Questions tagged [notation]

For questions about the usage of, origin of or reasons for specific notation used in the context of quantum computing.

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38 views

For two-qubit systems, do we have $\langle 01|01\rangle = \langle 0|0\rangle\langle 1|1\rangle$?

I am new to quantum computing and I want to know the following: If I have a 2 qubit system in state e.g. $\left|01\right>$ and I want to calculate the probability of measuring e.g. $\left<01\...
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57 views

When discussing error correction, what are the objects in the expression $PE_i^\dagger E_j P=\alpha_{ij} P$?

I've started reading the book "Quantum Computation and Quantum Information" by Michael A. Nielsen and Issac L. Chuang, specifically chapter 10 (about quantum error correction), and I'm ...
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41 views

What does $\langle\partial_i\psi(\theta)|\psi(\theta)\rangle$ mean when implementing the Quantum Fisher information matrix?

Following this paper, the quantum Fisher information matrix (QFIM) - $\mathcal{F}$ can be calculated as: $\mathcal{F}_{i, j}(\theta)=4 \operatorname{Re}\left[\left\langle\partial_{i} \psi(\boldsymbol{\...
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39 views

What is meant by a "projection operator" in the book "Quantum Computation and Quantum Information"?

I've started reading the book "Quantum Computation and Quantum Information" by Michael A. Nielsen and Issac L. Chuang, specifically chapter 10 (about quantum error correction), and I'm ...
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61 views

Is there any standard notation for two states being "equivalent up to a global phase"?

Is there any standard notation for two states "equivalent up to a global phase"? Currently, I use just an equal sign but I'd like to use a different symbol to distinguish between two phases ...
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123 views

Question regarding the notation of QFT

I have a question about the notation of QFT. I would like to present briefly what my problem is. So given is the QFT as a mapping with: $$|j_1,...,j_n\rangle \rightarrow \frac{(|0\rangle + e^{2\pi i 0....
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65 views

How to compute the measurement probabilities of $|\phi\rangle=\sum_k c_k |k\rangle$ in a rotated basis $V|k\rangle$?

I came across the following question and have some conceptual questions. Consider a general quantum state $|\phi\rangle$ of dimension $N$ spanned by some standard basis $\{|k\rangle,k=0,1,...N-1\}$. ...
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Is there a convention for denoting $Y$ eigenstates?

Two common shorthands for eigenstates of the $Z$ operator are $\{|0\rangle,|1\rangle\}$ and $\{|1\rangle,|-1\rangle\}$, where in the first case we have $Z|z\rangle=(-1)^z|z\rangle$ and in the second ...
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49 views

What is the best notation to write pairs of one-qubit ket states?

I am working on coming up with practice problems for a QC course. I have a problem that considers two qubits as so: $$|\psi_a\rangle = \alpha_a |0\rangle + \beta_a |1\rangle$$ $$|\psi_b\rangle = \...
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133 views

Elementary question: is $\langle a|b \rangle$ the same as $\langle a||b\rangle$?

In trying to learn more about quantum computing, one of the first things encountered is a definition of the inner product $\langle a|b\rangle$. If I interpret it correctly, and looking near the bottom ...
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166 views

What is a term for a basis state along with its corresponding complex amplitude?

For some arbitrary state $|\psi\rangle = c_0|x_0\rangle + c_1|x_1\rangle + c_2|x_2\rangle ... + c_{2^n}|x_{2^n}\rangle$, where each of $|x_i\rangle$ is a basis state, and each of $c_i$ is the ...
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47 views

What are QHT, QWT and QFT?

I am new to quantum computing. I am not able to understand QHT, QWT and QFT. I tried searching for a suitable material but still could not find them. It would be great if someone can explain and also ...
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62 views

How is a classical bipartite state written in quantum notation?

As in the title, is a classical bipartite state on $AA'$ given by $$\sum_{ij} p_{ij} \vert i\rangle\langle i\vert_A \otimes \vert j\rangle\langle j\vert_{A'}$$ with $\sum_{ij}p_{ij} = 1$. In ...
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422 views

What does it mean that 1 qubit can do the job of 1 ebit (entanglement bit)? (second Bennett's law)

I just came across Bennett's laws and I wonder what the second law mean. It states that 1 qubit "can do the job" of 1 ebit. However, the definition of ebit (entanglement bit, wiki just ...
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74 views

What is the "quantum phase" of a quantum state?

On this page IMBQ docs, until the sentence '..and since the global phase of a quantum state is not detectable..' I follow everything. However 'quantum phase' is introduced without any explaination? ...
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29 views

What is the correct notation to denote operations conditional on a measurement outcome?

What is the correct mathematical notation to describe the following setup? I have classical state in register $A$ which I can think of as $\sum_i p_i \vert i\rangle\langle i \vert_A$. I measure this ...
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41 views

What is a bipartite unitary?

