Questions tagged [linear-algebra]
For questions about vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc.
466
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2
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1
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Finding the "dual" basis of an overcomplete basis for Quantum State Tomography
This question is related to this stack exchange post: What does the POVM corresponding to single-qubit state tomography look like?
From what I understand, when we are interested in reconstructing a ...
- 21
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2
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57
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Why $\sqrt{\rho} = P \sqrt{\rho}$ in the proof of quantum error correction conditions in Nielsen & Chuang?
I have trouble understanding this proof in Nielsen & Chuang, specifically the identity in $(10.20)$, which reads $$ U_k^\dagger P_k F_l \sqrt{\rho} = U_k^\dagger P_k^\dagger P_k^\dagger F_l P \...
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1
vote
1
answer
71
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Problem with the mathematical definition of the eigenvalue algorithm on a specific exercise
I think I understand well how the eigenvalue algorithm works but when I try to define it mathematically I have problems.
Specifically I have the matrix U:
$$
U =
\begin{pmatrix}
0 & i \\
i & 0
...
- 35
2
votes
1
answer
147
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Problem with eigenvalue evaluation algorithm application on matrix $U$
Once I get to the end of the algorithm, I can't understand how to calculate the eigenvalue using formulas. Bear in mind that it is an exercise to be carried out with pen and paper.
the matrix of $U$ ...
- 35
2
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1
answer
196
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Finding the eigenvalues of a qutrit state
I am interested in the state:
$\frac{1}{\sqrt{2}} (\left|11\right> + \left|22\right>)$
If I find the density matrix of this, I find the $9 \times 9$ matrix $\rho$.
If I want to find the reduced ...
- 295
4
votes
1
answer
80
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What is known about the size of the spectral gap of unital quantum channels?
I am interested in the spectrum of unital quantum channels $\Phi$ (which act on finite dimensional spaces). The literature is extremely vast on such objects so I hope some experts can point me along ...
- 540
3
votes
1
answer
113
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Is possible to write a separable state as a finite or countable infinite sum of product states?
Let us consider the tensor product of two finite Hilbert spaces $\mathcal{H}_1\otimes \mathcal{H}_1$.
According to Watrous book, the set of separable states is the convex hull of the set of product ...
- 142
2
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0
answers
39
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Relationship between the eigenvalues of a Laplacian matrix and the eigenvalues of the Hamiltonian of a graph for Max-Cut
Is there a relationship between the eigenvalues of a Laplacian matrix of a graph and the eigenvalues of the Hamiltonian for Max-Cut?
It is shown here, that Max-Cut can be written as a maximization ...
- 21
2
votes
1
answer
62
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How is this step performed in Deutsch's algorithm?
I am reading Quantum Computing for Computer Scientists.
Given the circuit for Deutsch's algorithm:
Denoting $H|0\rangle$ as $|x\rangle$, the book says that
$|\phi_2\rangle = (-1)^{f(x)}|x\rangle\left\...
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2
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1
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31
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How to get the Kraus operator $M_0=\sqrt{1-p}\, I$ for the depolarizing channel, from its isometric representation?
I am confused as to how we get $M_{0} = \sqrt{1-p} I$ and the following $M_{1}, M_{2}, M_{3}$.
The above notes say that we should partially trace over the environment in the $|{0}\rangle, |{1}\rangle, ...
- 295
0
votes
1
answer
65
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Is there a general method for calculating expectation values for time-dependent wavefunctions?
Is there a general method for calculating expectation value? I have a workshop question, and I'm sure what a good process to follow is. It is given that $$|\psi(t = 0)\rangle = |0\rangle\,,\tag{1}$$ ...
- 11
3
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2
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123
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What do the values in a unitary matrix represent/what do they mean? How do you figure out what gate a unitary matrix represents?
I am trying to learn Qiskit on my own. I am struggling with unitary matrices. I understand what a unitary matrix is, and why a matrix is unitary. But, I don't understand what the values inside of the ...
- 31
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0
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schmidt coefficients are the square root of the eigenvalue of the two partial trace of a density matrix
Let $\psi\rangle_{AB} = \sum_{i=1}^{d}\lambda_{i}|i_{A}\rangle |i_{B}\rangle$ be a state vector of a pure bipartite syste.
