Does this argument refuting the existence of superintelligence work?

A quantum computer has a huge amount of internal state that even the machine can't get at directly. (You can only sample the matrix state.) The amount of that state goes up exponentially with each quantum bit involved in the system. Some operations get insane speedups from quantum computing: you just put the quantum wire through a quantum gate and you've updated the entire matrix at once.

Simulating a quantum computer with a classical one would take exponentially longer for each qubit. With several dozen qubits, the machine's computing power for some tasks couldn't even be approached by a normal computer, much less a human mind.

Relevant: my answer on To what extent can quantum computers help to develop Artificial Intelligence?

Note that with quantum computers, you've gone beyond the normal zeroes and ones. You then need a quantum Turing machine, which is a generalization of the classical one.

I believe this argument is based on the fact that intelligence is a single dimension when it really isn't. Are machines and humans really on the same level if a machine can solve a complex problem in a millionth of the time a human can?

It also assumes that the Turing machine is still the best computational model for the time period that you are in, which is not necessarily true for the future, it is just true until this point in time.

The flaw in your argument is that "surpass" doesn't just mean that you should be able to run all algorithms, it includes a notion of complexity, i.e. how many time steps you will take to simulate an algorithm.

How do you simulate an algorithm with a Turing machine? A Turing machine consists of a finite state machine and an infinite tape. A Turing Machine does run an algorithm, determined by its initial state and the state transition matrix, but what I think you are talking about is Universal Turing Machines (UTM) that can read "code" (which is usually a description of another Turing machine) written on a "code segment" of the tape and then simulate that machine on input data written on the "data segment" of the tape.

Turing machines can differ in the number of states in their finite state machines (and also in the alphabet they write on the tape but any finite alphabet is easily encoded in binary so this should not be the big reason for differences among Turing machines). So, you can have UTMs with bigger state machines and UTMs with smaller state machines. The bigger UTM could possibly surpass the smaller one if they use the same encoding for the "code" part of the tape.

You can also play around with the code used to describe the TM being simulated. This code could be C++, for example, or could be a Neural network with the synapse strength written down as a matrix. Which description is better for computation depends on the problem.

An example comparison among UTMs with different state machines: consider different compilers for the same language, say C++. Both of them will first compile C++ to assembly and then run another UTM which reads and executes assembly (your physical CPU). So, a better compiler will run the same code faster.

Back to humans vs computers, humans are neural networks that run algorithms like those you would write in C++. This involves a costly and inefficient conversion of the algorithm into hand movements. A computer uses a compiler to convert C++ to assembly that it can run natively, so it's able to do a much more efficient implementation of C++ code. Alternately, humans have a ton of neurons, and the neural code, i.e. synapse strength, is hard to read, so current computers cannot run that code yet.