Questions tagged [unitarity]

For questions related to the unitarity (unitary evolution) of quantum systems, as applicable to quantum computing or quantum information.

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2answers
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How can I fill a unitary knowing only its first column?

I have a unitary matrix that I want to construct. I only care what happens to the first computational state, so the first column is specified. So far, I've been assigning each question mark to a ...
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2answers
78 views

How to prove that a matrix is an arbitrary unitary?

My goal is to prove that I can synthesise arbitrary unitary from two components. In the end, I find a matrix with the form \begin{equation} \mathbf{W}_j=\begin{pmatrix} |\alpha|2\cos{(\phi_{...
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2answers
128 views

How should I understand the change of qubit's basis as a rotation?

I have a little difficulty with understanding. How do I properly visualize the change of qubit's basis as a rotation? Let's say that we have classical basis vectors, $|0\rangle$ and $|1\rangle$. Now, ...
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1answer
38 views

Explicit form for composition of Choi representation quantum channels

Let $|\Omega \rangle$ be the maximally entangled state over a bipartite system whose parts are each dimension $d$, i.e. $$ | \Omega \rangle \equiv \sum_i^{d}| ii \rangle $$ Then one way of writing ...
4
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2answers
45 views

What is the correct sign in the unitary evolution operator of a beam splitter?

I'm a bit confused about which is the correct sign in the unitary evolution operator of a beam splitter. In paper Digital quantum simulation of linear and nonlinear optical elements author uses the ...
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2answers
85 views

Circuit of a very trivial thing

I am writing to double check that if have a hamiltonian of the form $H = I_1 \otimes I_2$, when I seek to find the unitary, $e^{-i\gamma I_1 \otimes I_2}$, there really is no need to convert this into ...
0
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1answer
54 views

How can I build up an arbitrary quantum circuit given a certain unitary matrix operation?

Suppose I want to put a qubit whose initial state is $|0\rangle$ to the final state $\frac{1}{\sqrt{3}}|0\rangle + \sqrt{\frac{2}{3}}|1\rangle$. Well, in that case, the unitary matrix that performs ...
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0answers
30 views

Are there different orderings of the fifteen SU(4) generators in common use?

I've recently performed certain analyses (Archipelagos of Total Bound and Free Entanglement) pertaining to eq. (50) in Separable Decompositions of Bipartite Mixed States , that is \begin{equation} ...
4
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2answers
128 views

What unitary gate produces these quantum states from the computational basis?

Suppose that we have one-qubit unitary $U$ that maps $$ \left| 0 \right> \longmapsto \frac{1}{\sqrt{2}} \left| 0 \right> + {\frac{1+i}{2}} \left| 1\right> $$ and $$ \left| 1 \right> \...
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1answer
227 views

Why is a Hadamard gate unitary?

The Hadamard gate is a unitary gate, but how does the matrix times its own conjugate transpose actually result in the $I$ matrix? I am currently looking at it as a scalar, 0.707..., multiplied by the ...
4
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2answers
89 views

Prove that the state $\sum_{S\in P_n}(-)^{\tau(S)}|S\rangle$ is invariant up to a phase when changing the basis

I am trying to prove that the $|S_{n}\rangle$ is $n$-lateral rotationally invariant, where $|S_{n}\rangle$ is defined as $$|S_{n}\rangle=\sum_{S \in P_{n}^{n}} (-)^{\tau(S)}|S\rangle\equiv\sum_{S \in ...
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1answer
93 views

Trying to make irreversable operation in the quantum circuit

I want to make a 2 qubit circuit such that the non-unitary program will transform the regular basis in the way that: $|0 0\rangle \to |00\rangle$ $|0 1\rangle \to |01\rangle$ $|10\rangle \to |01\...
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3answers
198 views

Is there a quantum operation whose output is always orthogonal to the input?

I'm trying to show/convince myself the following statement is correct (I haven't been able to find any similar posts): "There is no reversible quantum operation that transforms any input state to a ...
1
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1answer
53 views

How can classical computations be non-unitary?

Given that classical physics emerges from quantum physics on a macroscopic scale, and all quantum operators are unitary, how are we able to perform non-unitary operations (such as setting a register ...
0
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1answer
67 views

Should a Pauli $X$ matrix equal the identity matrix to be unitary?