Now, $\rho_{AB} = |\psi\rangle\langle\psi| = \sum_{i=1}^{d} \lambda_{i}^{2}|...
- 510
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1
answer
40
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Quantum compilation algorithm with respect to other Shatten $p$-norm
In standard quantum compilation algorithms (such as the Solovay-Kitaev theorem), one approximates an arbitrary unitary using words from some universal gate set. The "approximation" here is ...
- 429
3
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2
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102
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How to know what eigenvalue corresponds to measurements of individual qubits in a multiqubit system?
I'm working through the book "Introduction to the Theory of Quantum Information Processing" by Bergou and Hillary, and I've encountered a scenario that I'm not sure how to approach. In ...
- 57
1
vote
1
answer
87
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How to create a maximally entangled state of two 4-level quantum mechanical systems?
Let's say that I have a 4-level quantum state, which is described by a linear combination of the following four eigenbases:
$$|\text{red}⟩ = \begin{bmatrix}
1 \\
0 \\
0 \\
0
\end{bmatrix} , |\text{...
- 23
1
vote
1
answer
78
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Why can $(0,0,3/5,0,0,0,4/5,0,0)$ be written as $\frac35|3\rangle+\frac45|7\rangle$?
Context. $\newcommand{\qr}[1]{\left|#1\right\rangle}$ A passage from a lecture by Scott Aaronson: "As an example, instead of writing out a vector like $$(0,0,3/5,0,0,0,4/5,0,0),$$ you can simply ...
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2
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45
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Does separability of a matrix implies the matrix is a density matrix?
Suppose I have a matrix that is unknown whether it is a density matrix and assume that finding the eigenvalues of it is difficult because the matrix is expressed generally.
However, suppose that this ...
- 510
2
votes
1
answer
84
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$\mathbb{C}^2 \otimes \mathbb{C}^2$ vs $\mathbb{C}^4$
Is there a difference between the following two Hilbert spaces: $H_1 = \mathbb{C}^2 \otimes \mathbb{C}^2$ and $H_2 = \mathbb{C}^4$?
Here's my confusion. For the following bases, $H_1 = H_2$ holds: $\...
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1
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Understanding the operation of commutation of stabilizer operators
I want to show that the stabilizer operators ($M_{0}, M_{1}, M_{2}, M_{3}$) for the 5-qubit quantum error correcting code:
If $M_{1} = [XXZIZ]$ and $M_{2} = [XZIZX]$
They commute iff $[M_{1},M_{2}]=0$....
- 295
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votes
1
answer
100
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Moments of Pauli coefficients of Haar-random states
I want to evaluate the quantity $\sum_{P\in \rm{P}^n}\text{Tr}^{\alpha}(\rho P)$, where $P$ is an element of n-qubit Pauli group $\rm{P}^n$ and $\rho$ is a density matrix of a Haar random state. It is ...
- 53
1
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0
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42
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How do I show that a reduced density matrix of $1$ is $\rho_{12}^{1} = Tr_{2}[\rho_{12}] = \sum_{i}\langle i_{2} | \rho | i_{2} \rangle$?
Let the system be a 2 - qubit system and let $\rho_{12}$ be a density matrix of some state for this 2 - qubit system.
How do I show that a reduced density matrix of $1$ is $\rho_{12}^{1} = Tr_{2}[\...
- 510
3
votes
1
answer
45
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A matrix that can be simultaneously diagonalisable can induce a decomposition of its space
Potentially "space" is not the correct word to use in the title, please correct if wrong
I am reading "Quantum Error Correction Via codes over Gf(4)" and I came across something I ...
- 295
0
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1
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37
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Does the fact that the elements of the normalizer group commute with elements of the stabilizer group imply that the normalizer is abelian?
The following question is from a paper I am reading called "Quantum Error Correction Via Codes Over GF(4)"
It says:
Let $E$ be the quantum error group.
Let $S' \leqslant E$ which specifies ...
