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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.

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$\langle Z \rangle_L$ in the Distance Two Surface Code

In an experimental realization of the distance 2 surface code, the codewords are: $$|0\rangle_L = \frac{1}{\sqrt{2}} (|0000\rangle + |1111\rangle), |1\rangle_L = \frac{1}{\sqrt{2}} (|0101\rangle + |...
clunky monkey's user avatar
1 vote
0 answers
23 views

Take kronecker product of non-adjacent qubits

Suppose I have 4 qubits and I have the density matrix on qubits 1, 3 and I want to take the tensor product with the identity on the 2nd and 4th qubits for example. What is the fastest way to code this?...
snickers_stickers's user avatar
1 vote
1 answer
22 views

Are peripheral eigenvalues of a completely positive map always semisimple?

It is known that all peripheral eigenvalues (i.e. all eigenvalues $\lambda\in\mathbb C$ such that $|\lambda|$ equals the spectral radius) of positive trace-preserving or positive unital maps are ...
Frederik vom Ende's user avatar
3 votes
0 answers
76 views

Probability that a quantum state is in the typical subspace of another quantum state

From the properties of the Typical subspace we already have the following theorem [1]: Theorem (Unit Probability, see [1] page 467): Suppose that we perform a typical subspace measurement of a state $...
IamKnull's user avatar
  • 483
1 vote
1 answer
59 views

Clarification about the Alberti's Theorem proof given by Watrous in his condensed lecture notes

In the John Watrous condensed TQI lecture notes, an alternative proof of the Alberti's Theorem is given. He use an auxiliary lemma that states; Lemma 4.9. Let $P \in Pos(X)$. It holds that $${inf}_{R\...
Lucas Brugger's user avatar
1 vote
2 answers
38 views

For stabilizer codes, why does the error syndrome not depend on the codeword?

While reading through some lecture notes on quantum error correction, I read the statement: "In particular, the syndrome doesn’t depend on the specific codeword, only on the Pauli error." I'...
Daniel Mandragona's user avatar
3 votes
2 answers
57 views

Is every pure 1-qubit state an eigenstate of $aX + bY + cZ$?

As stated in the question, I have seen this claim made that a pure state can be written as an eigenstate of $aX + bY + cZ$ for some $a,b,c$ where $X,Y,Z$ are Pauli matrices. Why is this true and what ...
qubit's user avatar
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3 votes
1 answer
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Do all Hermiticity-preserving maps generate completely positive maps?

I am confused about what kinds of maps are valid infinitesimal generators of completely positive maps. I know that any Markovian completely positive map can be written in the form $e^{t \mathcal{L}}$, ...
nlupugla's user avatar
1 vote
0 answers
28 views

How to approximate the time-dependent Hamiltonian in quantum adiabatic theory by the non time-dependent Hamiltonian?

Recently, I am learning how to solve the linear equation $A\left | x \right \rangle =\left | b \right \rangle $ using quantum adiabatic theory. In the solving process, people usually need to set the ...
user30173's user avatar
0 votes
1 answer
43 views

How to find the $+1$ eigenvectors of the stabilizers for the Shor code

I am currently working through chapter $3$ of "Stabilizer Codes and Quantum Error Correction" (Daniel Gottesman's thesis). I would like to know the general method for finding the $+1$ ...
am567's user avatar
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Variational Quantum Linear Solver

I'm studying quantum computing right now and trying to implement variational quantum linear solver to solve a system of linear equations. From what I have understood from the paper written by Carlos ...
Sangkyu Baek's user avatar
1 vote
1 answer
64 views

Can the spectral radius of a completely positive map exceed the spectral radius of its transition matrix?

Recalling the spectral radius $r(T):=\max_{\lambda\in\sigma(T)}|\lambda|$ of a linear map $T$ (where $\sigma(T)$ refers to the spectrum of $T$), it is known that every quantum channel $\Phi:\mathbb C^{...
Frederik vom Ende's user avatar
3 votes
0 answers
40 views

Alternative algorithm for quantum phase estimation problem

The Quantum Phase estimation problem is stated below: Input: Given $U$ as a unitary operator acting on an m-qubit register. If $| \psi \rangle$ is an eigenvector of $U$, then U$| \psi\rangle$ = $e^{ ...
Manish Kumar's user avatar
3 votes
2 answers
85 views

Commutation of $XX$ and $ZZ$ operators

It is known that the Pauli operators $XX$ and $ZZ$ commute. Consider the state $\vert{++}\rangle$ which is an eigenstate of $XX$. But we also know that $$ZZ\vert{++}\rangle = \vert{--}\rangle$$ so ...
Kieran's user avatar
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1 answer
51 views

To what extent is the normal form of the Pauli transfer matrix unique?

