# Tag Info

### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...

### 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 ...
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### 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 ...

### 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 ...

### 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 ...
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### 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 $|1\rangle$...

### 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 $|\psi_0\rangle$...
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### 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 ...

### 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 ...
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### Can the Kraus decomposition always be chosen to be a statistical mixture of unitary evolutions?

You cannot always find such a Kraus decomposition. Notice that any CPTP map $\mathcal E$ which does have a decomposition as a probabilistic mixture unitaries is unital, which is to say that it maps ...

### 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. ...

### 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 ...
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### 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&...
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### 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 ...

### 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) \...
More mathematically, because $\mathbb{R}^n$ with an $L^p$ norm is a Hilbert space only for $p=2$.
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: \...