**TL;DR: No, we do not have any ***precise* "general" statement about exactly which type of problems quantum computers can *solve*, in complexity theory terms. However, we do have a rough idea.

According to Wikipedia's sub-article on Relation to to computational complexity theory

The class of problems that can be efficiently solved by quantum
computers is called **BQP**, for "bounded error, quantum, polynomial
time". Quantum computers only run probabilistic algorithms, so **BQP** on
quantum computers is the counterpart of **BPP** ("bounded error,
probabilistic, polynomial time") on classical computers. **It is defined
as the set of problems solvable with a polynomial-time algorithm,
whose probability of error is bounded away from one half**. A
quantum computer is said to "solve" a problem if, for every instance,
its answer will be right with high probability. If that solution runs
in polynomial time, then that problem is in BQP.

BQP is contained in the complexity class #P (or more precisely in the
associated class of decision problems P^{#P}), which is a subclass of
PSPACE.

BQP is suspected to be disjoint from NP-complete and a strict superset
of P, but that is not known. Both integer factorization and discrete
log are in BQP. Both of these problems are
NP problems suspected
to be outside BPP, and hence outside P. Both are suspected to not be
NP-complete. There is a common misconception that quantum computers
can solve NP-complete problems in polynomial time. That is not known
to be true, and is generally suspected to be false.

The capacity of a quantum computer to accelerate classical algorithms
has rigid limits—upper bounds of quantum computation's complexity. The
overwhelming part of classical calculations cannot be accelerated on a
quantum computer. A similar fact takes place for particular
computational tasks, like the search problem, for which Grover's
algorithm is optimal.

Bohmian Mechanics is a non-local hidden variable interpretation of
quantum mechanics. It has been shown that a non-local hidden variable
quantum computer could implement a search of an N-item database at
most in ${\displaystyle O({\sqrt[{3}]{N}})}$ steps. This is slightly
faster than the $\displaystyle O({\sqrt {N}})$ steps taken by
Grover's algorithm. Neither search method will allow quantum computers
to solve NP-Complete problems in polynomial time.

**Although quantum computers may be faster than classical computers for
some problem types, those described above can't solve any problem that
classical computers can't already solve.** A Turing machine can simulate
these quantum computers, so such a quantum computer could never solve
an undecidable problem like the halting problem. The existence of
"standard" quantum computers does not disprove the Church–Turing
thesis. It has been speculated that theories of quantum gravity, such
as M-theory or loop quantum gravity, may allow even faster computers
to be built. Currently, defining computation in such theories is an
open problem due to the problem of time, i.e., there currently exists
no obvious way to describe what it means for an observer to submit
input to a computer and later receive output.

As for *why* quantum computers can *efficiently* solve BQP problems:

The number of qubits in the computer is allowed to be a polynomial
function of the instance size. For example, algorithms are known for
factoring an $n$-bit integer using just over $2n$ qubits (Shor's
algorithm).

Usually, computation on a quantum computer ends with a measurement.
This leads to a collapse of quantum state to one of the basis states.
It can be said that the quantum state is measured to be in the correct
state with high probability.

Interestingly, if we theoretically allow post-selection (which doesn't have any scalable practical implementation), we get the complexity class post-BQP:

In computational complexity theory, PostBQP is a complexity class
consisting of all of the computational problems solvable in polynomial
time on a quantum Turing machine with postselection and bounded error
(in the sense that the algorithm is correct at least 2/3 of the time
on all inputs). However, Postselection is not considered to be a
feature that a realistic computer (even a quantum one) would possess,
but nevertheless postselecting machines are interesting from a
theoretical perspective.

I'd like to add what @Discrete lizard mentioned in the comments section. You have not explicitly defined what you mean by "can help", however, the rule of thumb in complexity theory is that if a quantum computer "can help" in terms of solving in polynomial time (with an error bound) iff the class of problem it can solve lies in BQP but not in P *or* BPP. The general relation between the complexity classes we discussed above is **suspected** to be:

$\text{P $\subseteq$ BPP $\subseteq$ BQP $\subseteq$ PSPACE}$

However, P=PSPACE, is an open problem in Computer Science. Also, the relationship between P and NP is not known yet.

Hi Niel! There is actually a quantum version of PPSZ with Grover speed-up: https://digitalcommons.utep.edu/cgi/viewcontent.cgi?article=1256&context=cs_techrep

– Martin Schwarz – 2018-04-05T07:50:13.383@MartinSchwarz: Thanks, that's an excellent reference! :-) I've added it to the final remarks on 'helpfulness', which feels quite apt. – Niel de Beaudrap – 2018-04-05T08:47:15.320

Niel, admittedly, my math skills are a bit under par for understanding this answer, but am I correct in interpreting what you said to mean that when there's an underlying relationship between the data that is difficult to impose on classical algorithms, that is when quantum computers shine? So to test with an example, should quantum computers be fantastic for finding primes? – TheEnvironmentalist – 2018-04-06T06:36:10.607

1@TheEnvironmentalist: that could be considered a

necessarycondition for a quantum advantage, but it isn't sufficient. One also has to be able to see precisely how the structure might be accessible by other means. ('Accessible' here is relative: the HHL algorithm shows aspects of linear algebra which are efficency solvable classically, but even more accessible to quantum algorithms; and Grover's algorithm shows how quantum algorithms seem to obtaina little bitmore access to information about unstructured problems than classical algorithms can, but 'shine' is a strong word to use there.) – Niel de Beaudrap – 2018-04-06T07:45:02.410Very interesting answer. What is exactly meant by "

features that do not have a (provably) statistically significant relationship to the standard basis." ? – JanVdA – 2018-06-19T12:22:57.963@JanVdA: I mean mathematical properties which can be described in terms of a relationship between vectors and operators, which are relevant to a problem, but where there is no known way to show that this relationship holds for vectors which are close to being standard basis vectors for interesting instances of the problem. If that sounds vague, that's because it is: it is intended as a heuristic test for a mathematical property, rather than a sharp characterisation which can be used to cook up new examples. – Niel de Beaudrap – 2018-06-19T16:51:42.147