# Questions tagged [quantum-gate]

For questions regarding usage, performance, implementation, application or theory related to quantum gates.

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### What does a quantum NOT operation do to an entangled set of qubits?

Quantum computing is not my field, so answers understandable to a layman will be most useful. Please forgive any incorrect terminology in my question! Assume that a set of the states of N qubits ...
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### How is the joint state of these qubits derived?

Can someone show to me the steps to derive the joint state at the bottom of this image, please? I tried to follow his explanation but I didn't get the same results… This is taken from the lecture ...
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### Reversibility and irreversibility of logic gates (quantum vs classical)

I have been told that one of the great keys that unlock quantum computing's potential is the reversibility of quantum logic gates as for classical gates there's some loss of information, but I cannot ...
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### Qiskit CNOT-gate matrix mixup?

In the qiskit textbook chapter 1.3.1 "The CNOT-Gate" it says that the matrix representation on the right is the own corresponding to the circuit shown above, with q_0 being the control and ...
2k views

### How to make a circuit for the CNOT using $H$ and $CZ$ gates?

How can we draw a circuit that is based on the gates $H, CZ$ that implements $CNOT$. I know that the $H$ gate is like that: And also the $CZ$ gate is: But I'm not sure how to draw this with the ...
86 views

### Can a circuit map $|x,y\rangle$ to the reflection of $|y\rangle$ with respect to $|x\rangle$?

Statement of the problem. I want to consider/design a quantum circuit that takes as input two vectors $\vert x \rangle$  and $\vert y \rangle$. The output of this quantum circuit must contain the ...
94 views

### How to determine an unknown quantum gate if we know all other gates in the circuit and the inputs and outputs? [duplicate]

Suppose we have a quantum circuit like this. All the gates are known except for one. For any input of q and q, we know the corresponding output. I have provided the output state for four ...
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### How to prove that the query oracle is unitary?

The query oracle: $O_{x}|i\rangle|b\rangle = |i\rangle|b \oplus x_{i}\rangle$ used in algorithms like Deutsch Jozsa is unitary. How do I prove it is unitary?
511 views

### Number of gates required to approximate arbitrary unitaries

If I understand correctly, there must exist unitary operations that can be approximated to a distance $\epsilon$ only by an exponential number of quantum gates and no less. However, by the Solovay-...
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### Possibility of a "reset" quantum gate

I wish to have a "reset" gate. This gate would have an effect to bring a qubit to the $\mid0\rangle$ state. Clearly, such a gate is not unitary (and so I'm unable to find any reliable implementation ...
406 views

### Is the controlled-Hadamard gate in the Clifford group?

Is the controlled-Hadamard gate a member of the Clifford group? I understand that Controlled Pauli gates are in the Clifford group. If controlled Hadamard is Clifford member, then is a controlled-...
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### Construct Controlled-$G^{\dagger}$ from known Controlled-$G$

Let there be a known a scheme (quantum circuit) of Controlled-G, where unitary gate G has G$^†$ such that G≠G$^†$ and GG$^†$=I (for example S and S$^†$, T and T$^†$, V and V$^†$, but not Pauli and H ...
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### How does the CNOT gate operate when the control qubit is a superposition?

If a control qubit is in superposition, how it will affect target qubit if it is collapsed or in superposition? Is it true that CNOT works only if the control bit collapsed to 1? Also, is it possible ...
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### Most efficient way for general state generation

Assume we are given an $n$-qubit system and complex numbers $a_0, \ldots, a_{m-1}$ with $m = 2^n$. Assume further we start with the initial state $|0 \ldots 0\rangle$ and want to make the ...
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### Composing the CNOT gate as a tensor product of two level matrices

I don't understand, why is the control not gate used so often. As far as I understand it, if you apply two 2 level operations on two qubits then you get a 4 x 4 matrix by the tensor product. So how ...
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### How to check if a $n$-qubit unitary is the tensor product of single-qubit unitaries

Let's assume I give you the expression of a unitary matrix acting on two qubits that is: $$U=\sum_{i} A_i \otimes B_i$$ for some operators $A_i$ and $B_i$. Is there a simple criterion allowing you to ...
393 views

### Decomposing a $(w+1)$-qubit permutation gate into $w$-qubit permutation gates, SWAPs and NOTs

Say I have a quantum circuit of $w+1$ qubits with a permutation gate (mapping computational basis states to computational basis states) that does the permutation $(i, i+1)(i+4, i+5)$ on $w+1$ qubits ...
557 views

### Prove that adding any non Clifford gate to the Clifford group yields a universal gate set

I have seen it claimed in multiple places that adding any non Clifford gate to the Clifford group yields a universal gate set. It is, however, not easy to find an accessible proof of this fact. The ...
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### Does conjugation by a Clifford send each non-identity Pauli to every other non-identity Pauli with equal frequency?

I see here in Olivia DeMatteo's notes, she states: When we consider the action of the entire Clifford group on a single non-identity Pauli, it maps that Pauli to each of the $d^2 − 1$ other possible ...
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### Partial trace and SWAP in the basis of subsystems

I'm trying to derive equation $(1)$ on p.2 in Lloyd et al, 2013 which reads  \text{Tr}_A\left[\exp(-i\theta S_{AB}) (\rho_A \otimes \sigma_B) \exp(i\theta S_{AB}) \right] = (\cos^2 \theta) \sigma_B +...
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### Is the universality of a qubit based quantum computer different from the universality of a continuous-variable quantum computer?

I understand that a quantum computer is universal if it can compute anything that a quantum Turing machine can. Another way to think about universality is that any unitary transformation on, e.g., a ...
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### Question Regarding Quantum Period-Finding Fourier Transform Approximation

I am following the 5.4.1 Period-Finding Algorithm in Nielsen and Chuang as shown below: My confusion lies with the second expression of point 3 in the procedure. Why is the second expression an ...
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### If an auxiliary qubit is allowed, how to construct toffoli gate in easier way?

We know if we don't use auxiliary, the construction of Toffoli gate will be: However, if now you are allowed to use one auxiliary qubit, how to realize a CCNOT in a simplier way? (Can we only use X,Y,...
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### How to implement the CCH gate in quantum computers available in clouds?

How to implement CCH gate in quantum computers available in clouds? If there is not any gate directly available for it, what are the possible ways to represent CCH?
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### How to check if a two-qubit gate is entangling?

I would like to know if there's an analog for Schmidt rank that can tell me if a two-qubit unitary is entangling? Suppose I have a parametrized two-qubit unitary $U^{(2)}(\theta)$. I would like to ...
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### Custom gates on IBM Q

I realized that QASM supports custom gates. However, when I tried to create the gate, transpiling error appeared both on simulator and real quantum processor. I suspect that IBM has not implemented ...
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### What are the relations between the permutation group and the Clifford group?

I'm trying to understand the relation between the permutation group on all the $2^n$ bitstrings and the Clifford group. My question arises from the fact that the Toffoli gate (which can be thought of ...
My idea was to apply $Z$ operator 𝐭𝐰𝐢𝐜𝐞, which leads us back to the point where we started from, and also show that after applying the $Z$ operator just 𝐨𝐧𝐜𝐞 we are not at the same point ...