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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8
votes
2answers
126 views

How to recover the normalization constant of the HHL solution

HHL solves the linear equation $Ax=b$ by the quantum state $|x\rangle=A^{-1} |b\rangle$. However, the quantum state $|x\rangle$ is normalized and thus diffs a normalization constant from the solution ...
8
votes
1answer
113 views

Why is HHL the popular choice to solve QLSP and not the Childs et al. (2017) algorithm?

The Childs, Kthari, and Rolando (2017) (CKS) algorithm can solve the quantum linear systems problem (QLSP) in $\operatorname{poly}(\log N, \log(1/\epsilon))$ time while the HHL algorithm solves it in $...
6
votes
1answer
1k views

How are arbitrary $2\times 2$ matrices decomposed in the Pauli basis?

I read in this article (Apendix III p.8) that for $A\in \mathcal{M}_2$, since the normalized Pauli matrices $\{I,X,Y,Z\}/\sqrt{2}$ form an orthogonal matrix basis. $$A=\frac{Tr(AI)I+Tr(AX)X+Tr(AY)Y+...
5
votes
3answers
482 views

Why is the transpose of a density matrix positive and trace preserving?

With density matrix $\rho=\sum_{a,b=0}^1\rho_{a,b}|a\rangle\langle b|$ and it's transpose $\rho^T=\sum_{a,b=0}^1\rho_{a,b}|b\rangle\langle a|$. How to confirm that $\rho^T$ is positive and trace ...
5
votes
1answer
167 views

Closeness of purifications of states

Uhlmann's theorem states that if two states $\rho_A, \sigma_A$ satisfy $F(\rho_A, \sigma_A)\geq 1 - \varepsilon$, then there for any purification $\Psi_{AR}$ of $\rho_A$, one can find a purification $\...
5
votes
2answers
96 views

Prove that $\rho_{AB} \leq |B|(\rho_A\otimes I_B)$ for any bipartite state $\rho_{AB}$

I'm trying to prove the following statement but am lost on how to show it. For a quantum state $\rho_{AB}$ with marginal $\rho_A$, how can one show that $$ \rho_{AB} \leq|B|(\rho_A\otimes I_B)$$ where ...
5
votes
2answers
109 views

How to calculate the spectral norm of the density operator used in Molina et al. 2012 paper?

In Molina et al (2012)'s article on quantum money, the proof of security of Wiesner's quantum money scheme depends on the fact that the density operator $$Q = \frac{1}{4}\sum_{k \in \{0, 1, +, -\}}\...
5
votes
1answer
114 views

Stinespring dilation: Size of environment

Let $\mathcal{E}_{A\rightarrow B}$ be a quantum channel and consider its $n-$fold tensor product $\mathcal{E}^{\otimes n}_{A^n\rightarrow B^n}$. Any isometry $V_{A\rightarrow BE}$ that satisfies $\...
5
votes
1answer
133 views

Does a basis of maximally entangled states exist for two-qubit or two-qutrit system so that the density matrices of the basis states don't commute?

I want to find a basis of maximally entangled states $|\Psi_i\rangle$, for $\mathcal{H}^{2} \otimes \mathcal{H}^{2}$ and, $\mathcal{H}^{3} \otimes \mathcal{H}^{3}$ such that the density matrices of ...
5
votes
1answer
76 views

Prove that $|(\langle \psi|_{A} \otimes \langle \phi|_{B})|\theta\rangle_{AB}|^{2}<1$ for entangled $|\theta\rangle_{AB}$

I am trying to show that $|\langle \psi|_{A} \otimes \langle \phi|_{B}|\theta\rangle_{AB}|^{2}<1$ given $|\theta\rangle$ is an entangled state, and as such has schmidt rank >1. Decomposing it, ...
5
votes
1answer
222 views

Does the controlled Pauli Z gate cause entanglement?

I'm trying to understand the relationship between the factorability of a 2 qubit gate and that gate's ability to cause entanglement. I've begun by considering the controlled Pauli Z gate. After ...
5
votes
2answers
145 views

How are the eigenvalues of $\rho=\frac12(|a\rangle\!\langle a| +|b\rangle\!\langle b|)$ derived?

