Questions tagged [mathematics]
Use this tag for questions about mathematics relevant to quantum computing and/or quantum information theory. DO NOT use this tag for general mathematics questions.
101
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How to derive the CNOT matrix for a 3-qubit system where the control & target qubits are not adjacent?
In a three-qubit system, it's easy to derive the CNOT operator when the control & target qubits are adjacent in significance - you just tensor the 2-bit CNOT operator with the identity matrix in ...
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votes
2
answers
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Can arbitrary matrices be decomposed using the Pauli basis? [duplicate]
Is it possible to decompose a hermitian and unitrary matrix $A$ into the sum of the Pauli matrix Kronecker products?
For example, I have a matrix 16x16 and want it to be decomposed into something ...
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votes
2
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How does $\mathcal E(\rho)=\mathrm{Tr}_{env}[U(\rho\otimes\rho_{env})U^\dagger]$ turn into $P_0\rho P_0+P_1\rho P_1$?
In the Quantum Operations section in Nielsen and Chuang, (page 358 in the 2002 edition), they have the following equation:
$$\mathcal E(\rho) = \mathrm{Tr}_{env} [U(\rho \otimes \rho_{env})U^\dagger]$$...
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1
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How is Grover's operator represented as a rotation matrix?
I have seen that it is possible to represent the Grover iterator as a rotation matrix $G$. My question is, how can you do that exactly?
So we say that $|\psi\rangle$ is a superposition of the states ...
user4961
4
votes
3
answers
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Purity of mixed states as a function of radial distance from origin of Bloch ball
@AHusain mentions here that the purity of a qubit state can be expressed as a function of the radius from the center of a Bloch sphere. The state corresponding to the origin is maximally mixed whereas ...
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1
answer
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Is the Pauli group for $n$-qubits a basis for $\mathbb{C}^{2^n\times 2^n}$?
The $n$-fold Pauli operator set is defined as $G_n=\{I,X,Y,Z \}^{\otimes n}$, that is as the set containing all the possible tensor products between $n$ Pauli matrices. It is clear that the Pauli ...
5
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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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4
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2
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Why is the boundary of the set of states in the generalised Bloch representation comprised of singular matrices?
Consider an arbitrary qudit state $\rho$ over $d$ modes.
Any such state can be represented as a point in $\mathbb R^{d^2-1}$ via the standard Bloch representation:
$$\rho=\frac{1}{d}\left(\mathbb I +\...
glS♦
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3
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When can a matrix be "extended" into a unitary?
DaftWulie's answer to Extending a square matrix to a Unitary matrix says that extending a matrix into a unitary cannot be done unless there's constraints on the matrix. What are the constraints?
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3
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Nielsen & Chuang Exercise 2.2 - “Matrix representations: example” [closed]
Reproduced from Exercise 2.2 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition):
Suppose $V$ is a vector space with basis vectors $|0\rangle$ and $|1\...
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3
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4
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Can a Kraus representation act as the identity on any operator?
In the textbook “Quantum Computation and Quantum Information” by Nielsen and Chuang, it is stated that there exists a set of unitaries $U_i$ and a probability distribution $p_i$ for any matrix A,
$$\...
3
votes
3
answers
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How does one create the unitary sending $|0\rangle$ into a target quantum state?
The Hadamard gate allows us to construct an equal superposition of states. If one wants to construct an arbitrary superposition e.g. $\alpha\vert 0\rangle + \beta\vert 1\rangle + ..$, how does one ...
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2
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1
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Grover algorithm for more than one element
I am currently working on the Grover algorithm again. In many lectures and documents, as well as books, I noticed that there is always talk of looking for a single element of $N$ elements. Now I read ...
user4961
16
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1
answer
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Is there a closure property for the entire Clifford hierarchy?
TL;DR
Is the entire Clifford hierarchy (as opposed to any one level), a group?
Background.
The Clifford hierarchy (on $n$ qubits), is a collection of nested subsets $\mathcal C^{(1)} \subset \mathcal ...
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12
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1
answer
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General parametrisation of an arbitrary $2 \times 2$ unitary matrix
From Nielsen & Chuang's Quantum Computation and Quantum Information (QCQI):
Since $U$ is unitary, the rows and columns of $U$ are orthonormal, form which it follows that there exist real numbers $...
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2
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How can I calculate the inner product of two quantum registers of different sizes?
I found an algorithm that can compute the distance of two quantum states. It is based on a subroutine known as swap test (a fidelity estimator or inner product of two state, btw I don't understand ...
