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.

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38 views

How do I apply a matrix to a ket state?

If we have the following matrix: $$\frac{1}{\sqrt{2}}\begin{pmatrix}1&1&0&0\\ 1&-1&0&0\\ 0&0&1&-1\\ 0&0&1&1\end{pmatrix}$$ How do we find the output for ...
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1answer
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Unit vanishes in the Quantum Cramer-Rao Bound?

The Quantum Cramer-Rao Bound states that the precision we can achieve is bounded below by: $$(\Delta \theta)^2\ge\frac{1}{mF_Q[\varrho,H]},$$ where $m$ is the number of independent repetitions, and $...
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Modeling building blocks for quantum computation

If I would design library for quantum computation I would naively consider a sequences of entangled qudits with unit length as a building blocks. I.e., unit length elements from $$\mathbb{C}^{d_{1}}\...
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Is the No-Cloning Theorem Violated in $C^\ast$-Circuit Models?

In Cleve, et al., the authors discuss self-embezzlement of a catalyst state $\psi$, making the statement on page 2, [B]y local operations, state $\psi\otimes(\vert 0 \rangle \otimes \vert 0 \rangle)$ ...
3
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1answer
65 views

How do I show that $R_z(\theta)=e^{-iZ\theta/2}$?

I know that an $R_z (\theta)$ gate is equivalent to the unitary transformation $e^{-iZ * \theta/2}$ but I'm not sure how we get there. I know that for every Hermitian matrix there is a corresponding ...
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Why is it so important to have uniform chain lengths in a minor embedding?

Very brief background In quantum annealing, the discrete optimization problem we wish to solve (such as finding the minimum of $b_1b_2 - 3b_1 + 5b_3b_4$ for binary variables $b_i$) may have a ...
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0answers
34 views

What happens in the Cramer-Rao bound if the quantum Fisher information is zero?

The famous Cramer-Rao bound is $$\Delta\theta\ge\frac{1}{\sqrt {F[\rho,H]}}$$ But what happens if the denominator vanishes, i.e., $F[\rho,H]=0$ ($F[\rho,H]$ here stands for the quantum fisher ...
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1answer
50 views

Upper bounding a permutation invariant state

Let $\rho_{A^n}$ be a permutation invariant quantum state on $n$ registers i.e. $\pi(A^n)\rho_{A^n}\pi(A^n) = \rho_{A^n}$ for any permutation $\pi$ among the $n$ registers. If we trace out $n-1$ ...
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1answer
87 views

Math behind Rz Gate

After reading some books I'm very confused about one question. For example in Nielsen and Chuang chapter 4.2 the universal gate U could be used to construct the Rz gate, which means a turn around the ...
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1answer
50 views

How the arguments of $U_3$ gate are converted when they're not lying in the expected range?

From the qiskit documentation (here), a general form of a single qubit unitary is defined as $$ U(\theta, \phi, \lambda) = \begin{pmatrix} \cos\left(\frac{\theta}{2}\right) & -e^{i\lambda} \sin\...
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Why there're two axis of rotation when I'm trying to visualize this time-evolution?

This is a follow-up question of the problem I posted earlier. The following diagram illustrates my question: I'm trying to perform the time evolution of a random Hamiltonian. The green vector ...
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1answer
50 views

Why did I get two solutions to solve for the parameters of this $U_3$ gate? (I only expected one of them)

I have the following complex vector in $\mathbb{C}^2$: Vec= [[ 0.89741876+0.j] [-0.33540402+0.28660724j]] I'm trying to implement a $U_3$ gate to prepare this ...
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1answer
140 views

Adiabatic Quantum Computer e intermediate Hamiltonian evolves the state within the manifold

The Adiabatic Quantum Computer is implemented by slowly increasing the parameter s from 0 to 1 in the intermediate Hamiltonian $[\hat{H}(s) = \hat{H}_{input} + (1-s)\hat{H}_{init} + s\hat{H}_{circuit}]...
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How to calculate $\operatorname{Var}[H]$ in the context of VQEs?

Given a Hermitian operator $H$, I can calculate the variance of the Hamiltonian $Var[H]$ as $$ Var[H] = \langle H^2 \rangle -\langle H \rangle^2 $$ Now, $H$ can be decomposed as $$ H = \sum_i \alpha_i ...
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2answers
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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 $B$? ...
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1answer
94 views

How would I theorise a quantum query algorithm in O(1)?