Let's say I have a density matrix of the following form: $$ \rho := \frac{1}{2} (|a \rangle \langle a| + |b \rangle \langle b|), $$ where $|a\rangle$ and $|b\rangle$ are quantum states. I saw that ...
5
votes
1answer
141 views

What's the state-of-the-art to calculate $|Ab\rangle$, given a matrix $|A\rangle$ and a vector $|b\rangle$ in QRAM encoding

Assuming that we have a matrix $A\in \mathbb{R}^{m\times n}$ stored in a quantum superposition, i.e. $$|A\rangle= \frac{1}{\|A\|_F}\sum_{i,j=0}^{n-1}{a_{ij}}|i,j\rangle$$ and a vector $b\in \mathbb{R}^...
4
votes
2answers
435 views

Inverting the depolarizing channel

I have a depolarizing channel acting on $2^n \times 2^n$ Hermitian matrices, defined as $$\tag{1} \mathcal{D}_p (X) = p X + (1-p) \frac{\text{Tr}(X)}{2^n} \mathbb{I}_{2^n} $$ where $\mathbb{I}_{d}$ is ...
4
votes
2answers
148 views

In Stinespring dilation, can we always use a mixed state as the ancilla?

The Stinespring dilation theorem states that any CPTP map $\Lambda$ on a system with Hilbert space $\mathcal{H}$ can be represented as $$\Lambda[\rho]=tr_\mathcal{A}(U^\dagger (\rho\otimes |\phi\...
4
votes
2answers
229 views

If the eigenvalues of $Z$ are $\pm1$, why are the computational basis states labeled with "$0$" and "$1$"?

The computational basis is also known as the $Z$-basis as the kets $|0\rangle,|1\rangle$ are chosen as the eigenstates of the Pauli gate \begin{equation} Z=\begin{pmatrix}1 & 0 \\ 0 & -1\end{...
4
votes
1answer
244 views

What is the matrix for a SWAP operation on two qubits?

Say we want to swap qubits $a$, $b$ in the same register, where $a,b \in \left \{ 0, 1,\cdots, n-1 \right \}$. What would be the corresponding matrix. For those interested, I'm curious about this ...
4
votes
1answer
101 views

Cliffordness of the qutrit Hadamard gate

Consider a simple generalization of the Hadamard gate to qutrits, defined as follows. \begin{equation} \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\\ \frac{1}{\sqrt{2}} &...
4
votes
2answers
58 views

Are the two ways of interpreting the expression $(|a\rangle\otimes|b\rangle)(\langle c|\otimes\langle d|)(|e\rangle\otimes |f\rangle)$ equivalent?

Reading Nielsen and Chuang, I am under the impression that a linear operator on the tensor product can be written in two ways: \begin{equation} (\left|a\right> \otimes \left|b\right>)(\left<c\...
4
votes
1answer
240 views

How do I prove that $P_\pm=\frac12(1\pm U)$ if $U^2=I$?

Suppose I have an $n$-qubit Hermitian operator $U$ such that $U^2=I$. The projection operators with eigenvalue $+1$ and $−1$ are $P_+$ and $P_-$. How can I prove that $P_+=\frac{1}{2}(1+U)$ and $P_-=\...
4
votes
1answer
104 views

What is the structure of coefficients of $2^N\times 2^N$ unitary matrices decomposed in terms of the Pauli basis?

Any square $2^N\times 2^N$ matrix can be written as a sum of tensor products of pauli matrices. Eg a $8\times 8$ matrix can be written as $$U=\sum_{i_1,i_2,i_3}u_{i_1,i_2,i_3}\sigma_{i_1}\otimes\...
4
votes
2answers
154 views

Clarification on Watrous' proof of Alberti's theorem on the fidelity function

I am reading John Watrous' quantum information theory book. In the proof of Theorem 3.19 (practically the Alberti's theorem on the characterization of the fidelity function) he claims the following ...
4
votes
1answer
112 views

Conjugation of $R_x(\theta)$ with $CNOT$

Section 2.5 (4.3) of the Qiskit textbook, see here, discusses the conjugation of $R_x(\theta)$ by $CNOT$. The following expression is given: $$CX_{j,k}(R_x(\theta)\otimes 1) CX_{j,k}=\color{brown}{...
4
votes
1answer
257 views