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3
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2
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What is the probability of observing a search string $\omega$ after $r$ iterations of Grover's algorithm?
Per Wikipedia, in Grover's algorithm the probability of observing search string $\omega$ after $r$ iterations is:
$$\left|\begin{bmatrix}\langle \omega | \omega \rangle &\langle \omega | s \...
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2
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How do I apply the Hadamard gate to one qubit in a two-qubit pure state?
So in lectures I see lots of these:
And somehow I intuitively understand it (at least for the 1 qubit case), but I don't understand the math – especially for 2 qubits.
- 183
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2
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Automatic compilation of quantum circuits
A recent question here asked how to compile the 4-qubit gate CCCZ (controlled-controlled-controlled-Z) into simple 1-qubit and 2-qubit gates, and the only answer given so far requires 63 gates!
The ...
- 12.8k
13
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3
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Is acting with a positive map on a state not part of a larger system allowed?
In the comments to a question I asked recently, there is a discussion between user1271772 and myself on positive operators.
I know that for a positive trace-preserving operator $\Lambda$ (e.g. the ...
11
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2
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How to check if a quantum circuit can be constructed for a given matrix representation?
Let's say I have a matrix representation, e.g.
$$
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}.
$$
How ...
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9
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2
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331
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Is there an algorithm for determining if a given vector is separable or entangled?
I'm trying to understand if there is some sort of formula or procedural way to determine if a vector is separable or entangled – aka whether or not a vector of size $m$ could be represented by the ...
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2
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Minimum number of 2 qubit gates to build any unitary
Any unitary $U$ acting on $N$ qubits can be decomposed in a finite product $U=U_1U_2...U_n$ where every $U_i$ acts on only 2 qubits, for example through decomposition in CNOT, phase shifts and 1 qubit ...
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8
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3
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Is the tensor product of two states commutative?
I'm reading "Quantum Computing Expained" of David McMahon, and encountered a confusing concept.
In the beginning of Chapter 4, author described the tensor product as below:
To construct a ...
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7
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2
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When would I consider using an outer product of quantum states, to describe aspects of a quantum algorithm?
I know the inner product has a relationship to the angle between two vectors and I know it can be used to quantify the distance between two vectors. Similarly, what's an use case for the outer ...
- 1,179
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2
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What is a separable decomposition for the Werner state?
Consider the two-qubit Werner state, defined as
$$\rho_z = z |\Psi_-\rangle\!\langle \Psi_-| + \frac{1-z}{4}I,
\quad |\Psi_-\rangle\equiv\frac{1}{\sqrt2}(|00\rangle-|11\rangle),$$
for $z\ge0$.
Using ...
glS♦
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How do you represent the output of a quantum gate in terms of its basis vectors?
I'm stuck while trying to understand the Hadamard Gate in a more linear algebra understanding. (I understand the algebraic way). This is because I want to program a simulation of a quantum computer. ...
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6
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2
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449
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In Bell nonlocality, why does $P(ab|xy)\neq P(a|x)P(b|y)$ mean the variables are not statistically independent?
I've been working through the paper Bell nonlocality by Brunner et al. after seeing it in user glS' answer here. Early on in the paper, the standard Bell experimental setup is defined:
Where $x, y \...
- 3,988
5
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1
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Extending a square matrix to a unitary matrix
Suppose we have a square matrix $M$ of size $n\times n$. It is given that any element $M_{ij}$ of $M$ is a real number and satisfies $0 \leq M_{ij} \leq 1$, $\forall$ $i,j$. No other property for $M$ ...
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1
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What types of quantum systems use infinite values?
Background
I am curious to learn more about any work that has been done regarding quantum systems that deal with infinite values. I am primarily interested in photonic quantum computing; however I am ...
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3
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1
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How to translate the Hadamard gate matrix into Dirac notation?
Hadamard gate matrix is:
$$\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\...
- 209
1
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1
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Two-qubit Bell measurement matrix where the two qubits are not contiguouis
In the answer here, it is explained that where the measurement operates on only a subset of the qubits of the system (for example qubits 2 and 3 out of five), the matrix can be constructed using the ...
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2
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What happens if $|\psi\rangle$ = $|0\rangle$ or $|\psi\rangle$ = $|1\rangle$ is passed as an input to two Hadamard gates in sequence?
I'm a computer science student and soon I will have a math exam. I'm really struggling with this preparation question. Also, includes the following:
How does this demonstrate that we need the “ket” ...
- 13
25
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1
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How to understand the Haar measure from a quantum information perspective?
