Questions about the definition of the Quantum Turing Machine

I am trying to have a better understanding of the definition of the Quantum Turing Machine.

My questions:

1. If the output of a quantum program is the eigenvalue of the ground state of a Hamiltonian and is an irrational number, how do we finish writing it on the output tape?
2. How do we deal with the gates / unitary matrices when we are supposed to write the shortest description of a quantum program? We can always squeeze a deep quantum circuit into a single unitary matrix.
• 1. If the output of a classical program is $\sqrt{2}$, how do we finish writing it on the tape? Does that mean that $\sqrt{2}$ is uncomputable? Commented Sep 13, 2016 at 3:07
• 2. For any classical problem, can't you define a classical gate that has $n$ wires in, $n$ wires out, and takes the input to the output? Why don't we just use that gate rather than writing complicated algorithms? Commented Sep 13, 2016 at 3:08
• @PeterShor, yes $\sqrt{2}$ is computable. So, yes, now I understand that a quantum computer can print $n$ digits of the number and halt. Commented Sep 13, 2016 at 18:26
• Right. The same is true for quantum computers. Some definitions allow any unitary gate from two qubits to two qubits, but most interesting problems you want to solve involve maps from $n$ qubits to $n$ qubits. Commented Sep 15, 2016 at 14:51
• In the begining, you should check very well about quantum mechanics, before ansewring this question, because, there is tight relation ship between quantum mechanics and quantum turing machine Commented Apr 18, 2017 at 5:45