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0
answers
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Prove $\sum_{ij}(\mathcal{A_G})_{ij}(|\rho\rangle\!\rangle)_j|i\rangle\!\rangle={\cal A}_G|\rho\rangle\!\rangle$ in the Pauli-Liouville representation
Define the Pauli-Liouville representation of a (linear) map $\mathcal{G}$ as $\mathcal{A_G}$, which has components
\begin{equation}\label{2}
(\mathcal{A_G})_{ij}:=\mathrm{tr}[P_i\mathcal{G}(P_j)]
\...
- 657
0
votes
2
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Explanation of the 2.60 equation page 76 in the Nielsen and Chuang [duplicate]
In the Nielsen and Chuang book page 76, equation 2.60 says that we can rewrite the trace $$Tr(A \left|\psi\right>\left<\psi\right|)$$ as follow :
$$Tr(A \left|\psi\right>\left<\psi\right|) ...
- 57
4
votes
2
answers
277
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closeness between two unitaries on the bloch sphere
The fidelity between two (single-qubit) quantum states can be easily translated into the euclidean distance between the two states on the Bloch sphere (hilbert-schidmit distance). I'm curious if this ...
- 127
4
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1
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150
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Efficient way to calculate trace of product of Pauli string and matrix?
Basically the title, but more formally: is there a way to efficiently calculate the trace of the product of a Pauli string $P$ and a $2^n \times 2^n$ matrix $M$? That is, is there a way to calculate ...
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1
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Given that for every valid density matrix $\rho$, $\text{Tr}(M\rho) = 1$; what can we conclude about matrix $M$?
My intuition says that $M$ has to be the identity matrix, but I am not able to show it rigorously. I tried playing around using spectral decomposition. If
$$
\rho = \sum_i \lambda_i |\lambda_i \rangle ...
- 1,311
0
votes
1
answer
55
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For tetrapartite state, and another way of decomposition, is the Schmidt basis separable?
Consider two tetrapartite quantum states $|\phi\rangle^{AA^\prime BB^\prime}$ and $|\psi_1\rangle^{AA^\prime}|\psi_2\rangle^{BB^\prime}$ in a finite dimentional Hilbert space $\mathcal{H}^A\otimes\...
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3
votes
2
answers
279
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Sufficient conditions for a single-qubit unitary to be the identity
Say I have a unitary $U = e^{-iHt}$ where $H = \alpha X + Z$.
First, suppose $U = I$. Then it rotates a set of initial states to themselves. Say I'm working on a computational basis, then on the Bloch ...
- 127
4
votes
1
answer
148
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Can you twist a qubit?
Is it possible to operate on a single qubit by a map which has a nonzero degree?
Let $|c\rangle=c_0|0\rangle + c_1|1\rangle$ represent a qubit state where $c_0,c_1 \in \mathbb{C}$ and $|c_0|^2+|c_1|^2=...
- 279
2
votes
0
answers
37
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Mechanics of expanding projector operator (two - qubits) in basis of traceless Hermitian Paul operators
I am currently on a set of lecture notes which says that for a state vector $| \psi \rangle_{AB}$ describing a tensor product state, its density operator $| \psi \rangle \langle \psi |_{AB}$ can be ...
- 510
1
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1
answer
53
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Statevector from Density matrix of non-pure state
I have a state vector of a 16 qubit system. I want to get the wave function (in the form of a state vector) for half and quarter of this system. When I try to make a ...
0
votes
1
answer
110
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How to apply CNOT on a three qubit system, with two qubits already entangled?
I am trying to understand the math behind the following applications of gates on three qubits:
You can view this via the Quirk simulator here:
https://algassert.com/quirk#circuit={%22cols%22:[[%22H%...
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2
votes
2
answers
131
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Prove the fidelity equals $F( \rho , \sigma) = |\langle \psi_{\rho} | \psi_{\sigma}\rangle|^2$ for pure states
I am trying to learn by myself quantum computing and information and I have a very simple question concerning the demonstration of the following equality: $F( \rho , \sigma) = |\langle \psi_{\rho} | \...
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1
answer
79
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Upper bound on entanglement entropy of a Product State for any possible partition of the Joint System
Let $|\psi\rangle$ be an $n$ qubit quantum state on a line with Von Neumann entanglement entropy at most $r$ with respect to any bipartition of the qubits (does not have to be a contiguous bipartition)...