Difference between change of basis in bra-ket notation and matrix notation

In matrix notation, say I have the vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$. It is currently represented in the computational basis $\{\begin{bmatrix} 1 \\ 0\end{bmatrix}, \begin{bmatrix} 0 \\ 1\...
4
votes
1answer
154 views

Variational Quantum Linear Solver (Hadamard test): circuit question

Trying to understand the circuit/algorithm for VQLS and I found this diagram to show the high-level idea of doing the Hadamard test in this tutorial. But I am not quite sure why we need the two ...
4
votes
1answer
114 views

Creating orthogonal quantum states from a set of given (possibly linearly independent) quantum states

I want to understand how to orthogonalize a system of qubits. Suppose I have $n$ sets of quantum states like $$\{ |1_i\rangle|2_i\rangle|3_i\rangle \cdots|k_i\rangle \mid i=1 \dots n \}$$ where $i=1, \...
3
votes
3answers
660 views

How to show that Bell states are orthonormal

I was reading some material on QC online and I found some material that explains how to show that Bell states are orthonormal but without details. I understand that we need to check $\langle state1|...
3
votes
2answers
114 views

Can we express $\mathrm{tr}_A((A\otimes B)\rho_{AB})$ in terms of $A$, $B$, $\rho_A$ and $\rho_B$?

For a density matrix $\rho_{AB}$ and some operators $A, B$, is there a way to express $$\text{Tr}_A((A\otimes B)\rho_{AB})$$ using the reduced states $\rho_A$ and $\rho_B$ and operators $A$ and $B$? ...
3
votes
2answers
295 views

How to express real matrices as linear combinations of unitaries?

I am working on using Variational Quantum Linear Solver (VQLS) for some tasks. Here, we need to represent matrix A as a linear combination of unitaries. $$ {\bf A} = \Sigma^n_{i=1} c_iA_i $$ My ...
3
votes
2answers
245 views

Expected value of a Haar random quantum state multiplied by a unitary

Consider a quantity \begin{equation} \mathbb{E}\big[\langle z|\rho|z\rangle\big], \end{equation} where $\rho = |\psi \rangle \langle \psi|$ is a Haar-random state $n$-qubit quantum state and $z$ is ...
3
votes
1answer
160 views

Is there an identity for the partial transpose of a product of operators?

The partial transpose of an operator $M$ with respect to subsystem $A$ is given by $$ M^{T_A} := \left(\sum_{abcd} M^{ab}_{cd} \underbrace{|a\rangle \langle b| }_{A}\otimes \underbrace{|c \rangle \...
3
votes
1answer
88 views

HHL algorithm for linear systems with a real matrix and a real right side

HHL algorithm can be used for solving linear system $A|x\rangle=|b\rangle$. If we put $|b\rangle$ (to be precise its normalized version) into the algorithm and measuring ancilla to be $|1\rangle$ we ...
3
votes
1answer
52 views

How to calculate the overlap of the orthogonal state?

This is probably a very obvious question, but I am going through this problem set and I don't understand why in 1b) it says that it is obvious that $|\langle\psi_1^\perp|\psi_2\rangle|=\sin\theta$ ...
3
votes
1answer
136 views

Diamond norm distance bound on Stinespring dilations of channels

The diamond distance between two channels $\Phi_0$ and $\Phi_1$ is defined in this answer. $$ \| \Phi_0 - \Phi_1 \|_{\diamond} = \sup_{\rho} \: \| (\Phi_0 \otimes \operatorname{Id}_k)(\rho) - (\Phi_1 ...
3
votes
2answers
46 views

Are the states in the convex decomposition of a density matrix necessarily orthogonal?

In Nielsen and Chuang's QC&QI, I do not see a statement one way or another. In Steeb and Hardy's Problems and Solutions, orthogonality is asserted. If the $p_i$ in $\sum_i p_i |\psi_i\rangle\...
3
votes
1answer
37 views

Relation between symmetric subspaces and $n$-exchangeable density matrices

Let us consider $n$ elements, each taken from the set $\{1, 2, \ldots, d\}$ and let $S_n$ be the set of all permutations on these $n$ elements. Define a permutation operator on the set of $n$ qudits ...
3
votes
2answers
41 views

Normalization question in VQLS

I was studying VQLS in https://qiskit.org/textbook/ch-paper-implementations/vqls.html and run into the following normalization during the cost calculation. It says if $ |\Phi\rangle$ has a small norm....
3
votes
2answers
100 views

