Questions tagged [quantum-computation]

Quantum Computation deals with considering computation as fundamentally physical, as well as replacing the classical binary digit (bit) with the quantum binary digit (qubit). While the classical bit is either 0 or 1, the qubit can be in a superposition of these states. Computation systems that use quantum phenomena, such as superposition and entanglement, can solve certain complex problems very quickly.

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Generators of $U(2)$

I'm reading a book on quantum computing. In the book it says that any linear optical element (represented by the set of unitary matrices $U(2)$) is equivalent to a combination of balanced phase ...
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Condition for positive semidefinite operator in Hilbert space

In Nielsen and Chuang exercise 2.64, the following problem is given: Suppose Bob is given a quantum state chosen from a set $\{ \lvert \psi_1 \rangle, \ldots , \lvert \psi_m \rangle \}$ of linearly ...
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Is $\mathsf{SU}(2) \otimes \mathsf{SU}(2) \otimes \mathsf{SU}(2)$ unitarily conjugate to a subgoup of $\mathsf{SO}(8)$?

We know that there is an isomorphism $\mathsf{SU}(2) \otimes \mathsf{SU}(2) \to \mathsf{SO}(4)$ given explicitly by $M \mapsto Q^\dagger M Q$ where $$ Q=\frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 0 &...
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Exercise 10.2 Nielsen and Chuang Quantum Computation and Quantum Information

The question asks to show that the action of the bit flip channel described by the quantum operation $$\epsilon(\rho)= (1 - p) \rho + p X\rho X$$ can be given an alternative operator-sum ...
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Physical interpretation of operator-sum measurement

In Nielsen and Chuang, on page 362, in the attached reference, why do we have $$|e_k\rangle \langle e_k|U(P \oplus |e_0\rangle \langle e_0|)U^{\dagger} |e_k\rangle \langle e_k|$$ instead of $$|e_k\...
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How to find define the relative phase of a qubit

Given a complex, two-dimensional vector space, let the vector $ |V\rangle = r_0 e^{i\theta_0} |0\rangle + r_1 e^{i\theta_1} |1\rangle $ correspond to the state of a qubit where $r_0,r_1,\theta_0,\...
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Understanding the Irreps of a Particular Representation of Symmetric Group Product

I'm reading this paper on the element distinctness problem, and I'm having some trouble parsing Claim 2. I've recently been going through The Symmetric Group by Sagan (Chapters 1 and 2 so far). Here's ...
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Understanding a step in the proof of Solovay-Kitaev theorem

There is a step in the proof of the proof of Solovay-Kitaev theorem about the existence of a set containing words of at most length length $l_0$ that cover $SU(2)$ . The proof I'm reading in given in ...
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Condition number of a non-invertible matrix and solving with quantum linear systems algorithms

In this paper one of the things they do is solve the Poisson equation with periodic BCs by using the finite difference representation then using a quantum linear systems algorithm to solve the ...
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Find $\beta$ block matrix given $\tilde{E}_m\rho_j\tilde{E}_n^\dagger=\sum_k\beta_{jk}^{mn}\rho_k$

Given $\tilde{E}_1=I,\tilde{E}_2=X,\tilde{E}_3=-iY,\tilde{E}_4=Z$ and $ \rho_1=|0\rangle\langle 0|,\rho_2=|1\rangle\langle 0|=X\rho_1,\rho_3=|0\rangle\langle 1|=\rho_1X,\rho_4=|1\rangle\langle 1|=X\...
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Explicit computation of Choi matrix for a qubit channel

I'm struggling with the explicit computation of the Choi matrix of a generic quantum channel ${\Phi}:\mathbb{C}^2\to\mathbb{C}^2$. I know that I can write the channel in the Bloch representation as ${\...
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Cryptography or quantum computing

I’m currently a senior applied mathematics student, I love math, and plan to study master in France. I’m choosing whether to pursue master in cryptography or quantum computing. (In fact, I find some ...
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A question about product representation of quantum Fourier transform

In the Nielsen and Chuang's Quantum Computing and Quantum Information, the last step of proving the product representation of quantum Fourier transform is $$ \frac{1}{2^{n/2}}\bigotimes_{l=1}^{n} \...
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the quantum world: is $f$ balanced or constant? [closed]

Why here on the page 15, the fact whether $f$ is constant or balanced is derived from the value of $f$ at $0$ only ?
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Trouble proving approximation for quantum simulation using Zassenhaus formula

I would like to prove the following $e^{i(A+B)\Delta t} = e^{\frac{iA\Delta t}{2}}e^{iB\Delta t}e^{\frac{iA\Delta t}{2}} + \mathcal{O}(\Delta t ^3)$. I have tried to prove it using Zassenhaus formula ...
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Affine map of single qubit quantum operations

In my reference, Page 375, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that The Pauli matrices, along with the identity matrix $I$, form an orthonormal ...
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What does Tensor do to the values next to the vector?