My understanding is that any unitary matrix must have its inverse be equal to its conjugate transpose. Looking at the pauli x gate as shown here: $$\begin{bmatrix}0&1\\1&0\end{bmatrix}$$ It ...
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1answer
45 views

Find the local unitary that takes the bell state to a state phi that has an extractable bell state

I have a state $|p\rangle$ that has an extractable Bell state and I want to write it as a Bell state, $|b\rangle$, with a local unitary acting on one side. Basically I am trying to find a local ...
2
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1answer
61 views

Does the general form of a unitary operator define strict signs for the second column?

As per IBM's documentation for quantum circuits, the general unitary operator is defined as: $$\hat{U}=\begin{bmatrix}\cos(\frac{\theta}{2})&-e^{i\lambda}\sin(\frac{\theta}{2})\\e^{i\phi}\sin(\...
4
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1answer
40 views

What is the unitary operator realizing a given CPTP operator

Complete Positive Trace Preserving Map (CPTP) operator is the most general operation that can be performed on a quantum system. This post mentioned that a CPTP operator is nothing but a unitary ...
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1answer
85 views

Are CPTP operators and unitary operators the same thing?

I am reading some quantum papers (In particular, this one page 34) . One of the theorem statement reads, "For every CPTP operator M, we have that .... " I ...
5
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1answer
69 views

Finding a global phase that transform the Hadamard gate to an element of $SU(2)$ and propose an evoultion operator which implents the operation

I was looking back over an old assignment and I came across a question I wasn't quite sure how to do the problem statement is as follows: The Hadamard rotation is an element of the group $U(2)$. (i)...
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4answers
2k views

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&...
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2answers
658 views

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?
4
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1answer
119 views

Differentiate between local and global unitaries

Just like we have the PPT, NPT criteria for checking if states can be written in tensor form or not, is there any criteria, given the matrix of a unitary acting on 2 qubits, to check if it is local or ...
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3answers
378 views

Is it correct to say that we need controlled gates because unitary matrices are reversible?

I am new to quantum computing and saw this argument on this site but I don't understand it. First of all, I don't understand what is exactly meant by 'reversible'. Because even if you had a unitary ...
6
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1answer
752 views

General parametrisation of an arbitrary $2\times 2$ unitary matrix?

Following is an excerpt from QCQI: I can understand that this matrix satisfies a unitary matrix. Also, intuitively, I am able to understand it. However, what is the proof that any given Unitary ...
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0answers
61 views

What do you specify when you physically apply a unitary?

In the Environment and Quantum Operations in Nielsen and Chuang, section 8.2.2, they say that when you apply a unitary on a state, you expect the output as the just the state transformed by the ...
5
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1answer
227 views

Extending a square matrix to a unitary matrix

Suppose we have a square matrix $M$ of size $n\times n$. It is given that any element $M_{ij}$ of $M$ is a real number and satisfies $0 \leq M_{ij} \leq 1$, $\forall$ $i,j$. No other property for $M$ ...
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1answer
142 views

Understanding why the modular function part of Shor's algorithm is unitary

I've been struggling to understand the modular exponent bit of Shor's algorithm. My understanding is that it takes a register in the state $\frac{1}{\sqrt{Q}}\sum_{k=1}^{Q-1} |k\rangle |0\rangle$ to ...
4
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1answer
178 views

Unitary gate(s) from product of exponent

Often unitary gates are defined as a product of exponentials, with some parameter in the power-term. However, often it is not clear how to construct unitary gates from it, at least not for me that is. ...
3
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2answers
68 views

Unitary acting on standard qubit basis properties

If we have a $U$ (unitary with all real entries) such that: $U|0\rangle =a|0\rangle +b|1\rangle$ What is $U|1\rangle=?$ I know: the definition of what it means to be unitary ie. $U^\dagger U=UU^\...
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3answers
882 views

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 ...
6
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2answers
139 views

What is the smallest quantum circuit to produce two-qubit state (a,b,b,b)?

How can I synthesis a two-qubit quantum state of the state vector (a,b,b,b) using basic quantum-gate circuit (arbitrary single-qubit rotation and controlled $Z$ gate)? And further, can I know a given ...
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6answers
3k views

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 ...
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4answers
1k views

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 ...
22
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7answers
2k views

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 ...
16
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1answer
987 views

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 ...