What is a 'bipartite unitary'? I saw it appearing in a paper "Efficient verification of quantum gates with local operations" (https://arxiv.org/pdf/1910.14032.pdf) A reference to the ...
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53 views

How to get probability when the coefficient in wave function is a matrix?

Following this circuit: With $\mathcal{G}, A$ being unitary matrices and $|\psi\rangle$ the initial state. First, the system is: $\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\;\otimes|\psi\rangle$ Next: $\...
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78 views

Can Dirac notation be used with 2 or more gates?

Can Dirac notation be used with 2 or more gates? I've been trying to do the math with the $X$ and $Z$ ($X\otimes Z$) gates but I'm not getting the answer I should. In fact, the answer makes no sense. ...
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128 views

What is the notation for factoring a state when non-adjacent qubits are entangled?

Suppose I have the entangled state $$|\psi\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |110\rangle).$$ If i want to factor the non-entangled parts of this state out, I can easily write that down as $$|\...
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78 views

What is the name for the gate rotating around $Z$ by $\pi/8$?

I think similar to $R_z\big(\frac{\pi}{4}\big)$ gate named T gate, how to standardize the name $R_z\big(\frac{\pi}{8}\big)$?
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61 views

How does this expression from the VQLS paper result in a CZ-gate?

I am reading the VQLS paper and equation C2 on page 10 they have: $ \delta_{ll'}^j = \beta_{ll'} + \langle0|V^\dagger A_{l'}^\dagger U(Z_j \otimes I_\bar{j}) U^\dagger A_lV|0\rangle $ Here they define ...
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181 views

Is there any difference between "value of a qubit" and its "state"?

Value of a qubit and its state - is there any difference between these two terms in sense of terminology? For example, can we name this state of a qubit also a value of a qubit: $$ |\psi\rangle = \...
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217 views

What is the relation between POVMs and observables (as Hermitian operators)?

Let $\renewcommand{\calH}{{\mathcal{H}}}\calH$ be a finite-dimensional Hilbert space. An observable $A$ is here a Hermitian operator, $A\in\mathrm{Herm}(\calH)$. A POVM is here a collection of ...
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87 views

What is meant by $\log(m)$ qubits

I am reading a paper on Quantum Computing. What does it mean when authors says "$\log(m)$-qubits. I know what does $n$-qubit mean where $n$ is positive integer, but $\log(m)$ may not be an ...
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39 views

What does the term efficiency mean when talking about quantum memories?

Lots of papers about quantum memories talk about the efficiency generally or the storage/retrieval efficiency as a figure of merit. What exactly does this term mean? My intuition would be that it ...
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64 views

What does "large-scale universal quantum computation" mean?

I was reading this lecture notes by Sevag Gharibian. He mentioned the following statement: Large-scale universal quantum computer can be built. I do understand what quantum computer is, but I don't ...
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43 views

What does "bipartite system" mean? [duplicate]

I have seen this term appearing multiple times when discussing density matrices. For example here is an excerpt from the lecture notes: We are now ready to introduce the idea of a reduced density ...
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56 views

Are the two ways of interpreting the expression $(|a\rangle\otimes|b\rangle)(\langle c|\otimes\langle d|)(|e\rangle\otimes |f\rangle)$ equivalent?

Reading Nielsen and Chuang, I am under the impression that a linear operator on the tensor product can be written in two ways: \begin{equation} (\left|a\right> \otimes \left|b\right>)(\left<c\...
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59 views

Shouldn't the input state of Deutsh-Jozsa's algorithm look like $|0\rangle^{\otimes n}\otimes |1\rangle$ rather than $|0\rangle^{\otimes n}|1\rangle$?

According to this wikipedia page the initial state in Deutsch–Jozsa algorithm is written as follows: $$|0\rangle^{\otimes n} |1\rangle$$ shouldn't it look like this?: $$|0\rangle^{\otimes n} \otimes |...
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223 views

If the eigenvalues of $Z$ are $\pm1$, why are the computational basis states labeled with "$0$" and "$1$"?

The computational basis is also known as the $Z$-basis as the kets $|0\rangle,|1\rangle$ are chosen as the eigenstates of the Pauli gate \begin{equation} Z=\begin{pmatrix}1 & 0 \\ 0 & -1\end{...
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142 views

What is the relation between observables (as defined in the measure-theoretic framework) and POVMs?

A POVM is typically defined as a collection of operators $\{\mu(a)\}_{a\in\Sigma}$ with $\mu(a)\in\mathrm{Pos}(\mathcal X)$ positive operators such that $\sum_{a\in\Sigma}\mu(a)=I$, where I take here $...
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723 views

What does it mean to "measure an operator"?

I was reading a book and then I found this statement. I will put the text as well as a screenshot of the text. The expectation value of an operator is the mean or average value of that operator with ...
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71 views

Is there a name for the terms that make up a state?