In order to properly state the question let me be precise about the object at the core of this question's title. First, given any orthonormal basis of $G:=\{G_j\}_{j=1}^{n^2}$ of $\mathbb C^{n\times n}...
Frederik vom Ende's user avatar
1 vote
2 answers
55 views

In the QECC condition $\langle\psi|E_a^\dagger E_b|\phi\rangle=C_{ab}\langle\psi|\phi\rangle$, what is $C_{ab}$?

In this book, Theorem 2.7 has the QECC conditions. I attach a snippet here Theorem 2.7 (QECC Conditions). $(Q, \mathcal{E})$ is a $Q E C C$ iff $\forall|\psi\rangle,|\phi\rangle \in Q, \forall E_a, ...
Polya's user avatar
  • 13
2 votes
0 answers
57 views

How to systematically find the kernel of a channel from its Kraus operators?

A quantum channel is a completely positive trace-preserving map. Given a quantum state $\rho$ and channel $N$, let the output be $N(\rho)$. Given the Kraus operators of the channel, how can one find ...
user1936752's user avatar
  • 3,043
2 votes
1 answer
115 views

Is $\rho = \sum_{j} p_j|n_j\rangle\langle n_j|$ a valid construction for any mixed state?

I have a mixed state $\rho$ and its hamiltonian $H$. Firstly, I find the eigenvalues $\{p_j\}$ of $\rho$, and orthonormal basis of $H$. I write $\rho$ in terms of $H$'s eigenstates and $\rho$'s ...
Việt Nguyễn's user avatar
0 votes
2 answers
30 views

Does $\frac{I + K}{2}\otimes\frac{I+L}{2} = \frac{I+K\otimes L}{2}$ hold for operators $K,L$ acting on different subsystems?

Let's index $i= 1, 2$ be the index representing different systems. A $Z_i$ projective measurement has projectors $P_i = \{ \frac{I+Z_i}{2}, \frac{I-Z_i}{2} \}$. One can verify that the measurement $...
user1936752's user avatar
  • 3,043
1 vote
0 answers
64 views

How to express a traceless matrix in Pauli basis

This question is probably too obvious, so sorry beforehand. We know that the generalized Pauli elements $P\in \mathcal{P}_d \setminus {\mathrm{Id}_d}$ in Sylvesters representation, hence not Hermitian,...
relativeentropy's user avatar
4 votes
1 answer
107 views

Can we find a product of single-qubit states that is orthogonal to a stabilizer codespace?

Let's consider a single-qubit pure state $$\rho=\rho(x, y,z)=\frac{1}{2}(I+xX+yY+zZ)$$ with $x^2+y^2+z^2=1$. Can we show the following statement is true: For any $n$-qubit tensor product states $\rho^{...
Yunzhe's user avatar
  • 761
5 votes
1 answer
100 views

Can quantum computers help to solve questions of general relativity theory?

My question is rather straightforward: Can quantum computers be used to solve problem within general relativity theory? To put more context. As GR is based on solution of rather complicated systems of ...
Martin Vesely's user avatar
1 vote
1 answer
98 views

If $\text{tr}_B \rho \in A$, then $\rho \in A \otimes B$?

Let our Hilbert space be $H = (A \otimes B) \oplus (A \otimes B)^{\perp}$. If $\rho \in A \otimes B$, then we have $\text{tr}_B \rho \in A$. Is the converse true: if $\text{tr}_B \rho \in A$, then $\...
karavan's user avatar
  • 21
6 votes
1 answer
295 views

Clifford group without the phase gate

The Clifford group is generated by the Hadamard gate $H$, the phase gate $S=\sqrt{Z}$, and the $\text{CNOT}$ gate. I was wondering what happens if we dropped $S$, so that all matrices are real. I ...
Jun_Gitef17's user avatar
2 votes
0 answers
33 views

What is the rank of a superoperator of the form $\Xi (\cdot) = \sum_i^n U_i^\dagger {\cdot}\, U_i$?

Given a superoperator $\Xi$ as $\Xi (\cdot) = \sum_i^n U_i^\dagger \cdot U_i $ where $U_i$ are unitary. What can I say about the image of this map or about the rank of $\Xi$? Also, do you have some ...
relativeentropy's user avatar
1 vote
2 answers
75 views

What is the action of $CCZ$ on $X \times I \times I$?

