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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Computational basis absolute value coefficient of stabilizer state in STIM [duplicate]

Given a stabilizer state $|\psi\rangle$, I write it in the computational basis $|\psi\rangle = \sum_{n=0}^{2^N-1} c_n |n\rangle$. I was wondering if there is a way to compute the exact $|c_n|^2 = \...
archxrk's user avatar
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1 answer
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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
  • 19
4 votes
1 answer
272 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
1 vote
0 answers
28 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
71 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
  • 569
4 votes
2 answers
69 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
44 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
39 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
45 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
50 views

How to represent general isometries and unitaries:

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
42 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
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8 votes
3 answers
248 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
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1 vote
0 answers
43 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
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1 answer
64 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
39 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
3 votes
2 answers
134 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
31 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
110 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
57 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
41 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
42 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
47 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
61 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
68 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
239 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
105 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
0 votes
1 answer
123 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
145 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
  • 507
2 votes
2 answers
201 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
4 votes
2 answers
403 views

What unitary commutes with all local Pauli operators?

I was thinking about this problem of identifying a set of unitary operations (other than the identity operation) that commute with local pauli $\sigma_X$ and $\sigma_Z$ matrices, i.e. find $U$ such ...
Mohan's user avatar
  • 161
1 vote
2 answers
96 views

What happens to $|y\rangle \sum_{x}|x\rangle|f(x) + g(y)\rangle$ when we throw away the first register?

Let's suppose, that applying $\mathbf{H}$ (Hadamard operator) to the first register of the state $c \cdot \sum_{x}|x\rangle|f(x)\rangle$ ($f$ is a permutation, $c$ is a normalization factor), and ...
Georgy Firsov's user avatar
1 vote
1 answer
47 views

Simplification of a generic quantum state

We are given a generic 2-qubit density matrix $$\rho=\frac{1}{4}\left[I_4+\Sigma_i a_i \sigma_i \otimes I_2 + \Sigma_i b_i I_2 \otimes \sigma_i + \Sigma_{i,j} c_{ij} \sigma_i \otimes \sigma_j\right]$$ ...
Anindita Sarkar's user avatar
-2 votes
1 answer
53 views

help understanding gate to hamiltonian and representation

So I have this question: Given an operator, find some Hamiltonian implementing this operator/gate. I have realized that this is a swap gate and I know the matrix for it. I also know that $U = \text{...
George's user avatar
  • 1
1 vote
1 answer
58 views

How to take partial trace of a $n - 1$ qubit subsystem from a $n$ qubit system

I would like to calculate the expression $$ \text{Tr}_2\left\{R^z \sigma\right\}\,, $$ where $$ \sigma = \rho \otimes |0\rangle \langle0|^{{\otimes}(n-1)}\,. $$ Here $$ R = \sum{\theta_m}G_m\,,$$ ...
Sudhir Kumar's user avatar
5 votes
1 answer
35 views

Upper bound on $\Vert U_1 \otimes U_2 \otimes \cdots \otimes U_k - V_1 \otimes V_2 \otimes \cdots \otimes V_k \Vert$

Let $U_i$ and $V_i$ be unitaries that act on the same subsystems. Can we upper bound the difference between the tensor products of these unitaries, i.e. $\Vert U_1 \otimes U_2 \otimes \cdots \otimes ...
Mohan's user avatar
  • 161
1 vote
1 answer
105 views

show that $\mathrm{tr}_A \left[\rho_A \lvert \phi^+ \rangle_{AB} \langle \phi^+ \rvert\right]$ equals to $\rho_B^T$

I have difficulties calculating with partial traces in terms of quantum operations. For me it is not clear how to derive the equality stated in the question title for a quantum mechanical system whose ...
gehbiszumeis's user avatar
2 votes
1 answer
56 views

Explicit calculation for multiplying two projection operators

Can someone explain the explicit calculations for: $$(I \otimes ( |00\rangle \langle 00| + |11 \rangle \langle 11| ) ) \times ( (|00 \rangle \langle 00| + |11 \rangle \langle 11|) \otimes I) = |000 \...
am567's user avatar
  • 569
2 votes
1 answer
250 views

Why does the definition of the oracle in Deutsch's algorithm fail to specify its action on superpositions?

