POWER OF QUANTUM COMPUTERS

It is clear that when it comes to solving numerical search problems like Integer Factorization, quantum computers allow us to find the solution(s) instantly.
We just setup the problem (multiply two unknown integers and get an unknown integer result, set the unknown result to a result we want) and instantly the input integers become known.
So quantum computers are infinitely more powerful than regular computers for solving numerical search problems.

But we use regular computers also for symbolic calculation.
(CAS (Computer Algebra System) software like Mathematica, Maple etc.) What more quantum computers could provide when it comes to symbolic calculation?

I think they could provide the same benefit as for numerical calculation. Meaning instantly solving symbolic search problems.
Imagine if we could just setup an equation expression string as input, then quantum computer sets the output string (with unknown value) to a general solution expression (known value), if such solution really exists/possible.
For example:
1)
Input string: "a*x^0+b*x^1=0"
String value search problem: "x=?"
Output string: "-a/b"
2)
Input string: "a*x^0+b*x^1+c*x^2=0"
String value search problem: "x=?"
Output string: "(-b+(b^2-4*a*c)^(1/2))/(2*a)"

I think using quantum computers for symbolic calculation should allow us solving many important such problems which we cannot solve with regular computers in a practical time.
I am guessing those would even include some Millenium Prize Problems like finding (all) general solution expressions for Navier-Stokes equations (and proving Riemann Hypothesis?).

I think, assuming we will have a general purpose suitable quantum computer someday, only issue is figuring out exactly how to express and solve symbolic calculation problems like the two examples above.

Let's try to solve the first problem using a quantum computer:
Assuming quantum computer symbolic calculated the solution (expression string E), how we could test it to be correct or not?
How about creating an equation that would be true only if E is a valid solution, which is the input equation itself, then:
"a*E^0+b*E^1=0" or "a+b*E=0"
Then I think the solution algorithm for the quantum computer would be:
Start with unknown values E, a, b.
Calculate a+b*E (not numerical calculation but symbolic expression calculation, using an expression tree).
Set the unknown calculation result to 0.
Unknown string E collapses to the answer: "-a/b"

And if we consider how we could do the symbolic calculation step above using a regular computer, which requires manipulating an expression tree using stack(s), then we need figure out how to create a quantum stack using a quantum computer.
(Imagine a stack that can do any number of push/pop operations instantly, to collapse into its final known state instantly.)
(If we could do quantum stacks, then we also could do quantum queues.)
(And then quantum versions of other standard programming data structures would also be possible.)

What could be the most practical way to build a large scale quantum computer?

I think currently building a quantum computer is really hard because our physical world is highly noisy at quantum scale.
Imagine using single atoms/molecules as qubits.
Imagine cooling them close to absolute zero in vacuum environment that needs to be perfectly maintained.

Could there be a better way?

What if we create a quantum computer in a different level of reality, which does not have noise?

Think about our regular digital computers.
Could we think of the bit values in memory of a working regular computer, like a different level of reality of quasiparticles, which does not have noise?

Can we create an extrinsic-semiconductor-based quantum computer chip, that creates and processes qubits as quasiparticles?
(And the quantum computer designed and operated like a Cellular Automata, similar to Wireworld?)

https://en.wikipedia.org/wiki/Quasiparticle
https://en.wikipedia.org/wiki/Electron_hole
https://en.wikipedia.org/wiki/Extrinsic_semiconductor
https://en.wikipedia.org/wiki/Cellular_automaton
https://en.wikipedia.org/wiki/Wireworld