# How does a quantum circuit capture the "non-halting ability" of classical Turing machines?

It is claimed that a quantum Turing machine is "computationally" equivalent to the circuit model. Quantum computing also includes all classical computing. However, we know that there are classical Turing machines that do not halt on certain inputs.

How does a quantum circuit capture this non-halting "ability" of classical Turing machines? The way we draw them, every quantum circuit inevitably halts when we perform the measurements at the end.

• How do you capture a classical one? If the same as what you stated, the classical one, instead of using abstract Turing Machine we use the more concrete classical circuit, will also halt(except we have for loop? If so, the question reduced to for loop in quantum computers?) Dec 1, 2021 at 10:08
• Yeah you're right! Dec 2, 2021 at 3:30

$$t$$ steps of a quantum Turing machine running on an input of length $$n$$ can be simulated by a uniformly generated family of quantum circuits with size polynomial in $$t$$, and a polynomial-time uniformly generated family of quantum circuits can be simulated by a quantum Turing machine running in polynomial time.