Questions tagged [mathematics]
DO NOT use this tag. Use more specific tags such as [linear-algebra] instead.
241 questions
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How to calculate spherical angles for Bloch vector
I am doing the 5th exercise on https://qiskit.org/textbook/ch-states/representing-qubit-states.html#Quick-Exercise (all the way at the bottom).
Which states find the angle for the vector $\frac{1}{\...
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1
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Is the Haar measure invariant under conjugation?
Denote the Haar measure on the unitary group $U(\mathcal X)$ by $\eta$. Does this equation hold (assuming the integral exists):
$\int d\eta(U) f(U) = \int d\eta(U) f(U^\dagger)$?
Intuitively this ...
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Nielsen & Chuang Exercise 2.55: Prove that $\exp \left[ -\frac{iH(t_2 - t_1)}{\hbar} \right]$ is unitary
$\newcommand{\expterm}[0]{\frac{-iH(t_2 - t_1)}{\hbar}}
\newcommand{\exptermp}[0]{\frac{iH(t_2 - t_1)}{\hbar}}$Nielsen & Chuang (10th edition, page 82) states that $H$ is a fixed Hermitian ...
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Equivalence of two ways to recover a map from its Choi state
Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a quantum channel, $\Phi:\mathrm{Lin}(\mathcal X)\to\operatorname{Lin}(\mathcal Y)$.
We define its Choi representation as the operator $J(\Phi)\in \...
glS♦
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When we do a linear fit, what is the correlation coefficient of the estimated parameters?
In Google's quantum supremacy experiment, supplementary Section VIIIH, they calculate the correlation coefficient of the linear fit coefficients $p_0$,$p_1$. I can't figure out the definition of this ...
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Nielsen & Chuang Exercise 2.32: Show that the tensor product of two projectors is a projector
$\newcommand{\bra}[1]{\left<#1\right|}
\newcommand{\ket}[1]{\left|#1\right>}$Here is what I tried:
Given that we have two projectors:
$$
A = \sum_i \ket{i} \bra{i}, \hspace{2em}
B = \sum_j \ket{...
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3
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1
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127
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Eigenvectors and eigenvalues of the gate $U_a:|s\rangle\mapsto|sa \bmod N\rangle$
I'm studying Shor algorithm. This is a demostration about the eigenvectors and eigenvalues of $U_a$ gate:
Can somebody explain me from the third step to the last?
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1
answer
53
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Create this state using CIRQ Coding language [closed]
I needed help with CIRQ coding as I'm quite new to Quantum Computing.
I read the tutorials on CIRQ but don't really understand it.
I'd be very thankful if someone could help.
A program to create the ...
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2
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1
answer
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Asymmetry in distributing phase change across components
The quantum computing text books and theory in general seems to have added an asymmetry in the distribution of change in phase across the components in the context of a qubit. Is there any reason for ...
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1
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86
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How is $S(\rho)=H(p_{i})+\sum_{i}p_{i}S(\rho_{i})\le \log(d)$ possible if $\rho_{i}$ are not pure states?
I know how this can be proved using the quantum relative entropy. However, even with this proof, and am still confused about how this emerges.
Say I have a source that produces two states $\rho_1$ and ...
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4
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1
answer
153
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Deriving $\left( A | v \rangle \right)^\dagger = \langle v | A^\dagger$ without using $A^\dagger=\left(A^* \right)^T$
From Nielsen & Chuang (10th edition), page 69:
Suppose $A$ is any linear operator on a Hilbert space, $V$. It turns
out that there exists a unique linear operator $A^\dagger$ on $V$ such
that for ...
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1
answer
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Confused about associativity of outer product notation
Consider this expression where $A$ and $B$ are matrices, $|i \rangle$ is a ket (column vector) and $\langle j |$ is a bra (row vector) :
$$
A | i \rangle \langle j | B \tag1\label1
$$
Due to the ...
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3
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Why can every $|X\rangle\in H_1\otimes H_0$ be written as $|X\rangle=(X\otimes I_{H_0})|\Omega \rangle$ for some $X\in\mathcal L(H_0,H_1)$?
In A theoretical framework for quantum networks is proven that a linear map $\mathcal{M} \in \mathcal{L}(\mathcal{H_0},\mathcal{H_1})$ is CP (completely positive) iff its Choi operator $M$ is semi ...
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Why does the Hadamard gate satisfy $H|x\rangle=\frac{1}{\sqrt2}\sum_{z\in\{0,1\}}(-1)^{xz}\lvert z\rangle$?
I'm studying Deutsch–Jozsa algorithm and I don't understand this passage about Hadamard gate result:
$$\newcommand{\ket}[1]{\lvert #1\rangle}H\ket x=\frac{1}{\sqrt2}(\ket0+(-1)^x\ket1)=\frac{1}{\sqrt2}...
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1
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How can you decompose Grover's diffusion operator into gates?
I know how Grover's diffusion operator works ($U_s = 2|s\rangle\langle s|-I$) with the inversion around the mean. However, I want to implement it in simpler gates, to use the algorithm. How can I do ...
