Toffoli and Hadamard: simulating quantum calculation with integer numbers?

Question

I see that Toffoli and Hadamard gates are "universal for quantum calculation". This paper by Aharonov says that this universality has a slightly wider meaning than usual. This is clear from the fact that Toffoli and Hadamard gates are represented by matrices composed by real numbers, so they do not know anything about phase rotations and other gates that have imaginary numbers in their matrix. This is explained e.g. in this previous answer.

The matrices representing Toffoli and Hadamard gates are not only represended by real numbers, but, if we neglect a global real normalization factor, they are only composed by 0, 1, and -1. So we could work with them using amplitudes represented by integer numbers (times a global real normalization factor).

However, I still do not understand the extended definition of universality.

i) In the paper of Aharonov, the "universality" is still defined as the ability of approximating any other gate (definition 3). This is puzzling for me.

ii) On the other hand, this post suggests that the "universality" of Toffoli and Hadamard gates means that such circuits are able to perform BQP calculations. It is a weaker but very interesting statement.

iii) There is yet another interpretation, i.e. that Toffoli and Hadamard gates can implement any gate, if provided with suitable ancillae.

Probably my interpretation of (i) is wrong. (ii) says that we can perform the decision in BQP using circuits involving only real and integer numbers. Fascinating, but is it true? Probably (iii) is true, but it does not tell us too much about the possibility of quantum calculations with integer numbers, because the ancillae contain complex numbers.

Answer 1

There is no discrepancy among (i), (ii) and (iii).

The basic idea is that $\{T, H\}$ is computationally universal and you can simulate any quantum circuit with it though overhead qubits may be necessary. However, that isn't the same thing as approximately decomposing an arbitrary quantum gate into some sequence of $\{T, H\}$ gates.

If you look at Aaronhov's paper (theorem 2), they clearly mention that they're simulating $\Lambda(P(i))$ which has complex entries by converting it to its real version using overhead qubits.