I am currently attempting to solve a problem from Nielsen-Chuang, and I can't seem to figure out how I would do this; I'm trying to implement Grover's algorithm to solve the problem of differentiating ...
3
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1answer
77 views

How can I generate a quaternion using Qiskit?

I noticed there's a Quaternion class in qiskit docs (Here). I've seen there're a couple of methods such as norm and ...
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1answer
51 views

On what basis can we write a positive operator as $A=\sum_k\lambda_k|k\rangle\langle k|$?

In Nielsen & Chuang's book equation 2.172 says $$A=\sum_{i}|\widetilde{\psi_i}\rangle \langle \widetilde{\psi_i}| = \sum_j |\widetilde{\phi_j}\rangle \langle \widetilde{\phi_j}|.$$ Then it makes ...
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1answer
46 views

Why is the subscript like this in the equation $\sum_i |\psi_i\rangle \langle\psi_i| = \sum_{ijk} u_{ij} u_{ik}^{*}|\phi_j\rangle \langle\phi_k|$?

In Nielsen's book when proving "Unitary freedom in the ensemble for density matrices"(Theorem 2.6): $$\text{Suppose }|\widetilde{\psi_i}\rangle = \sum\limits_{j}u_{ij} |\widetilde{\phi_j}\...
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1answer
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From mathematical notation to quantum circuit, in general

I am learning the basics of quantum computing using Qiskit and I encountered a problem when I tried to solve some of our course exercises. I feel like I am missing an invisible step, the step from ...
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2answers
66 views

How does the sum of two operators act on a two-level system of qubits?

I am confused how the sum of N operators will act on an N-level system of qubits. Here, lets say N=2 so the state is $|00⟩_{CD}$. Then how will this operator $ X_{C} + Z_{D} ⊗ I_{C} + X_{D}$ act on ...
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3answers
40 views

How to prove that the transpose operation maps an arbitrary qubit to its complex conjugate?

How to prove that the transpose operation maps an arbitrary qubit to its complex conjugate, $|\psi^*\rangle \rightarrow |\psi\rangle$
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1answer
50 views

Show algebraically that $U|0\rangle\otimes |0\rangle+U|1\rangle\otimes|1\rangle=|0\rangle\otimes U^T|0\rangle+|1\rangle\otimes U^T|1\rangle$

Suppose that Alice applies a unitary operator $U$ with real entries to her qubit in an EPR pair $|\beta\rangle=\frac{1}{\sqrt 2}(|00\rangle+|11\rangle)$. Is this the same as having Bob apply $U^T$ to ...
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1answer
39 views

How would I compute a density matrix of a complex qubit mixed state?

I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed state, $$ \frac{1}{9}\begin{bmatrix} 5 & 1 & −i \\ 1 & 2 &...
2
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1answer
46 views

How would I compute a density matrix of a 2 qubit mixed state?

I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed states, how would I do this? $$ |00> \;with \;probability \; 2/4 \\ ...
2
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1answer
39 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 ...
3
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1answer
67 views

What does the identity operator represent when computing $\langle\varphi|I\otimes Z|\varphi\rangle$?

Consider a single qubit state $|\varphi\rangle$ and a hamiltonian $H = Z$. Evaluating $\langle \varphi | H | \varphi \rangle$ corresponds to a measurement of $|\varphi\rangle$ in the computational ...
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1answer
181 views

Condition that a tripartite/multipartite qubit state does/does not admit a Schmidt decomposition?

I saw answers such as this and this which provide examples of tripartite system that don't take a Schmidt decomposition, but I wonder if there's an explicit condition that can tell whether a state is ...
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0answers
47 views

Will there be any difference in solution for a weighted and unweighted graph? I mean is there any relation between weight and solution?

I was working on max-cut problem. To do I have been given a graph of unweighted version, and I have to convert it into weighted version. I did it using qiskit. Now I was playing around the code, hence ...
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2answers
59 views

Is there a simplified formula for the adjoint of the outer product of ket and bra?

I was reading about measurements and got to some operator like this: $$\left| 0\rangle \langle 0\right| $$ Is there any form I can apply when I have to calculate $$ \left( \left| 0\rangle \langle 0\...
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2answers
70 views

Find the unitary implementing the transformation $|0\rangle\to\frac1{\sqrt2}(|0\rangle+|1\rangle),|1\rangle\to\frac1{\sqrt2}(|0\rangle-|1\rangle)$ [closed]

I have found a question for finding the Unitary operator for the following transformation: I found the solution as well. But I didn't understand how they got the solution!
4
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2answers
52 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\...
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3answers
376 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 ...
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1answer
77 views

Why isn't $\{1,2,3\}$ well ordered? [closed]

I was reading the book "Quantum Computing Since Democritus". "The set of ordinal numbers has the important property of being well ordered,which means that every subset has a minimum ...
1
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1answer
32 views

Formula for a single unit gate acting on a lexicographically represented state

I've been trying to find in textbooks a discussion on the action of arbitrary single qubit gates on a lexicographic state. That is, given an operator $G_{l}= 1\otimes 1...\otimes G \otimes ... \...
3
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1answer
85 views

How do I determine if a given pure two-qubit state is separable?