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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 ...
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How are eigenvectors and eigenvalues expressed in the Bloch sphere?
I'm relatively new to the subject of quantum computing, and I recently came across the idea of eigenvalues and eigenvectors. I believe I understand the relationship between the two, where eigenvalues ...
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1
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131
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Alternative derivation of $P(\text{First qubit}=0)$ for the swap test
I'm trying to derive $P(\text{First qubit}=0) = \frac{1}{2} + \frac{1}{2}|⟨a|b⟩|^2$ for the swap test.
The wiki page shows one way, but the result should also be obtainable via direct expansion of ...
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0
answers
106
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Why an element of SU(2) acts as a rotation for Majorana representation of states?
I know that for a given spin-j quantum state, say $\vert\psi\rangle = (\psi_0 , \psi_1 , \cdots , \psi_{2j})$, we can construct a polynomial as follows
$
w(z) = \sum_{k = 0}^{2j} (-1)^k \psi_k \sqrt{\...
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Prove that QFT and Walsh-Hadamard gates give the same output when acting on $\lvert x\rangle\lvert 0\rangle$ [duplicate]
I know that $QFT_n|0\rangle$ is equivalent to $H_n|0\rangle$
(mathematical proof).
And it is also easy to prove that $QFT_1$ is equivalent to $H_1$ (applied to one QuBit).
From looking at the circuit ...
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2
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525
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Meaning behind obtaining a hermitian operator for measurement in another basis?
If
$$P_{+} = |+\rangle\langle+|=\frac{1}{2}(|0\rangle\langle0|+|0\rangle\langle1|+|1\rangle\langle0| +|1\rangle\langle1|)$$
and
$$P_{-} = |-\rangle\langle-|=\frac{1}{2}(|0\rangle\langle0|-|0\rangle\...
- 1,232
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2
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What is the set of all functions from ℤ to ℤ?
In David Deutsch's classic paper Quantum theory, the Church-Turing principle and the universal quantum computer (1985), Deutsch writes on p. 99:
(I thought that this might be in typo in the original ...
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127
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How well defined is $\log(P)$ for $P$ projection?
Whenever we calculate entropy we make use, for example, $\log(P)$ for $P$ a projection defined for some arbitrary finite dimensional Hilbert space.
But for projection operators this is not well-...
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Can a Kraus representation act as the identity on any operator?
In the textbook “Quantum Computation and Quantum Information” by Nielsen and Chuang, it is stated that there exists a set of unitaries $U_i$ and a probability distribution $p_i$ for any matrix A,
$$\...
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0
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54
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Step-by-step passages in calculation
I would like to better understand some passages in a paper (Appendix A):
Properties of Tensor Product
Bilinearity: $A\otimes(B+ C) = A \otimes B + A \otimes C $
Mixed-product property: $(A\otimes B)(...
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3
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Is the tensor product of two states commutative?
I'm reading "Quantum Computing Expained" of David McMahon, and encountered a confusing concept.
In the beginning of Chapter 4, author described the tensor product as below:
To construct a ...
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2
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695
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Find the probability of a measurement outcome in terms of the coefficients of the state
Suppose we have a quantum state $|\psi \rangle$ of $n$ qubits, where $|\psi\rangle=\sum_{x∈\{0,1\}^n}\alpha_x |x\rangle$,and we measure the first qubit of $|\psi\rangle$ in the computational basis. ...
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Computing variance under the action of a unitary operator
I wish to calculate the expectation and variance for an observable on a particular qubit of a multi qubit quantum state. I'm using a quantum computing simulation library which allows me to apply ...
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How do you decompose an arbitrary quantum state into its corresponding projection subspaces such that their direct sum is the quantum state?
I understand that every Hilbert space $H$ can be decomposed into two mutually orthogonal subspaces $H_1$ and $H_2$ whose direct sum is $H$.
Therefore, every vector $v\in H$ can be decomposed into $v_1\...
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1
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Proof of QFT for a Periodic Function
For Mosca Keynes, ex 7.1.5:
You are asked to prove:
$\text{QFT}^{-1}_{mr}|\phi_{r,b}\rangle = \frac{1}{\sqrt{r}}\sum_{k=0}^{r-1}e^{-2\pi i \frac{b}{r}k}|mk\rangle$
where
$|\phi_{r,b}\rangle = \...
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What are boost and shift operators and why are they called so?
In some texts I see $X$ and $Z$ Pauli operators as being said as boost and shift operators respectively.
But I came across some text that defines its own operators, namely:
$$
X \vert j\rangle = \...
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How to get the stabilizer group for a given state?
Let's say we have the GHZ state with 3 qubits:
$$ |\mathrm{GHZ}\rangle = \dfrac{1}{\sqrt{2}}\Big(|000\rangle + |111\rangle \Big)$$
I want to find the stabilizer group of this state, that is, the $...
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93
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How do I add 1+1 using a photonic computer?
