What are my options for computing symbolic results for quantum circuits?
I have identified some possibilities:
SymPy (Python library, see also this), unfortunately, the software forces you to encode the matrix of each gate explicitly instead of working with circuits;
IBM Qiskit v0.25 operator framework (Python library), which claims
Qiskit operators fully support parameterization
even if it seems not completely true since the same error of this older question holds and this issue is still open.
The asker has clarified in this comment that they want a symbolic computation software in which the user does not "have to manually define circuits in terms of matrix multiplications", and the question says that the software cannot "force you to encode the matrix of each gate explicitly". This unfortunately does not exist.
In all symbolic computation software, one will have to encode the matrix of each gate (X,Y,Z,CNOT,H,etc.) and use tensor products (i.e. Kronecker products) and matrix multiplications to get the effect of the unitary operation of a quantum circuit; or find some add-on where someone else did exactly that and made it available to others. Luckily, it is not hard to define the gates manually, and to do matrix multiplications, in any symbolic computation software.
The most popular long-standing symbolic computation packages which are still actively being developed are:
There's others too, which people are welcome to suggest I add, but seriously anything other than the first five are not really "popular" in the circles that I know.
Since none of the above symbolic computation software has quantum circuit functionality built in to the extent where the user does not have to define the gates or explicitly involve matrix multiplications commands *natively *without relying on a third-party add-on where someone else did exactly the same thing, the asker says again in the same comment as above:
"any software for symbolic computation having dedicated features/facilities for quantum computing is accepted."
I will therefore point out that of that list of 8 major symbolic computation software packages, the one with the largest volume of quantum add-ons is Mathematica (also by far the most popular and highly developed package out of all of the above, for symbolic computation):
Quantum Mathematica: includes the ability to do quantum circuits in 2D and 3D, some Adiabatic Quantum Computing functionality, and the Quantum Fourier Transform. Unfortunately this has not been revised since 2010, according to that link, and since 2011 according to this, though the authors wrote a paper about it in 2015.
QDENSITY: Less well documented than Quantum Mathematica, but published more recently (August 2017).
Using Mathematica for Quantum Mechanics: This is also published in a book, however it's more geared towards "quantum mechanics" than "quantum computing".
There's also 378 quantum mechanics related Mathematica demonstrations on the Wolfram Demonstrations Project, but most if not all of them are geared towards quantum mechanics and not so much quantum computing.
Therefore "Quantum Mathematica" and "QDENSITY" are your best options with Mathematica, and unfortunately these will require a Mathematica license, which will be very expensive if you don't have a discount or free access through (for example) a university.
Although Maple has declined in popularity in recent years, they also do support Dirac notation as seen here and here, but there's nothing that can do entire quantum circuits as far as I know.
Just use SymPy or Sage (both open-source and Python based) and define the gates (X,Y,Z,CNOT,H,etc.) manually and do the matrix multiplications, since this will just take a couple hours. You can also read the above papers on how it was done in Mathematica, to give you ideas on how to efficiently save time in doing it with SymPy or Sage (i.e. how to make it so that you don't have to repeat the identity matrix so many times, for example).