Checking whether a state is almost orthogonal to permutation invariant states

Let us consider \begin{equation} |T\rangle = |\psi \rangle^{\otimes m} \end{equation} for an $n$-qubit quantum state $|\psi\rangle$. Let $\mathcal{V}$ be the space of all $(m + 1)$-partite states that ...
3
votes
1answer
71 views

Mistake in using dirac notation when applying $X$ gate to vector

The X gate is given by $\big(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\big)$ in the computational basis. In the Hadamard basis, the gate is $X_H = \big(\begin{smallmatrix} 1 &...
2
votes
2answers
83 views

Properties of composition of isometry and a perturbed adjoint

Suppose $\vert\Phi\rangle_{AR} = \frac{1}{\sqrt{|D|}}\sum_{i\in D} \vert ii\rangle_{AR}$ is the maximally entangled state. Let $V_{A\rightarrow BE}$ and $\tilde{V}_{A\rightarrow BE}$ be two isometries ...
2
votes
1answer
87 views

Why can any density operator be written this way? (quantum tomography)

From page 24 of the thesis "Random Quantum States and Operators", where $(A,B)$ is the Hilbert-Schmidt inner product: \begin{aligned} \rho &=\left(\frac{1}{\sqrt{2}} I, \rho\right) \frac{...
2
votes
1answer
118 views

What is the best way to write a tridiagonal matrix as a linear combination of Pauli matrices?

I'm looking for an algorithm to write an arbitrarily sized tridiagonal matrix as a linear combination of Pauli matrices. The tridiagonal matrix has the form, for example, \begin{pmatrix} 2 & -1 &...
2
votes
1answer
92 views

Depolarization of density operator with zeros in diagonal

I suppose a quantum state with density matrix like the following is not valid. $$ \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}. $$ Now, let's say I have a valid density operator representing ...
2
votes
1answer
112 views

Permutation covariant channels and their Stinespring dilations

I am interested in a quantum channel from $A^{\otimes n}$ to $B^{\otimes n}$ denoted as $N_{A^{\otimes n} \rightarrow B^{\otimes n}}(\cdot)$. Let $\pi(\cdot)$ be a permutation operation among the $n$ ...
2
votes
2answers
87 views

What is $\sum_{i}\langle i \vert U \vert j\rangle$ for unitary $U$?

The question is basically the title but given a unitary operator $U$ and a computational basis, can we say anything about the complex number below? $$c = \sum_{i}\langle i \vert U \vert j\rangle$$ I ...
2
votes
1answer
90 views

How to find the eigenstates of a general $2\times 2$ Hermitian matrix?

Given a measurement operator in the general Hemitian form $$ M = \begin{pmatrix} z_1 & x+iy \\ x-iy & z_2\end{pmatrix}, $$ where $x,y,z_1,z_2 \in \mathbb{R}$, show that the eigenvalues are $$ ...
2
votes
1answer
41 views

Is the quantum mutual information variance bounded from above?

The relative entropy variance between two quantum states $\rho$ and $\sigma$ is defined to be $$V(\rho\|\sigma) = \text{Tr}(\rho(\log\rho - \log\sigma)^2) - D(\rho\|\sigma)^2,$$ where $D(\rho\|\sigma)$...
2
votes
1answer
128 views

Prove that any Hermitian Matrix is a real linear combination of Pauli operators [duplicate]

This is an important result in Quantum Computing because it means that the Hamiltonian of a Quantum System can be encoded as a sequence of real numbers and their corresponding Pauli Operator. How do ...
2
votes
1answer
157 views

Finding Eigen Values from Quantum Phase Estimation - Using qiskit

I am trying to use the quantum phase estimation(EigsQPE) of qiskit to find the eigen values of a matrix. As I am new to quantum computing so I am confused what to measure in the circuit to derive the ...
2
votes
1answer
45 views

How do I represent my 3-qubit state in the computational basis?

I have taken the tensor product of $|0\rangle \otimes |-\rangle \otimes |+\rangle$ which resulted in the matrix $$\begin{bmatrix} 1/2\\ 1/2 \\ -1/2 \\ -1/2 \\ 0 \\ 0\\ 0\\ 0\\ \end{bmatrix}.$$ How ...