A similar question has been previously asked & has an excellent answer discussing half, full & ripple carry adders. I am curious to find out how these adders would be constructed in the ...
- 3,342
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Pauli Identity Using Tensor Network Notation
I am trying to understand the meaning of the equation shown in the above image taken from this paper, but I am unfamiliar with the tensor network notation. My current strategy is trying to write down ...
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Why does the state fidelity satisfy $\operatorname{tr}|\sqrt{\rho}\sqrt{\sigma}|=\operatorname{tr}\sqrt{\sigma^{1/2}\rho\sigma^{1/2}}$?
Given the the two states $\rho$ and $\sigma$ of a quantum system, with $|\psi\rangle$ and $|\varphi\rangle$ as their purification respectively, the fidelity is defined as:
$$F(\rho,\sigma)=\max_{|\...
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1
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Rearrangement of qubits in Quantum Teleportation Protocol
I have been reading Quantum Teleportation (Pg. 27) from Nielsen and Chuang and noticed that after the Hadamard operation, the state obtained was re-written by regrouping the terms to obtain the ...
7
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1
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848
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Correct Formulation of N&C Exercise 4.11 and other textbooks misquoting
Inspired by the comments in this question How to approximate $Rx$, $Ry$ and $Rz$ gates?, there is the errata for question 4.11 pg 176 in N&C. The original form states that for any non parallel $m$ ...
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Are superpositions of an infinite number of states realizable?
I first encounted this idea in Constructing finite dimensional codes with optical continuous variables where it mentions "superpositions of an infinite number of infinitely squeezed states" in the ...
- 3,342
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What does the minus sign in the four bell states represent?
I am in grade 11, so answers as simple as possible. I understand that in quantum teleportation, the bell measurement must be made on the teleportee and the sender, and I understand that yields one of ...
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Expressing a term of an $n$-qubit Hamiltonian in terms of Pauli operators
Consider a $2^n\times 2^n$ Hermitian matrix $M$ containing up to two non-zero elements, which are $1$ (so, either $M_{ii}=1$ for some $i$, or $M_{ij}=M_{ji} = 1$ for some $i$ and $j$). Each such ...
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Understanding proof of joint concavity of the (root) fidelity
I have some problem in understanding the proof of the concavity of root fidelity given in Chapter 9.2 of Mark M. Wilde's "Quantum Information Theory". Here, the fidelity is defined by $F(\...
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1
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Universality and coverage of irrational multiples of $2\pi$ In $[0, 2\pi)$
This related to the proof of universality (pg 196),and partially related to the question Why is Deutsch's gate universal?, however i'm trying to workout a more rigorous proof and understanding of ...
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Effect of Pauli X gate on minus state using bloch sphere
As I understood, the X gate flips the state around : $X(|0\rangle) = |1\rangle$. It can also be visualized with a $\pi$ rotation around the $x$ axis in the Bloch sphere. I have no problem with that.
...
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2
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389
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Is it right to think of superposition as just angle?
Based on my current understanding, a qubit is represented as a vector $(a, b)$ which satisfy $a^2 + b^2 = 1$. Classical bit one can be represented as $(0, 1)$ and bit zero can be represented as $(1, ...
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2
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360
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Prove that the partial trace is a quantum operation, finding its Kraus representation
I am referring to Nielsen and Chuang Quantum Computation and Quantum Information 10th Anniversary Edition Textbook, Chapter 8.3.
A linear operator $E_i:H_{QR}\longrightarrow H_Q $ is defined by:
$$...
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2
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Prove that for one-qubit unitaries $\text{Tr}|U-V|=2\max_\psi\|(U-V)|\psi\rangle\|$
Given two 1-qubit rotations $U=R_n (\theta)$ and $V=R_m(\phi)$ with $n$ and $m$ vectors defining a rotation and $\theta, \phi$ angles, define $D(U,V)=Tr(|U-V|)$ where $|U-V|=\sqrt{(U-V)^\dagger (U-V)}$...
- 555
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1
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106
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How to correctly define $U_\omega$ for Grover's search algorithm
I am working on Grover's algorithm and I am trying to implement the algorithm. I am following the Microsoft quantum katas and I finished and did everything correctly. I am trying to implement the ...
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Is Quantum Computing a problem for Cryptography?
How can we use Quantum Computing to break a Cryptosystem like RSA or AES-256?
Can we use Quantum Computing to solve difficult mathematical problems like Discrete Logarithms or Prime Number ...
- 153
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vote
1
answer
175
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Projection is trace-decreasing?
I'm studying Mark Wilde's "Quantum Information Theory" and the author sometimes use the inequality $\mathrm{Tr}(\prod_\mathcal{H'}Y) \leq \mathrm{Tr}(Y)$ where $Y\in \mathcal{H}'$ is a density matrix ...
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2
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1
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510
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What is infinite squeezing?
I am working my through the Strawberry Fields documentation & the section on state teleportation states:
Here, qumodes $q1$ and $q2$ are initially prepared as (the unphysical) infinitely squeezed ...
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