Confused about the action of the $CCZ$ gate on Pauli operators: I understand the action of the $CZ$ gate: $$CZ: XI \rightarrow XX$$ $$CZ: IX \rightarrow IX$$ $$CZ: ZI \rightarrow ZI$$ $$CZ: IZ \...
am567's user avatar
  • 597
4 votes
2 answers
71 views

What's the trace distance between $|0\rangle^{\otimes n}$ and $\frac{1}{\sqrt{2}}\big(|0\rangle^{\otimes n} + |1 \rangle^{\otimes n} \big)$?

I'm trying to figure out the trace distance between the states $\rho_1$ and $\rho_2$, where $$ \begin{align}\rho_1 &= (|0\rangle \langle 0|)^{\otimes n}\,,\\ \rho_2 &= \dfrac{1}{2}(|0\rangle^{\...
Sean Thrasher's user avatar
1 vote
0 answers
49 views

What is the intuition behind achieving Quantum advantage in simulating non-hermitian dynamics using Quantum computer?

There have been several works on simulating ODE for classical systems like here and here. They are quantum techniques to solve the ODE related to classical systems. A generic methodology is: To solve ...
Manish Kumar's user avatar
1 vote
1 answer
43 views

Why can't the eigenvalues of a unitary matrix have the form $e^{i\theta}$?

The textbook says that since $U$ is a unitary matrix, its eigenvalue should be of the form $e^{2 \pi i \theta}$. The thing I don't understand is why it's not $e^{i \theta}$ because it also lies on the ...
Nir Sharma's user avatar
2 votes
1 answer
63 views

Under what conditions are two sets of Pauli operators Clifford-equivalent?

Suppose I have two set of $N$-qubit Pauli operators $\mathcal{S} = \{P_1,\ldots,P_K\}$ and $\mathcal{T} = \{Q_1,\ldots,Q_K\}$. In this context, a Pauli operator is a Hermitian element of the Pauli ...
Solarflare0's user avatar
1 vote
1 answer
63 views

Can any isometry $V$ be written as $U(I\otimes |\psi\rangle)=V$ for some unitary $U$ and vector $|\psi\rangle$?

I have the following exercise: Let $V : H_A → H_A ⊗ H_E$ denote an isometry and $|ψ_E⟩ ∈ H_E$ a normalized vector. Show that there exists a unitary $U : H_A ⊗ H_E → H_A ⊗ H_E$ such that $$U(1_{H_A} ⊗ |...
Pink Elephants's user avatar
3 votes
0 answers
47 views

Trying to prove Theorem 4.1 from Neilsen and Chuang algebraically

Background Theorem 4.1 of Neilsen and Chuang (10th Anniversary Edition) states how a universal single-qubit unitary can be constructed from Y and Z rotations. Suppose $U$ is a unitary operation on a ...
kaddy's user avatar
  • 31
8 votes
3 answers
303 views

In Schur-Weyl's duality, why is the commutant of $\pi_k(S_k)$ spanned by $U(d)^{\otimes k}$ matrices?

I'm reading this tutorial paper about quantum state certification. However, I'm confused about the concept of Schur-Weyl duality, explicitly Theorem 35 of the paper. Let $S_k$ denotes the symmetric ...
Sherlock's user avatar
  • 695
1 vote
0 answers
46 views

Relationship between Symplectic Transvections and Their Associated Matrices

I'm studying symplectic transvections and their properties in the context of quantum information theory. I came across the definition of a symplectic transvection, $\tau_h$, which maps $\mathbb{F}_{2}^...
hiiii's user avatar
  • 21
0 votes
1 answer
77 views

Left-canonical matrix product state

A pure quantum state $$\tag{1}|\Psi\rangle=\sum_{j_1,\ldots,j_N=1}^{d}\psi_{j_1j_2\ldots j_N} |j_1, \dots, j_N\rangle\,,$$ can be represented exactly in the MPS form \begin{equation}\tag{2} |\Psi\...
jayjay's user avatar
  • 111
2 votes
1 answer
75 views

QuTiP ptrace function results do not recreate original composite system

I have a qutip density matrix fullsystem for a system composed of two quantum systems with ...
Bebotron's user avatar
  • 425
4 votes
2 answers
166 views

What can we say about the eigendecomposition of quantum channels?

It is known that quantum channels, being CPTP maps, map density operators to density operators. And thus, they can be seen as superoperators. Similar to operators, where eigenstates and eigenvalues ...
ironmanaudi's user avatar
1 vote
0 answers
35 views

How to obtain this measurment result for arbitrary qubits in $X$ basis?