I’m trying to understand the Deutsch algorithm. I can see that the math shows the algorithm to be correct, but I don’t understand how the math represents the given conditions. The oracle is supposed ...
Jon Vote's user avatar
  • 123
1 vote
1 answer
60 views

How to find projection operators for spectral decomposition

I am a little bit confused about the spectral decomposition for the observable $Z_{1}Z_{2}$ in Section $10.1$ of Nielsen and Chunag's "Quantum Computation and Quantum Information". The idea ...
am567's user avatar
  • 569
3 votes
2 answers
182 views

Equal partial traces

Given an arbitrary state $\rho_{AB}$, is it always possible to construct an extension $\rho_{ABC}$ such that $$Tr_B(\rho_{ABC}) := \rho_{AC} = \rho_{AB} := Tr_C(\rho_{ABC})$$ If yes, does there exist ...
user1936752's user avatar
  • 2,913
1 vote
1 answer
238 views

How is Quantum Computing expressed in the language of abstract algebra?

I've lately been taking further coursework in abstract algebra, and it has struck me as fairly reminiscent of quantum computing. Of course, Pauli matrices, etc. have relevant roots within abstract ...
Nurdick's user avatar
  • 21
1 vote
1 answer
92 views

How doesn't combining two eigenvectors that have the same eigenvalue for a specific matrix represent every vector left in the plane?

If we have a 2D plane and the hermitian matrix $L$ where: $$L|\lambda_1\rangle=\lambda|\lambda_1\rangle$$ $$L|\lambda_2\rangle=\lambda|\lambda_2\rangle$$ Given that $|\lambda_1\rangle$ and $|\lambda_2\...
zizaaooo's user avatar
2 votes
1 answer
95 views

How do you work out the matrix for controlled-U operations?

I see this equations all over for controlled-U operations: $$ \left|{0}\right>\left<{0}\right| \otimes \mathbf{1} + \left|{1}\right>\left<{1}\right|\otimes U = \begin{pmatrix} \mathbb{1} &...
grepgrok's user avatar
3 votes
0 answers
77 views

What properties of a local Hamiltonian are basis-(in)dependent?

Some properties of a Hamiltonian are unique to its spectrum and are basis-independent. For example, I think whether the Hamiltonian's gap remains constant as $n$ goes to infinity, or whether the ...
Mark Spinelli's user avatar
2 votes
1 answer
196 views

How to find the eigenvectors and eigenvalues of a hermitian operator?

While reading Theoretical Minimum by Leonard Susskind, I came across the exercise 3.4 where he asked to find the eigenvalues and the eigenvectors of the matrix that represents the $\sigma_{n}$ ...
zizaaooo's user avatar
2 votes
1 answer
54 views

unitary that transforms one Hilbert space to another Hilbert space

Let $H = A \otimes B$. If there exists a unitary operator $U$ that transforms the Hilbert space $H$ into another Hilbert space $H' = A' \otimes B'$ (meaning that $U$ maps each basis of $H$ to each ...
Mohan's user avatar
  • 161
0 votes
1 answer
62 views

What does "the eigenvectors of a Hermitian operator are a complete set" mean?

I read in my book that the eigenvectors of a Hermitian operator are a complete set. What does the author mean by that?
zizaaooo's user avatar
3 votes
1 answer
114 views

Asymptotic purity from the spectrum of the Choi matrix?

I have a completely positive map $T$ and a sequence of $d\times d$ states $S_1,S_2,\ldots$ obtained by applying $T$ repeatedly to the identity matrix. I'm interested in quantifying what happens to ...
Yaroslav Bulatov's user avatar
2 votes
0 answers
73 views

Given three quantum states, how to compute the triple product of amplitudes $\sum_i u_i v_i w_i$?

Assume I have three quantum states $|u\rangle$, $|v\rangle$ and $|w\rangle$ which can be obtained with three quantum circuits $U$, $V$ and $W$. We know that we can easily estimate the inner product $\...
francler's user avatar
  • 181
0 votes
1 answer
126 views

How to find $p_x$ and $p_y$ components on the Bloch sphere?

Consider an arbitrary state: $$|\psi\rangle = a|0\rangle+b|1\rangle,$$ where $a=\cos\left(\frac{\theta}{2}\right), b=\sin\left(\frac{\theta}{2}\right)e^{i\phi}$ (neglecting global phase), $\phi$ is ...
Curious's user avatar
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