Suppose I have the state $\frac{1}{2}(|0\rangle + |1\rangle)$, how can I refer to ket 0 and ket 1 in general within a state? Would "term 1" and "term 2" work?
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What does $|j\rangle$ mean, when $j > 1$?

I am trying to understand the state given here: $$ \frac{1}{\sqrt{n}} \sum_{j=0}^{n-1} |j\rangle |j\rangle $$ Suppose $n = 4$, I would interpret this state as being $1/2(|00\rangle + |11\rangle + |22\...
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49 views

What does the notation $|\psi(0)\rangle = |0\rangle|n=0\rangle$ mean?

Let us take the initial state with the particle located at the origin $|n=0\rangle$ and the coin state with spin up $|0\rangle$. So, $$ |\psi(0)\rangle = |0\rangle|n=0\rangle, $$ where $|\psi(0)\...
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245 views

Difference between change of basis in bra-ket notation and matrix notation

In matrix notation, say I have the vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$. It is currently represented in the computational basis $\{\begin{bmatrix} 1 \\ 0\end{bmatrix}, \begin{bmatrix} 0 \\ 1\...
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42 views

What are the physical meanings of the outer product when writing expressions for unitary gates?

I'm really confused with the interpretation of those equations: $1.$ The evolution of states under unitary operations can be expressed as $$ U = \sum_k\exp(i\phi_k)|\psi_k\rangle\langle\psi_k| $$ $2.$ ...
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45 views

Is it wrong to say that $a$ and $b$ are the square roots of the detection probabilities in a qubit state $|\psi \rangle = a|0 \rangle +b|1 \rangle $?

Is it wrong to say in $a$ and $b$ are the square roots of the probability of the qubit being in the state 0 and 1 when measured for a qubit in the state $|\psi \rangle = a|0 \rangle +b|1 \rangle $? ...
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94 views

What is the difference between the states $i|1\rangle$ and $|+i\rangle$?

I am new to Quantum computing. I see $|\mbox{+}i\rangle$ state maps to y-axis on bloch sphere ($\theta = 90$ degree and $\phi = 90$ degree) while $i|1\rangle$ maps on x-axis, $i|1\rangle$ is stated as ...
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104 views

How to represent the state vector form of a qubit in density matrix representation? [duplicate]

While I'm studying state vector and density matrix. I wonder how to write qubit state as density matrix. qubit state can be represented with state vector form. But how about density matrix?
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796 views

How to translate the Hadamard gate matrix into Dirac notation?

Hadamard gate matrix is: $$\frac{1}{\sqrt 2}\begin{bmatrix}1 && 1 \\ 1 && -1\end{bmatrix}$$ The Dirac notation for it is: $$\frac{|0\rangle+|1\rangle}{\sqrt 2}\langle0|+\frac{|0\...
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77 views

What is the "phase" of a state in terms of the Bloch sphere?

I am pretty new to Quantum Computing and am not exactly sure what the phase is. Could you please explain in terms of the Bloch sphere and point out how to mathematically calculate and represent the ...
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106 views

How does the graphical notation used to denote doubly-controlled gates work?

$\qquad$ $\qquad$ What is the difference between solid and hollow? How to express the corresponding matrix of these figures? In addition, if they are not adjacent, what should be done in the middle of ...
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122 views

What does the notation $U(B,\beta) = \prod_{j =1}^n e^{-i \beta \sigma_j^x} $ mean in the context of QAOA?

In the article Quantum Observables for continuous control of the Quantum Approximate Optimization Algorithm via Reinforcement Learning, the following notation is used to describe an Unitary operation ...
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120 views

What does the notation $\sigma_j^z$ mean for Pauli matrices?

In multiples papers or online article on the QAOA algorithm (such as this one), I found notation for the Hamiltonian similar to this one : $$ \sum_{ij} \frac{1}{2} (I-\sigma_i^z \sigma_j^z)$$ I don'...
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90 views

Confusing notation in Wikipedia's quantum channel article

In the Wikipedia's Quantum channel article, it is said that a purely quantum channel $\phi$ (it's not exactly the same phi calligraphy but it's close), in the Schrodinger picture, is a linear map ...
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74 views

How should I understand the link between $x$ and $| x \rangle $?

In the paper "Quantum Expectation-Maximization for Gaussian Mixture Models" I encountered the following proposition : Consider two vectors x,y and $\theta$ the angle between x and y, $\theta < \...
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244 views

In Dirac notation, why do we have $\langle cf|g\rangle = c^*\langle f\vert g\rangle$?

A Hilbert Space has this property $$\langle cf,g\rangle=c\langle f,g\rangle$$ where $f$ and $g$ are the vectors in the Hilbert Space and $c$ is a complex number. In Dirac Notation, $$\langle cf|g\...
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108 views

What are $|+\rangle$ and $|-\rangle$?

On a view places, I've seen kets that look like this $\left|+\right>$ or this $\left|-\right>$ but I don't seem to find any explanation of this base online. Is it just a different notation for $...