I have seen the following claim. Suppose one has two qubits $\vert\psi\rangle = a\vert 0\rangle + b\vert 1\rangle$ and $\vert\phi\rangle =a'\vert 0\rangle + b'\vert 1\rangle$ and both are measured in ...
Lorraine's user avatar
2 votes
1 answer
114 views

Minimizing $1 - \text{Tr}(\Phi(\rho,U)^2)$

I am looking for a computationally efficient way to minimize the following function. Let $$\Phi(\rho, U) = \text{Tr}_2(U\rho U^\dagger)$$ be a reduced density matrix where $\rho = \overline{\rho}_1 \...
Silly Goose's user avatar
1 vote
2 answers
67 views

Is $\text{Tr}(\text{Tr}_\mathcal{E}(\rho)) = \text{Tr}(\rho)$?

Let $\rho$ be a density matrix over some composite Hilbert space $\mathcal{H}_S \otimes \mathcal{H}_{\mathcal{E}}$. Is partial trace full trace preserving? I.e., is $$\text{Tr}(\text{Tr}_\mathcal{E}(\...
Silly Goose's user avatar
1 vote
1 answer
49 views

Projective measurement notation

I don't understand how projective measurements work; I think my confusion comes from the notation. How would this be written in matrix notation? Is it just [w row vector] [Ma matrix] [w column vector]^...
researcher101's user avatar
1 vote
0 answers
50 views

working of Variable time amplitude amplification (VTAA) compared to amplitude amplification (AA)

I read the paper by Ambianis on variable time amplitude amplification to improve the $\kappa$ (condition number) dependency for the Quantum linear system algorithm by Childs et al.. I can see VTAA ...
Manish Kumar's user avatar
3 votes
2 answers
51 views

Why is the linear combination of Pauli matrices $G =I-XX-YY-ZZ$ PSD?

Define $$G = I \otimes I - X \otimes X - Y \otimes Y - Z \otimes Z,$$ where $X,Y$ and $Z$ denote the Pauli matrices, and $I$ the identity. I can plug this matrix in my computer and note that $$G = \...
Matteo's user avatar
  • 161
1 vote
1 answer
67 views

Inner product as unitary operation

Inner products of two states $\psi$ and $\phi$ are usually performed at the end of a quantum algorithm where we measure the final state, e.g. using the swap test. However, this operation is not ...
Medulla Oblongata's user avatar
0 votes
0 answers
79 views

Show that the Choi of a tensor product is the tensor product of the Chois

I have the following problem. Let $N:L(H_A)\rightarrow L(H_A)$ be a quantum superoperator. The quantum state corresponding to this operator (via Choi-Jamiolkowski Isomorphism) is $\Gamma_A^{N}=id\...
Piotr Masajada's user avatar
4 votes
2 answers
263 views

Prove that if Kraus operators of $\Phi$ form an ONB then $\Phi$ is the replacement map

This problem is from a "passing remark" in this lecture notes. With the help of some colleagues I managed to find a way for this supposedly elementary fact, but I would like to see if there ...
Evangeline A. K. McDowell's user avatar
4 votes
1 answer
167 views

Optimal dependency of HHL (or any QLSP) algorithm on condition number $\kappa$

This is conserning the optimal dependency on condition number for Quantum linear system problem (QLSP). For solving QLSP, the HHL (algorithm) paper mentions any polylog($\kappa$) quantum algorihm ...
Manish Kumar's user avatar
1 vote
1 answer
136 views

Quantum Phase Estimation answers distribution

Suppose I have a random unitary matrix, known eigenvectors and eigenvalues. I know that exact eigenvalue for the given matrix is $0.5491617699847768+0.835716070437315j$. From here, if I'm not mistaken ...
Марина Лисниченко's user avatar
2 votes
2 answers
150 views

Does independence from the input state imply a tensor product structure for the unitary?

Let $U$ be a unitary acting on Hilbert space $\mathcal H = \mathcal H_A \otimes \mathcal H_B$ such that $$\mathrm{Tr}_A(U \vert \psi \rangle \langle \psi \vert_A \otimes \vert 0 \rangle \langle 0 \...
SescoMath's user avatar
  • 549
2 votes
2 answers
206 views

Prove that $\text{Tr}(M|ψ\rangle\langleϕ|)=\langleϕ|M|ψ\rangle$

Question: I am studying alone, and I found p.76 of the book quantum computation and quantum information of nielsen &c huang that: $$\text{Tr}(M |\psi\rangle \langle\psi)=\langle\psi| M |\psi\...
OffHakhol's user avatar
  • 155

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