I found it a little difficult to understand it using Wikipedia and some mathematical documents. How to understand the Haar measure from a quantum information theory perspective? Are there any ...
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6
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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 ...
- 4,702
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2
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Rigorous security proof for Wiesner's quantum money
In his celebrated paper "Conjugate Coding" (written around 1970), Stephen Wiesner proposed a scheme for quantum money that is unconditionally impossible to counterfeit, assuming that the issuing bank ...
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2
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What does it mean for a density matrix to "act on a Hilbert space $\mathcal{H}"$?
For a Hilbert space $\mathcal{H}_A$, I have seen the phrase
density matrices acting on $\mathcal{H}_A$
multiple times, e.g. here.
It is clear to me that if $\mathcal{H}_A$ has finite Hilbert ...
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Why does representation theory often arise in the context of quantum algorithms for the hidden subgroup problem?
I noticed that approaches for finding quantum algorithms the hidden subgroup problem for both Abelian groups ($(\Bbb Z_n\times \Bbb Z_n, +)$, $(\Bbb R, +)$, etc.) and non-Abelian finite groups like ...
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How to prove that antipodal points on the Bloch sphere are orthogonal?
I started by assuming two antipodal states
$$
|(\theta,\psi)\rangle = \cos\dfrac{\theta}{2}|0\rangle + \sin\dfrac{\theta}{2}e^{i\psi}|1\rangle\\
|(\theta+\pi,\psi+\pi)\rangle= \cos\dfrac{\theta+\pi}{2}...
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2
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Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere
The Bloch sphere is homeomorphic to the Riemann sphere, and there exists a stereographic projection $\Bbb S^2\to \Bbb C_\infty$. But this only holds for pure states. To quote Wikipedia:
Quantum ...
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7
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1
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Does conjugation by a Clifford send each non-identity Pauli to every other non-identity Pauli with equal frequency?
I see here in Olivia DeMatteo's notes, she states:
When we consider the action of the entire Clifford group on a single non-identity Pauli, it
maps that Pauli to each of the $d^2 − 1$ other possible ...
- 1,331
7
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2
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How to show a density matrix is in a pure/mixed state?
Say we have a single qubit with some density matrix, for example lets say we have the density matrix $\rho=\begin{pmatrix}3/4&1/2\\1/2&1/2\end{pmatrix}$. I would like to know what is the ...
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How to check if a two-qubit gate is entangling?
I would like to know if there's an analog for Schmidt rank that can tell me if a two-qubit unitary is entangling?
Suppose I have a parametrized two-qubit unitary $U^{(2)}(\theta)$. I would like to ...
- 5,770
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votes
1
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Understanding a quantum algorithm to estimate inner products
While reading the paper "Compiling basic linear algebra subroutines for quantum computers", here, in the Appendix, the author/s have included a section on quantum inner product estimation.
Consider ...
- 505
6
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1
answer
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Definition of locality in Bell experiments
Continuing from my previous question on Brunner et al.'s paper; so given a standard Bell experimental setup:
where independent inputs $x,y \in \{0, 1\}$ decide the measurement performed by Alice &...
- 3,988
6
votes
1
answer
505
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Are nearly all pure two-qubit state entangled?
I am using the code below, utilizing QETLAB's RandomStateVector(4) and IsPPT, to generate a random state and to judge whether the state is entangled or separable:
...
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1
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What kind of mathematics is common in quantum computing? [closed]
From what I have seen so far, there is a lot of linear algebra. Curious what other kinds of maths are used in QC & the specific fields in which they are most predominately invoked.
- 3,272
6
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1
answer
681
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Can we process infinite matrices with a quantum computer?
Can we process infinite matrices with a quantum computer?
If then, how can we do that?
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votes
1
answer
307
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In the Clifford group, is the center of $ \overline{\text{Cl}_n} \equiv\text{Cl}_n/U(1)$ trivial?
My question:
Is the center of $ \overline{\text{Cl}_n} $ trivial?
Recall that the algebra generated by the Pauli group is the full matrix algebra. So any matrix that commutes with the Pauli group must ...
- 2,556
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3
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Writing state $ |\Psi⟩ =\frac{1}{\sqrt{2}}|00⟩+\frac{i}{\sqrt{2}}|01⟩$ as separate qubits (qiskit textbook)
While going through the IBM qiskit textbook online, I came across the following question in section 2.2:
Write the state: $ |\Psi⟩ =\frac{1}{\sqrt{2}}|00⟩+\frac{i}{\sqrt{2}}|01⟩$ as two separate ...
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