enter image description here Like you can see in the picture, there is a 1/SQR(2) What exactly happens to this value when you use tensor. I assumed that you would just multiply the fractions leaving ...
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Prove that these two vectors represent the same state

Two vectors represent the same state if the relative phase is the same from what I've learned. However, I do not understand the approach in proving if these two vectors represent the same state ...
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How to efficiently compute logarithm of det of a mixed state density matrix

Given a strictly mixed state density matrix $\rho$, produced by an ensemble of quantum states {$\{p_i,\left|\psi_i\right\rangle\}$}, how may I efficiently compute or approximate the value $\log\det(I+\...
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Prove if $E_k=(I\otimes\langle e_k|)U(I\otimes|e_0\rangle)$ then $U=\begin{bmatrix}[E_1]&.&.\\ [E_2]&.&.\\ [E_3]&.&.\\ .&.&.\end{bmatrix}$

In my reference, Page 365, Operator-sum representation, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that We have a principal system $Q$ and an ...
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Learning Bernoulli Parameter (problem related to a Quantum Algorithm)

The LGZ algorithm for Topological Data Analysis via computing Betti Numbers is a quantum algorithm that relies on sampling random eigenvalues of an appropriate linear operator. The expected fraction ...
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Connection between vector scalar product and trace of matrix product

Given 2 vectors $\vec{s^{\prime}}$, $\vec{t^{\prime}}$ and $S^{\prime}=\vec{s^{\prime}} \cdot \vec{\sigma}$, $T=\vec{t^{\prime}} \cdot \vec{\sigma}$. The following equations are given \begin{equation} ...
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How to obtain the unitary operator to get specific partial trace?

Is there a unitary $U_{AB}$ such that, for any density operator $\rho$, we have $${\rm {Tr}}_A [U_{AB} (\frac{I_A}{2} \otimes \rho_B)U_{AB}^{\dagger}]= \frac{\rho_B}{2}+\frac{I_B}{4}$$ $${\rm {Tr}}_B [...
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Prove that $tr\Big(\sum_k E_k^\dagger E_k\rho\Big)=1$ for all $\rho$ implies $\sum_k E_k^\dagger E_k=I$

In my reference, Page 360, Operator-sum representation, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that \begin{align} 1&=tr\Big(\mathcal{E}(\rho)\Big)...
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Trace function on the quantum expectation

In the article "Barren plateaus in quantum neural network training landscapes", the objective function $E(\theta)$ is defined as $$ E(\theta) =i\langle{0|U(\theta)^\dagger H U(\theta)|0\...
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Relation between tensor product of Hilbert spaces and the Kronecker product of their elements

Consider a task of calculating the basis coefficients of a vector belonging to a tensor product of two Hilbert spaces. Some definitions first. Take two Hilbert spaces $(A,(\cdot,\cdot)_A)$ and $(B,(\...
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What does CPTP mean to the $N^2 \times N^2$ matrix form of quantum channels?

This question might be basic knowledge, but I found nothing on the internet. So any reference would also be highly appreciated! I have the quantum channel regarded as an $N^2 \times N^2$ matrix (when ...
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Finding probabilities of outcomes of a qubit

Given a qubit in state $\mid 0\rangle$, first find the probabilities of the outcomes from measuring it in the basis of $\mid a\rangle =\dfrac{1}{\sqrt{2}}(1,i)$ and $\mid b\rangle =\dfrac{1}{\sqrt{2}}(...
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How is the basis $\{\frac{1}{\sqrt{2}}(1, i),\frac{1}{\sqrt{2}}(1, -i)\} $ orthonormal

How are the bases $\frac{1}{\sqrt{2}}(1, i)$ and $\frac{1}{\sqrt{2}}(1, -i)$ orthonormal? I know for a basis to be orthonormal it must have a dot product of zero and a norm of $1$. Yet, while the dot ...
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Equate $e^{-i|Ψ⟩⟨Ψ|Δt}e^{-i|x⟩⟨x|Δt}$ to $(\cos^2Δt/2-\sin^2Δt/2ψ.z)I\\-2i\sinΔt/2[\cosΔt/2\frac{ψ+z}{2}+\sinΔt/2\frac{ψ\times z}{2}].\vec{σ}$

In my reference, Page 259, Quantum Computation and Quantum Information by Nielsen and Chuang it is given that Given the unitary operator $U(\Delta t)\equiv \exp(-i|\psi\rangle\langle\psi|\Delta t)\...
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2 votes
1 answer
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Precise relationship between classical and quantum Fourier transform for a finite abelian group

$\newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ket}[1]{|#1\rangle} \newcommand{\bra}[1]{\langle#1|}$ I asked this question on the quantum computing SE site a few weeks ago, but ...
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What does the dot dot dot mean in math?

