27
votes
If all quantum gates must be unitary, what about measurement?
Unitary operations are only a special case of quantum operations, which are linear, completely positive maps ("channels") that map density operators to density operators. This becomes obvious in the ...
- 2,457
23
votes
Accepted
How can classical bits be copied if qubits cannot be copied?
TL;DR: The ban on copying is not nearly as universal as it might seem. No-cloning theorem actually allows copying as long as it is limited to orthogonal states. Classical information is the type of ...
- 24.2k
17
votes
Accepted
Why are quantum gates unitary and not special unitary?
Even if you only limit yourself to special-unitary operations, states will still accumulate global phase. For example, $Z = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}$ is special-unitary ...
- 40.8k
17
votes
Accepted
If quantum gates are reversible how can they possibly perform irreversible classical AND and OR operations?
Let's say we have a function $f$ which maps $n$ bits to $m$ bits (where $m<n$).
$$f: \{0,1\}^{n} \to \{0,1\}^{m}$$
We could of course design a classical circuit to perform this operation. Let's ...
- 17.7k
17
votes
Accepted
General parametrisation of an arbitrary $2 \times 2$ unitary matrix
What is the proof that any given unitary matrix can be converted as above?
Let $U$ be an arbitrary $2\times 2$ unitary matrix. This is equivalent to the rows/columns of $U$ being orthonormal bases.
...
glS♦
- 26.4k
15
votes
If all quantum gates must be unitary, what about measurement?
Short Answer
Quantum operations do not need to be unitary.
In fact, many quantum algorithms and protocols make use of non-unitarity.
Long Answer
Measurements are arguably the most obvious example of ...
glS♦
- 26.4k
15
votes
What is a Haar random quantum state?
Typically this is a slight abuse of notation. One can have a unitary operator $U$ chosen from some Haar measure, such as the circular unitary ensemble. Then, taking some fiducial state $|\psi_0\rangle$...
- 4,569
14
votes
Why are quantum gates unitary and not special unitary?
The fact that quantum gates are unitary, is rooted in the fact that the evolution of (closed) quantum systems is by the Schrödiner equation. For a time interval in which we are trying to realise ...
- 12.3k
13
votes
If all quantum gates must be unitary, what about measurement?
At risk of going off-topic from quantum computing and into physics, I'll answer what I think is a relevant subquestion of this topic, and use it to inform the discussion of unitary gates in quantum ...
- 1,097
13
votes
Accepted
What do they mean by "qubit can't be copied"?
All operations on quantum states are unitary operations. We don't make the rules, this is just how nature seems to work. So if you want to define an operation that copies a qbit, it has to be a ...
- 4,178
13
votes
Accepted
Implementing "Classical AND Gate" and "Classical OR Gate" with a quantum circuit
Your construction by gueswork in this answer is OK but not really elegant. Moreover, it's a convention to start in the state $|0\rangle$; we usually don't initialize a qubit with the state $|1\rangle$...
- 17.7k
13
votes
Accepted
How to check if a quantum circuit can be constructed for a given matrix representation?
Correct, unitarity is a sufficient and necessary condition. From Nielson and Chuang page 18:
Amazingly, this unitary constraint is the only constraint on
quantum gates. Any unitary matrix specifies a ...
- 2,533
12
votes
How to prove that the query oracle is unitary?
Apply it twice:
$$
O_xO_x|i\rangle|b\rangle=O_x|i\rangle|b\oplus x_i\rangle=|i\rangle|b\oplus x_i\oplus x_i\rangle=|i\rangle|b\rangle
$$
Hence, $O_x$ is its own inverse, and therefore reversible.
To ...
- 60.8k
10
votes
Accepted
Quantum states are unit vectors... with respect to which norm?
Born's rule states that $|\psi(x)|^2 = P(x)$ which is the probability of finding the quantum system in the state $|x\rangle$ after a measurement. We need the sum (or integral!) over all $x$ to be 1:
\...
10
votes
What do they mean by "qubit can't be copied"?
As already mentioned in the other answers, the crucial point is that copying means implicitly that the state of the original qubit is unknown, i.e. given a qubit in an unknown state, you want to ...
