For questions related to the unitarity (unitary evolution) of quantum systems, as applicable to quantum computing or quantum information.
Questions tagged [unitarity]
237 questions
39
votes
8 answers
If all quantum gates must be unitary, what about measurement?
All quantum operations must be unitary to allow reversibility, but what about measurement? Measurement can be represented as a matrix, and that matrix is applied to qubits, so that seems equivalent to the operation of a quantum gate. That's…
auden
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22
votes
6 answers
Quantum states are unit vectors... with respect to which norm?
The most general definition of a quantum state I found is (rephrasing the definition from Wikipedia)
Quantum states are represented by a ray in a finite- or infinite-dimensional Hilbert space over the complex numbers.
Moreover, we know that in…
Adrien Suau
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22
votes
4 answers
Why are quantum gates unitary and not special unitary?
Given that the global phases of states cannot be physically discerned, why is it that quantum circuits are phrased in terms of unitaries and not special unitaries? One answer I got was that it is just for convenience but I'm still unsure.
A related…
wdc
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18
votes
5 answers
Implementing "Classical AND Gate" and "Classical OR Gate" with a quantum circuit
Quantum cNOT Gate (Classical XOR Gate)
A "Controlled NOT (cNOT) Gate" flips the 2nd qubit if the 1st qubit is $\left|1\right>$, and returns the 2nd qubit as-is if the 1st qubit is $\left|0\right>$. The 1st qubit is simply not changed.
The net effect…
Siu Ching Pong -Asuka Kenji-
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18
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1 answer
If quantum gates are reversible how can they possibly perform irreversible classical AND and OR operations?
Quantum gates are said to be unitary and reversible. However, classical gates can be irreversible, like the logical AND and logical OR gates. Then, how is it possible to model irreversible classical AND and OR gates using quantum gates?
Sanchayan Dutta
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15
votes
3 answers
What do they mean by "qubit can't be copied"?
What does it mean by ''qubit can't be copied'' ?
In a note, it is saying that:
Copying a qubit means $$U|\psi\rangle_A|0\rangle_B=|\psi\rangle_A|\psi\rangle_B$$
i.e; applying a unitary transformation on the qubit state. It is explained as, if the…
tarit goswami
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15
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1 answer
What is a Haar random quantum state?
Can somebody please explain me what is a Haar random state? I am not able to find any friendly resource to read about it.
Shweta Aggrawal
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14
votes
1 answer
What is the probability $\Pr(\|U-I\|_{\rm op}<\varepsilon)$ of a Haar-random unitary being close to the identity?
If one generates an $n\times n$ Haar random unitary $U$, then clearly $\Pr(U=I)=0$. However, for every $\epsilon>0$, the probability
$$\Pr(\|U-I\|_{\rm op}<\varepsilon)$$
should be positive.
How can this quantity be computed?
Calvin Liu
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13
votes
1 answer
General parametrisation of an arbitrary $2 \times 2$ unitary matrix
From Nielsen & Chuang's Quantum Computation and Quantum Information (QCQI):
Since $U$ is unitary, the rows and columns of $U$ are orthonormal, form which it follows that there exist real numbers $\alpha$, $\beta$, $\gamma$ and $\delta$ such that $$…
Tech Solver
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13
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1 answer
How can classical bits be copied if qubits cannot be copied?
The no-cloning theorem of quantum mechanics tells us there can be no general quantum circuit that can copy arbitrary qubit states, i.e. a quantum gate or circuit cannot send $|0\rangle |\psi\rangle\mapsto|\psi\rangle |\psi\rangle$ for arbitrary…
MaximusIdeal
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12
votes
2 answers
How to check if a quantum circuit can be constructed for a given matrix representation?
Let's say I have a matrix representation, e.g.
$$
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}.
$$
How can I determine whether a quantum circuit can be constructed given said matrix representation?…
Dorijan Cirkveni
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10
votes
2 answers
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?
Divyat
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10
votes
1 answer
How efficient is Qiskit's unitary decomposition?
In Qiskit's extension package we have the UnitaryGate module that you can initialize using a unitary matrix and then add it to your circuit. How efficiently is this decomposition done under the hood?
Also, if I wanted to do the decomposition myself,…
Dani007
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9
votes
2 answers
Are SU($n$) operations enough for quantum computation?
Usually we want a quantum computer that can perform all foreseeable unitary operations U($n$).
A quantum processor that can naturally perform at least 2 rotation operators $R_k(\theta)=\exp(-i\theta\sigma_k/2)$, where $\sigma_k$ are the Pauli…
Mauricio
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8
votes
1 answer
If a quantum algorithm requires a measurement, how can we use that as a subroutine in another quantum algorithm?
Some algorithms (like period finding), use one or more measurement step. The post measurement state is then acted upon by another set of gates to complete the algorithm.
If I imagine this as blackbox algorithm $f$ which takes $x$ as input and…
ssj009
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