Can someone please explain what the three dots mean? I knew this at some point but I have memory issues so I have to relearn things a lot. U (θ) = UL(θL) ··· U2(θ2)U1(θ1) |ψ〉 = ] φp ∈{φ} aφ1···φη |φ1 ·...
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Sum of square of Bessel functions

I'm working on the 1-D continuous quantum walk and I encounter the following problem: Let $J_n(x)$ be the Bessel function of the first kind of order $n$. (In this post, I consider only real $x$ and ...
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Show that $(1-it)|\psi\rangle-it\langle x|\psi\rangle|x\rangle$ is $|\psi\rangle$ rotated into the $|x\rangle$ direction

Here $|\psi\rangle$ is an $n$ component unit vector with $|x\rangle$ being one of the unit basis vectors, i.e., $|\psi\rangle=a_1|x_1\rangle+a_2|x_2\rangle+\cdots+a_x|x\rangle+\cdots+a_n|x_n\rangle$ ...
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Proof of the tensor contraction formula for the quantum circuit

Could you please verify a derivation of the tensor contraction formula (e.g. see wiki on Partial Trace), made in the style of Vannucci, Tensor algebra and analysis page 9 equation above (2.4). Unlike ...
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Accuracy of Classical Counting problem

Consider a classical algorithm for the counting problem which samples uniformly and independently $k$ times from the search space, and let $X_1, ... ,X_k$ be the results of the oracle calls, that is, $...
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Why does this circuit for Heisenberg XXZ model look like this?

So for the XXZ Heisenberg model taken form https://www.tensorflow.org/quantum/api_docs/python/tfq/datasets/xxz_chain: why is the X interaction part shown as such (with XX, HI, and CNOT): I wrote out ...
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Set of Density Operators with Separable Eigenbasis

Let $\mathbb{H}^A$ and $\mathbb{H}^B$ be finite dimensional Hilbert spaces. Consider the set $S$ of all bipartite density operators $\rho \in D(\mathbb{H}^A \otimes \mathbb{H}^B)$ such that every ...
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Grover's Algorithm Trigonometric doubt

The Grover's operator $G$ in Quantum Computing, has the following effect $$ G|\Phi\rangle=(4\sin^2\Delta -1)|\Phi\rangle-2\sin\Delta|z\rangle $$ where $|\Phi\rangle=\frac{1}{\sqrt{2^n}}(|x_1\rangle+|...
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Trace distance of tensor products of operators.

Let us focuse on the set of positive operators of trace 1 acting on some infinite dimensional Hilbert space, call it $S(H_{1})$ where $H$ is the mentioned Hilbert space. Let $S(H_{2})$ be another such ...
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Discrete Logarithm Problem as Period finding of a function

The discrete logarithm problem (DLP) : Find $b$ knowing $s,a$ and $p$ such that $$b=a^s\mod p$$ where $p$ is a prime number and $a$ is a generator of the group defined by $p$. It is stated that the ...
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Justify $P(q|s_i)\le\dfrac{1}{q}$ given $q$ is any prime and $s_i\in\{0,1,\cdots,r-1\}$

The probability that $s_1$ and $s_2$ have no common factors is given by $$ 1-\sum_q P(q|s_1)P(q|s_2)\ge 1-\sum_q \frac{1}{q^2} $$ where $q$ is any prime number and $s_1,s_2$ are chosen uniformly at ...
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Difference between Quantum Mechanics and Quantum Computing

Currently I'm reading Wolfgang Scherer's Mathematics of Quantum Computing, An Introduction. I am really enjoying this book as it treats the subject from a mathematical point of view, which for me is ...
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determine degree of boolean polynomial given as black box

I searched a lot but couldn't find a good resource that addresses this question. Given a boolean polynomial with $n$ boolean variables as a black box, what is the most efficient way to compute its ...
1 vote
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Approximating an exponent of non-commutative matrices as a product of exponents

The book Quantum Computation and Quantum Information chapter 4.7.1 presents the following equation. \begin{equation} e^{i(A+B)\Delta t} = e^{i A \Delta t}e^{i B \Delta t} + O(\Delta t^2) \end{equation}...
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log det on density matrix plus identity

A very naive question: given a pure quantum state $|\phi\rangle$, and the associated density matrix $\rho=|\phi\rangle\langle\phi|$, does there exist an efficient quantum operator/procedure that gives ...
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Is a unital completely positive map taking this form trace-increasing?

I just came across this problem, in which we consider a CP map $$ \Phi(\rho)=\sum_i K_i\rho K_i^\dagger. $$ Now, let us transform $\Phi$ into a unital map, by composing it with a Kraus rank-1 map $\...
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Unitary operator times projection operator

In this paper, the authors claim that for $C$ a unitary operator and $P$ a projection operator, if $CP \propto P$, then the constant of proportionality must be one. I don't see why this must be the ...
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1 vote
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Does quantum logic describe quantum logic gates?

From what I've gathered, there are multiple quantum computational logics. But, I've been having difficulty figuring out whether they subsume quantum logic or how much overlap there may be. I even ...
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Doubts regarding Local Hamiltonian Problem

I have background in CS but not in Quantum Computing/Quantum Complexity Theory. I am trying to understand the Local Hamiltonian Problem (the formal definition as below): Local Hamiltonians or Q5SAT: ...

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