- 1,251
0
votes
1
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93
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How to compute the partial trace of the state $|\psi\rangle = \sum_{k}c_k |k\rangle\otimes|k\rangle\otimes |k\rangle$?
Suppose we have a quantum system defined on a Hilbert space of $H=H_A\otimes H_B\otimes H_C$, and there is a state defined in $H$ of the form:
\begin{eqnarray}
|\psi\rangle = \sum_{k}c_k |k\rangle\...
- 199
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votes
2
answers
62
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Can we write a sum of unitary gates as an equivalent product of unitary gates?
If I did it correctly, in an answer to a question about Bell's states preparation, I found true that:
$$CNOT \space \space (H \otimes I) \space CNOT = ( X \otimes X + Z \otimes I ) \frac{1}{\sqrt{...
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0
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What is the value of $p(+)$?
I know the formula is $p = \left<\psi\right|M_{m}^{\dagger} M_{m}\left|\psi\right>$, where $\left|\psi\right>= \alpha\left|0\right>+\beta\left|1\right>$ and $M_m=\left|+\right> \left&...
- 1
2
votes
1
answer
286
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Prove that the eigenvectors of a Hermitian operator form a basis
While I was reading the book Quantum Mechanics The Theoretical Minimum, the author said that if a vector space is $N$ dimensional, an orthonormal basis of $N$ vectors can be constructed from ...
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0
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1
answer
60
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What is the correct order to multiply vectors?
I saw the equation: $$|A\rangle =\sum_i\alpha_i|i\rangle$$
Which could be also written like that:$$|A\rangle=\sum_i|i\rangle\langle i|A\rangle$$
I know that $\langle i|A\rangle=\alpha_i$, but can I ...
- 47
2
votes
2
answers
199
views
Is the inner product operation commutative or associative?
I am currently reading Quantum Mechanics The Theoretical Minimum by Leonard Susskind.
In the second lecture he says that for a given state of a spin $|A\rangle = a|u\rangle + b|d\rangle$: The ...
- 47
1
vote
1
answer
25
views
Obtaining the unitary matrix of a gate in analytic form in qiskit
Is there any way to obtain the unitary matrix of single way in analytic form?
In numerical form, I could use a workaround like this (i.e. for an X Gate):
...
- 121
0
votes
0
answers
37
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How to predict state vector of a quantum circuit in IBM qiskit
How to Predict the state vector output of the following quantum circuit?
...
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1
answer
56
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Why is the following exponential ignored (or equals 1) in the probability amplitude?
I'm reading Ronald de Wolf's lecture notes and when explaining Shor's algorithm on page 40 after applying a QFT to
$$
\frac{1}{\sqrt{m}} \sum_{j=0}^{m-1} |jr+s\rangle
$$
the following expression is ...
- 123
3
votes
0
answers
44
views
How to achieve the controlled rotation in the HHL algorithm
I'm trying to implement the HHL algorithm generally for any 2 x 2 hermitian matrix, but I'm having trouble with the implementation of the controlled rotation of the ancilla qubit. I've read very many ...
- 31
3
votes
1
answer
102
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Schur's lemma for quantum states
I am trying to understand Lemma 2 in this paper.
Consider a state $\tau_{H^n}=\int \sigma^{\otimes n}_{H} \mu(\sigma)$ where $\mu(\sigma)$ is the measure on the space of density operators on a single ...
- 2,597
1
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0
answers
31
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Is there a normal form for completely positive superoperators with rotationally symmetric spectra?
Let $d$ be a natural number. Given $A_1,\dots,A_r\in M_d(\mathbb{C})$, define a linear operator
$\Phi(A_1,\dots,A_r):M_d(\mathbb{C})\rightarrow M_d(\mathbb{C})$ by letting
$\Phi(A_1,\dots,A_r)(X)=...
- 111
2
votes
1
answer
72
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Efficiently computing weight enumerators of quantum codes
Are there any packages that efficiently compute weight enumerators (and dual weight enumerators) of quantum error correcting codes?
I'm interested in a general method that works for both stabilizer ...
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