- 201
10
votes
Accepted
Doing non-unitary operations on quantum computer
Yes, the operation you have just described (projecting a qubit onto the fiducial state $|0\rangle$) is indeed an example of a non-unitary operation. Even though all quantum gates are unitary, the ...
- 1,232
9
votes
When can a matrix be "extended" into a unitary?
A necessary and sufficient condition is that, given an $n\times n$ matrix $M$, you can construct a $2n\times 2n$ unitary matrix $U$ provided the singular values of $M$ are all upper bounded by 1.
...
- 60.8k
9
votes
Quantum states are unit vectors... with respect to which norm?
Some terminology seems a little bit jumbled here. Quantum states are represented (within a finite dimensional Hilbert space) by complex vectors of length 1, where length is measured by the Euclidean ...
- 60.8k
9
votes
How to check if a quantum circuit can be constructed for a given matrix representation?
Right. But when you build a quantum computer, you want to have a certain set of gates that you want to implement, and all other gates (unitary matrices) can be built from that set of gates. This is ...
- 14k
9
votes
Accepted
How efficient is Qiskit's unitary decomposition?
For 2x2 unitary, it is just a U3-gate.
For 4x4 unitary, TwoQubitBasisDecomposer is used. TwoQubitBasisDecomposer implements KAK ...
- 11.6k
9
votes
Why do quantum gates have to be unitary?
Time Evolution Postulate: A pure state $|\psi(t_0)\rangle$ in a Hilbert Space $ \mathbb{H}$ evolves to another state $|\psi(t)\rangle$ is given by the time evolution operator $U(t,t_0)$
$$ |\psi(t) \...
- 14k
9
votes
Accepted
When can pairs of states be transformed into other pairs of states via unitary mapping?
There is a unitary that maps $\{|\psi_1\rangle,|\psi_2\rangle\}$ to $\{|\phi_1\rangle,|\phi_2\rangle\}$ if and only if
$$
\langle\psi_1|\psi_2\rangle=\langle\phi_1|\phi_2\rangle.
$$
Thy "only if&...
- 60.8k
9
votes
Accepted
Is effective quantum cloning possible, given that any classical function can be implemented as a quantum circuit?
No-cloning theorem suggests no such thing. A closer look at the theorem's proof reveals a loophole for orthonormal states. The theorem says that there is no unitary $U$ such that
$$
U|\psi\rangle|0\...
- 24.2k
9
votes
Accepted
What unitary commutes with all local Pauli operators?
TL;DR: The only $U$ that commutes with all $\sigma_{X,i}$ and all $\sigma_{Z,i}$ is a scalar multiple of identity. This follows from the Schur's lemma, but can also be shown using elementary linear ...
- 24.2k
8
votes
If all quantum gates must be unitary, what about measurement?
There are several misconceptions here, most of them originate from exposure to only the pure state formalism of quantum mechanics, so let's address them one by one:
All quantum operations must be ...
- 1,853
8
votes
Quantum states are unit vectors... with respect to which norm?
More mathematically, because $\mathbb{R}^n$ with an $L^p$ norm is a Hilbert space only for $p=2$.
- 189
8
votes
Implementing "Classical AND Gate" and "Classical OR Gate" with a quantum circuit
Simulating Classical "AND/NAND/OR/NOR/XOR/XNOR" Gates
With the help of this answer from Blue, constructing a matrix for a classical gate is just a matter of following the steps.
Here is the combined ...
8
votes
Accepted
Why is a Hadamard gate unitary?
The Hadamard gate is described by this matrix
\begin{equation}
H=\frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & 1 \\
1 & -1
\end{pmatrix}
\end{equation}
Conjugate transpose of $H$ is again $H$. Hence ...
- 14.8k
8
votes
How can I fill a unitary knowing only its first column?
Take your vector $\frac{1}{\sqrt{5}}(0, 1, 1, 1, 1, 1)^T$ and five other arbitrary ones but at the same time these vectors have to be linearly independent. After that apply Gram-Schmidt process which ...
- 14.8k
8
votes
Accepted
If a Hamiltonian is quadratic in the ladder operator, why is its time evolution linear in the ladder operator?
Hint: Instead of using the BCH formula in the form usually presented, for example at the top of this Wikipedia page, use this consequence of Hadamard's Lemma:
$$\tag{1}
e^{iHt}\hat{a}e^{-iHt} = \hat{a}...
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