# Re-initializing qubits in a circuit without losing superposition

So recently I was building some circuit made of subcircuits. My subcircuits happen to use auxiliary qubits, that are always initialized in the state $$|0\rangle$$ and are not used after the calculation. Hence, I wanted to use the same set of auxiliary qubits for all the subcircuits by reinitializing them between subcircuits. As I code on Qiskit, I used reset gates to do so.

But I have a problem : my circuit is correct on each possible input state that belongs to the basis $$\{|0\rangle,|1\rangle\}^{\otimes n}$$, but the reset gates make it fail for any superposed state, henceforth losing all quantum advantage.

I'll explain the problem with a simplified example : let's say that I have two subcircuits, C and C', that operate on 1 main qubit (that I want to change) and use 1 auxiliary qubit. When C takes as input $$|0\rangle$$ for the main qubit, it outputs the main qubit in the state $$|0\rangle$$ and leaves the auxiliary qubits in the state $$|0\rangle$$, and when it takes as input $$|1\rangle$$ for the main qubit, it outputs the main qubit in the state $$|1\rangle$$ and gets the auxiliary qubits in the state $$|1\rangle$$. My full circuit resets the auxiliary gate between C and C' (with the Qiskit reset gate) and then C' is applied to get the wanted output on the main qubit. Now, if I input the state $$\frac{|0\rangle+|1\rangle}{\sqrt{2}}$$ on the main qubit, which is overall (taking into account the auxiliary qubit) the input state $$\frac{|0\rangle+|1\rangle}{\sqrt{2}} \otimes |0\rangle$$, after C we get : $$\frac{|00\rangle+|11\rangle}{\sqrt{2}}$$ But then after the reset gate we don't get : $$\frac{|00\rangle+|10\rangle}{\sqrt{2}}$$ But $$|00\rangle$$ 50% of the time and $$|10\rangle$$ 50% of the time. As superposition is broken, the main circuit gets flawed.

This behavior can be reproduced with the following code:

qc=QuantumCircuit(2)
qc.h(0)
qc.cx(0,1)
qc.reset(1)
backend=Aer.get_backend('statevector_simulator')
plot_histogram(execute(qc,backend,shots=1024).result().get_counts())


And by the way, in my understanding, this is in disagreement with this answer - which motivated me to ask my question, even if it may seem redundant at first sight.

Is there a way I could fix that ?

I am pessimistic, as in my previous example, the operation I'd like to apply isn't unitary. But I know that I can find on this site people that are amazing at quantum computing, so I have hope that a solution can be found.

Thanks a lot,

Thomas

• Hi Thomas! I can't reproduce your issue, if I'm running your code above I'm getting 50% $|00\rangle$ and 50% $|10\rangle$. Can you try installing the latest version of Qiskit and see if the behavior persists? Apr 8 at 21:22
• Hello Cryoris ! Well we then have the same behavior running this code :) And here is my problem, because I wanted to keep the superposition, and it is lost. My problem isn't a bug in Qiskit, it is that what I used to perform my resetting qubits action didn't go as I wanted. I am not saying that Qiskit reset gates behave wrongly, I'm saying that they do not fit my needs, as explained in the example. Thanks for your help :) Apr 9 at 6:15

The important point to understand is that you lose the superposition and obtain a classical mixture of states $$|0\rangle$$ and $$|1\rangle$$ on the first qubit. If you look at the circuit diagram below:

• Before applying the reset you're in a Bell state (green).
• As you apply the reset, you effectively remove the qubit from the system, you can think of it as tracing it out. If you trace out system B in the Bell state you obtain a maximally mixed state (blue).
• Finally, you add a new qubit in state $$|0\rangle$$ that has no correlation whatsoever with qubit A, so you end up in a product state of a mixed state and the zero state (red).

This means your final state is not a pure state where the first qubit is in state $$\frac{|0\rangle + |1\rangle}{2}$$ and the second one, reset, in state $$|0\rangle$$. Instead, it is a mix of classical probabilities $$\rho_{AB} = (p_1 \rho_1 + p_2 \rho_2) \otimes |0\rangle\langle 0|$$ where $$p_1 = p_2 = 1/2$$ and $$\rho_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \rho_2 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.$$

If you now perform a statevector simulation of this circuit, you have a 50% probability of measuring state $$|00\rangle$$ and 50% of measuring $$|10\rangle$$. Running this in Qiskit

from qiskit import QuantumCircuit, Aer, execute

circuit = QuantumCircuit(2)
circuit.h(0)
circuit.cx(0, 1)
circuit.reset(1)

print(circuit.draw())

backend = Aer.get_backend('statevector_simulator')
statevector = execute(circuit, backend).result().get_statevector()
print('Statevector')
print(statevector)


will print [1, 0, 0, 0] in half the cases and [0, 1, 0, 0] in the other half. It is a classical combination of quantum states, not a superposition like [0.707, 0.707, 0, 0].

What's a bit confusing now is that counting shots gives the same histogram for the classical mix of quantum states as for superposition. That's because the probability to measure either $$|0\rangle$$ or $$|1\rangle$$ is the same for both. In Qiskit code:

circuit = QuantumCircuit(2)
circuit.h(0)
circuit.cx(0, 1)
circuit.reset(1)
circuit.measure_all()

backend = Aer.get_backend('qasm_simulator')
counts = execute(circuit, backend).result().get_counts()
print('Counts')
print(counts)


Will give on average {'00': 512, '10': 512}. But the pure state $|\phi\rangle = \frac{|0\rangle + |1\rangle}{2} \otimes |0\rangle$ will give the same histogram:

circuit = QuantumCircuit(2)
circuit.h(0)
circuit.measure_all()


A reset is a very "brutal" non-unitary operation and will destroy the pure state you have.

Note: you can derive the same result by computing the density matrices in the circuit from @Egretta.Thula, I just skipped it to make the diagram easier.

• Overall, your answer makes sense to me, but there's something I don't really understand. You said $\mathbb{1}_2/2 \otimes |0\rangle \langle 0 |_B$ to be a state, and I don't follow you on that. From what I know of quantum mechanics (I'm still a grad student, so maybe it's incomplete), you are taking the tensor product of two operators, and thus you get an operator and not a state. As a result, I don't know whether your answer proves that in a general manner, there is no way I can reset my qubits as I initially wanted. Could you please clarify that ? Thanks a lot for your time Apr 10 at 13:34
• This is a density matrix representation (in fact all $\rho$ I used are) of a quantum state. A density matrix can represent more general states, such as mixed states, while the "ket"/vector representation only works for pure states. So $\mathbb{1}/2 \otimes |0\rangle\langle0|$ is a tensor product of two states (in density matrix representation), not operators. See e.g. en.wikipedia.org/wiki/Density_matrix for more details on density matrices. Apr 10 at 22:02
• Ok, I think that I understand your answer now, thanks a lot. It definitely answers my question, thanks a lot for all the details you provided. Apr 11 at 12:24
• Sure thing, it was a good question! Apr 11 at 12:39

According to this link, IBM does not implement Reset instruction as a swap gate between the target qubit and a new ancilla qubit in the $$|0⟩$$ state. It is implemented as a

not-gate conditioned on the measurement outcome of the qubit

Which means your circuit is equivalent to the following circuit:

This should explain the results you have obtained.