I'm trying to self-study some topics about quantum computing and I came across a topic of state separability. Talking about that, I wanted to determine separability on the following state (from Qiskit ...
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1answer
77 views

what matrix operations have better known time complexity on a quantum computer?

I'm exploring quantum computers for a semester project. I'm mainly interested in making faster matrix calculations than a regular computer. I was wondering what arithmetic operations (irrespective of ...
2
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2answers
129 views

Qiskit order of multiplication and tensor product

Section 2.3 of the Qiskit textbook shows us that $$ CNOT|0+\rangle \ = \ \frac{1}{\sqrt2}(|00\rangle + |11\rangle)$$ which I was able to translate to a circuit as such: ...
3
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3answers
78 views

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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502 views

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 ...
2
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1answer
52 views

How is $\sum_i\langle i|M|i\rangle$ correlated to $\mathrm{tr}(M)$?

In the book Quantum computation and quantum information, it says to evaluate $tr(A|\psi\rangle\langle\psi|)$ using Gram-Schmidt procedure to extend $|\psi\rangle$ to an orthonormal basis $|i\rangle$ ...
3
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2answers
71 views

Is a projective measurements over a superposition of eigenstates possible?

All observables admit a spectral decomposition in terms of projectors $P_m$ into the eigenspace corresponding to the eigenvalue $m$. So given for example a collection of kets $|0\rangle, |1\rangle,...,...
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4answers
114 views

Derivation of the identity $\sum_j p_j \langle \psi_j|M|\psi_j \rangle = \sum_j p_j \operatorname{tr}\left(|\psi_j \rangle \langle \psi_j|M\right)$

For measurement, we know $$\langle M \rangle = \sum_j p_j \langle \psi_j|M|\psi_j \rangle = \sum_j p_j \operatorname{tr}\left(|\psi_j \rangle \langle \psi_j|M\right).$$ My question is, how can we go ...
2
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2answers
96 views

What is the relation between observables (as defined in the measure-theoretic framework) and POVMs?

A POVM is typically defined as a collection of operators $\{\mu(a)\}_{a\in\Sigma}$ with $\mu(a)\in\mathrm{Pos}(\mathcal X)$ positive operators such that $\sum_{a\in\Sigma}\mu(a)=I$, where I take here $...
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1answer
59 views

Uniqueness of Density Operator

I have been reading "Introduction to Quantum Information Science" by Masahito Hayashi, Satoshi Ishizaka,Akinori Kawachi, Gen Kimura and Tomohiro Ogawa; Springer Publication. I'm currently in ...
4
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3answers
121 views

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 ...
2
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2answers
41 views

How to find the normalization factor of the eigenvectors of the $\sigma_x$ Pauli gate?

I'm trying to calcaute the eigenstates for the $\sigma_x$ gate, and I can follow the process up to finding eigenvalues $\pm 1$, but I don't understand where the $\frac{1}{\sqrt{2}}$ coefficient comes ...
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1answer
51 views

Compute the squared overlap between different given qubit states

I was checking this problem from the book. And here is an example, but I think it's wrong. If it is not wrong can you please explain how did they derive it? As per my workout, it should be one. But It ...
0
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0answers
29 views

RAC for XOR functions

I need the optimal encoding protocol for 3 $\rightarrow$ 1 Classical RAC such that the receiver is able to retrieve any one of the initial bits, as well as the XOR combinations of those bits. ( If a, ...
5
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1answer
127 views

In Nielsen and Chuang, how can $\frac{1}{2(e-1)}$ result from $\frac12\int_{e-1}^{2^{t-1}-1}dl\frac{1}{l^2}$?

From Nielsen and Chuang's book: $\textit{Quantum computation and quantum information}$, how can (5.34) equal (5.33)? I.e. $$\dfrac{1}{2} \int_{e-1}^{2^{t-1}-1} dl \dfrac{1}{l^2} = \dfrac{1}{2(e-1)}.$$...

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