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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Tensor product and Dirac notation

Can someone shows me how to proof this equality: $\frac{1}{\sqrt2}(\alpha|000⟩+\alpha|011⟩ + \beta|100⟩ + \beta|111⟩ )$ = $ \frac{1}{2\sqrt2}[(|00⟩+|11⟩) \otimes (\alpha|0⟩+\beta|1⟩) + (|01⟩+|10⟩) \...
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2answers
52 views

Is ($|+⟩$$⟨0|$ + $|-⟩$$⟨1|$ ) similar to ($|0⟩$$⟨+|$ + $|1⟩$$⟨-|$ )?

Is ($|+⟩$$⟨0|$ + $|-⟩$$⟨1|$ ) similar to ($|0⟩$$⟨+|$ + $|1⟩$$⟨-|$ ) ? Can we just reversed it this way when doing Dirac manipulation? I try to calculate HZH = X and i need to reverse the second H
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1answer
41 views

Steps to apply Hadamard gate to $n$ qubits

Can someone shows me, step by step, how to apply Hadamard and output the result?
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2answers
250 views

What are the conditions ensuring a two-qubit density matrix is positive semidefinite?

I've seen some papers writing $$\rho=\frac{1}{4}\left(\mathbb{I} \otimes \mathbb{I}+\sum_{k=1}^{3} a_{k} \sigma_{k} \otimes \mathbb{I}+\sum_{l=1}^{3} b_{l} \mathbb{I} \otimes \sigma_{l}+\sum_{k, l=1}^{...
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0answers
26 views

Simplifying equation for two qubit syndrome extraction code

In the paper Quantum Error Correction: An Introductory Guide, the author gives the following formula for a simple two qubit code (eq. 19 on the paper). $$ E|\psi\rangle_L|0\rangle_A \xrightarrow{\text{...
2
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2answers
163 views

How to understand combination states vs pure/mixed states?

I've learned that representing a combination of two states, I simply need to take the tensor product of the states. For example: $$\left|\Psi\right>=\alpha_0\left|0\right>+\beta_0\left|1\right&...
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1answer
53 views

Finding the measurement basis for single qubit with given probability of outcome $0$

I have the general state of a single qubit $|\psi \rangle = \alpha|0\rangle + \beta|1\rangle $. Assume I am given a probability $p$ such that $0 < p <1$. Now I need to find the basis in which ...
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3answers
107 views

Is every single-qubit unitary just a rotation around some unit vector on the Bloch sphere?

I remember reading this somewhere... Is there an elegant proof for this?
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0answers
17 views

Penalty Function for XOR gate

I was reading a paper on Gates for Adiabatic Quantum Computer. In the paper, there were different penalty functions already given in the form of the following table: I do not quite understand the ...
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1answer
62 views

Strange binomial formula for operators?

Does the binomial formula $(a+b)^n=\sum_i C_n^ia^ib^{n-i}$ still work when $n$ is replaced by operator $\hat{n}$(an operator), where $a$ and $b$ are numbers? Since it's not the normal binomial formula ...
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0answers
67 views

Is decomposing high-dimensional states in terms of Pauli matrices impossible?

I've been trying to decompose a 3x3 density matrix with 3-dimensional Pauli matrices but it doesn't work for all matrices. For example, the density matrix of the state $|0\rangle + |1\rangle + |2\...
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1answer
19 views

In the context of block-encoding, what does $|0\rangle\otimes I$ represent?

New to quantum and ran into the block-encoding. Having a bit of trouble understanding $|0\rangle \otimes I$. $|0\rangle$ is just a vector but $I$ is an $n$ by $n$ matrix? Not clear how vector can be ...
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1answer
136 views

What is the eigenvalue distribution of arbitrary unitary matrices?

I had a question regarding the nature of the eigenvalue distribution of unitary matrices. Searching for the answer I found that the unitary matrices which are sampled randomly have a defined ...
2
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1answer
93 views

How is the probability of success for Simon's algorithm determined?

