# Question About How Qiskit Reset Gate Affects Other Entangled Qubits

I am trying to understand how the reset gate in Qiskit affects qubits its entangled with. Consider the following circuit with qubits $$q_0$$ and $$q_1$$:

Where circuit240 takes $$|0\rangle$$ to $$a|0\rangle + b|1\rangle$$ and circuit 244 takes $$|0\rangle$$ to $$c|0\rangle + d|1\rangle$$. Right before the reset gate on qubit $$q_1$$ the state of this circuit is $$\frac{1}{\sqrt{2}}(c|1\rangle + d|0\rangle)|0\rangle + \frac{1}{\sqrt{2}}(a|1\rangle + b|0\rangle)|1\rangle \tag{1}$$

I'm not quite sure how to mathematically represent what this quantum reset gate does to the quantum entangled state. For example, I tried a few tests with this circuit where I played with the values $$a$$ and $$c$$ and tested how the quantum reset gate affects the measurement of the qubit $$q_0$$. It seems that whether or not the quantum reset gate is added, it doesn't affect the measurements of the other entangled qubit. Does this generally hold?

Furthermore, when I take the qubit expression above and just reset qubit $$q_1$$to $$|0\rangle$$ I get the following:

$$\frac{1}{\sqrt{2}}(c|1\rangle + d|0\rangle)|0\rangle + \frac{1}{\sqrt{2}}(a|1\rangle + b|0\rangle)|0\rangle \tag{2}$$

$$= \frac{1}{\sqrt{2}}((a + c)|1\rangle + (b + d)|0\rangle)|0\rangle \tag{3}$$

But, mathematically, the probability of measuring $$q_0$$ as $$|0\rangle$$ in $$(3)$$ is not the same as the probability in qubit expression $$(1)$$(even though the tests show that removing the reset gate did not change the probability of measuring a $$|0\rangle$$ in the qubit $$q_0$$. What is the correct way to represent what the qubit reset gate does to an entangled qubit?

A reset gate is equivalent to a swap gate between the target qubit and a new ancilla qubit in the $$|0\rangle$$ state. So you can replace your question with "how does swapping a qubit Q with an fresh ancilla qubit affect the qubits Q is entangled with?" or "how does discarding Q affect the qubits Q is entangled with". And the answer is that, for all intents and purposes, it doesn't affect them at all.
• Perhaps I misphrased the comment above. Lets say I have a circuit that takes $|v\rangle$ to $U|v\rangle$ where U is a circuit that has some reset gates. Now, I want to make another circuit $U^{\dagger}$, that takes $U|v\rangle$ to $|v\rangle$. In this case, how could I construct the circuit $U^{\dagger}$? I would need to know the value of each of the qubits before they were initially reset in order to do so right? Jul 10 '20 at 5:16
• Typically you'd implement a $U^\dagger$ that would work for any value, not just one. Jul 10 '20 at 10:13
• Right, which means that reversing a circuit with the reset gate isn't a straightforward task, correct (as opposed to reversing other more standard gates)? Let's say I have a circuit that is just the reset gate, taking any $|v\rangle$ to $|0\rangle$. There is no easy way to uncompute this circuit, right? (other than having a copy of $|v\rangle$ to begin with and using it) Jul 10 '20 at 19:19