable for the core concepts and insights behind quantum computing.
If you're interested in the physical implementation of a quantum computer (i.e. what does it _look_ like inside), then Mike 'n' Ike is the only one that will come close to satisfying you. The real world is so damn messy, and quantum hardware is no exception. QC tech moves fast and the money is still on the table as to just what that tech might look like inside the million qubit quantum computer of the future. Mike 'n' Ike does discuss some specific types of qubits, but I'm not aware of a book providing a truly comprehensive and up to date description of today's most promising approaches.
To second rjurney’s comment, I’ve also recently had the pleasure of working with Mike Loukides, and his feedback has always been hugely insightful and valuable.
https://www.oreilly.com/library/view/programming-quantum-com...