In step 3 of Simon's algorithm, we are told to "Repeat until there are enough such $y$’s that we can classically solve for $s$." It then goes on: The above are from this course notes. I am ...
2
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1answer
45 views

How to sample vectors close to the minimum eigenvector of a unitary matrix?

Say that we have an unknown $2^{n}\times2^{n}$ unitary matrix $U$ with eigenvectors $|v_{i}\rangle$ and eigenvalues $e^{2\pi j \theta_{i}}$and we want to sample a vector, say $|\phi \rangle$. Since ...
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0answers
35 views

Marginal output probability of first bit for constant-depth circuits

Consider a constant depth $1\text{D}$ quantum circuit, which is applied to the input state $|0^{n}\rangle$, and whose output is measured in the standard basis. You can assume that the gates of the ...
4
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1answer
94 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, \...
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0answers
97 views

Upper bound on the distance between two distinct orthonormal vectors

I need to prove that if $\phi$ and $\psi$ are distinct vectors of an orthonormal set then $|| \phi - \psi|| \leq \sqrt{2} $. Going by the definition of norm, $|| \phi - \psi||^2$ is the inner product $...
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0answers
110 views

Who was the first to call the phase gates $P(\pi/2)$ and $P(\pi/4)$ the $S$ and $T$ gates, and were they motivated by generators of the modular group?

Within the theory of quantum gates, a common pair of single-qubit phase gates are the $P(\pi/2)=S$ and $P(\pi/4)=T$ gates, with $$S= \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix},\:T = \begin{...
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0answers
12 views

Can you make anyons in 3 dimensions using rings?

I heard that anyons can only be made in 2 dimensions because when you visualize the spacetime diagram of a 2-dimensional system with point particles, you can get braids, but if you do the same with a ...
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0answers
43 views

What is a projection operation and how does it work?

I am reading about the Quantum Pigeon Hole Principle and having trouble understanding how the states are measured. Specifically from this paper. From equation (4) through equation (7).
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1answer
47 views

How to compute the eigenvector of this complex matrix in Grover's algorithm?

We know that SO(3) matrix stands for the proper rotation in 3D space. But when I read this paper, there is a SO(3) matrix stands for the general query matrix of Grover's algorithm in SO(3) form: $$ \...
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2answers
80 views

Why is there no angle for the $z$ axis in the Bloch sphere?

I see that in Bloch spheres, there is an angle for the $x$ and $y$ axes but not for the $z$ axis. Why?
2
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1answer
53 views

Mathematics Behind Deutsch-Jozsa Algorithm

I am currently learning from Nielsen and Chuang and I am currently learning about Deutsch-Jozsa algorithm. However, I am stumped with the mathematics of the algorithm at the following section: I ...
3
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1answer
155 views

A question from Aaronson 2004 paper

In Aaronson's paper about the efficient simulation of a stabilizer circuit (https://journals.aps.org/pra/pdf/10.1103/PhysRevA.70.052328), I have a problem with finding the reason why the following ...
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0answers
65 views

Expectation value of a quantum circuit [closed]

The expectation value of an operator $A$ is defined by this equation $\langle A \rangle_\psi = \sum_j a_j |\langle \psi | \phi_j \rangle|^2 $. My first question is does it mean that the expectation ...
2
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0answers
38 views

Can Grover's algorithm be applied to differential equation solving?

As I understand Grover's algorithm, given the output of a black-box function, can be used to find the corresponding input (or set of inputs if the function is not one-to-one). It is therefore ...
0
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2answers
76 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 ...
3
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1answer
40 views

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 $...
3
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0answers
50 views

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}}\...
4
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0answers
92 views

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
66 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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0answers
40 views

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 ...
2
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0answers
39 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 ...
4
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1answer
51 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
90 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
52 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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0answers
68 views

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 ...
4
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1answer
53 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 ...
1
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1answer
145 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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0answers
55 views

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 ...
3
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2answers
100 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
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1answer
99 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
79 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
53 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 ...
2
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1answer
51 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}\...
3
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1answer
67 views

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
86 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, $|\psi^*\rangle \rightarrow |\psi\rangle$
2
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1answer
53 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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