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 ...

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Why is Frobenius norm related to the inner product of characters?

This is a continuation of my question asked here. I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the ...
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Probability of measuring the label of representation in quantum Fourier transformaton

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the following function. $$ f : G \to \mathbb{C} $$ Then ...
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21 views

Commutation of Tensor Products as operators

Suppose I have unitary operators $$A: \mathbb{C}^{2^k} \rightarrow \mathbb{C}^{2^k}$$ $$B: \mathbb{C}^{2^j} \rightarrow \mathbb{C}^{2^j}$$ For some $k,j \in \mathbb{Z}, j,k \ge 0$. How do show that ...
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Commutative diagram for hidden subgroup representation of graph automorphism

The hidden subgroup representation of the graph automorphism problem is defined in the section 10.2 of QUANTUM ALGORITHMS FOR PROBLEMS IN NUMBER THEORY, ALGEBRAIC GEOMETRY, AND GROUP THEORY. It is as ...
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69 views

Quantum Mechanics Project Ideas!!! [closed]

I am in my first year in uni and I have to write a project in Quantum Mechanics. But I have been struggling with an idea for the project since I have recently started studying quantum mechanics and my ...
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16 views

Rotations on Bloch Sphere

I am trying to convince myself that the operators $R_x(\theta)$, $R_y(\theta)$, $R_z(\theta)$, are indeed the rotation operators on the Bloch sphere. Lets say we have a state vector $$|\psi \rangle = ...
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53 views

Exercising elementary toolkit for quantum computing

One of the major challenges for me (and I expect for many as a beginning students with only general maths skills) in studying quantum computation is that while the background and calculations required ...
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44 views

Difference between Schmidt decomposition and singular value decomposition

Schmidt decomposition of an operator is a useful tool of quantum information theory nowadays. Let $O$ be an operator acting on the Hilbert space $\mathcal{H}_{d_1} \otimes \mathcal{H}_{d_1}$. ...
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23 views

Books to learn tensor product on hilbert spaces

I have just started to work on Quantum Computing. I have began to read a paper which deals with tensor product on hilbert spaces. I have a had a course in functional analysis. So I don't have an ...
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32 views

Optimal Strategies in a Quantum Game

I've been playing around with problems involved in introductory quantum game theory, but I am having problems figuring out strategies in this one game. For background, consider the 2x2 Pauli spin ...
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23 views

Efficient curve fitting, and quantum computers

I have a two part question concerning curve fitting to N parameters using computers. First, is the time to find a curve fit to N data points proportional to N or is it worse? Second, is this class of ...
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36 views

Fourier transformation of the symmetric group $S_3$

I am trying to compute the Fourier transformation of the symmetric group $S_3$ following the section 4 of Quantum Computing and the Hunt for Hidden Symmetry. The multiplication table of $S_3$ is as ...
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57 views

Having Problem With Kronecker and Outer Product

I'm having an issue with some outer & Kronecker products where I am doing two different processes which should result in the same answer, but I'm getting a different answer for each. Can anyone ...
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1answer
76 views

Is this 3 Qubit state Entangled?

Is this 3-Qubit state entangled? http://prntscr.com/8vsg0b $|X\rangle=\frac{1}{\sqrt 2}~~|000\rangle+ \frac{i}{\sqrt 2}|111\rangle$ I've worked with 2-Qubit states and you can turn them into ...
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21 views

Qubit in Hilbert space in its SU(2) representation.

A qubit is normally defined as $$|q\rangle = \binom{\alpha}{\beta}= \binom{\phi_0 e^{i\theta_0}}{\phi_1 e^{i\theta_1}}$$ in a two dimensional complex Hilbert Space where $\alpha$ and $\beta$ are ...
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25 views

Probability of a qubit state

Everything I've read says that you can get the probability of a qubit state by squaring the state's component in the amplitude vector. For instance $[1/\sqrt{2}, 1/\sqrt{2}]$ and $[1/\sqrt{2}, ...
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55 views

How to plot a qubit on the Bloch sphere?

I've been reading pages such as this one: http://comp.uark.edu/~jgeabana/blochapps/bloch.html Which talk about the Bloch sphere, but I've been unable to figure out how to plot states on the sphere ...
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28 views

mapping of local Pauli operators

Let $A, B \subseteq P_n$, 2 finite sets of k-local commuting Pauli operators from the Pauli group $P_n$. Can we always a finite depth unitary $U$ such that $U^ \dagger AU=B$?
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63 views

Dimension of $\left(\lambda |\psi\rangle \langle\psi| +(1-\lambda)\frac{\mathrm{I}}{2}\right)^{\otimes N}$

I have the $N$-fold tensor product of a convex combination of a pure state, i.e. $|\psi\rangle\langle\psi|$ with $|\psi\rangle$ a unit vector in a complex Hilbert space of dimension two, and the ...
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83 views

Ignoring the workspace in quantum computation

In his book Quantum Computer Science, Mermin says that, although we'll need lots of "workspace" qubits in addition to those in the input and output registers, we can essentially ignore these in our ...
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58 views

Matrix for Control-Z gate

Can someone help me to prove that this Matrix: I - 2|11...1><11...1| represent (n-1) Control-Z gate (Z operator is applied to n-th qubit only if all remaining qubits are state |1>)
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109 views

a matrix metric

Let $U_1,...,U_n$ and $V_1,...,V_n$ be two sets of $n$ unitary matrices of the same size. We'll denote $E(U_i,V_i)= \max_v \, |(U_i -V_i)v|$ (max over all the quantum states), $U=\prod_i U_i$ and ...
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Would a quantum computer solve the Riemann hypothesis? [closed]

I heard that a quantum computer can give many results as one computation step. Does it mean that it would be just a brute force search for a quantum computer to solve for example the Riemann's ...
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101 views

A particular decomposition of a CPTP map

Let $\mathfrak{D}$ denote the set of $n\times n$, trace-one, positive semi-definite matrices (known as density matrices in quantum information theory). Consider a Completely Positive Trace Preserving ...
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114 views

Example of using the Hadamard's matrix to determine the superposition

I've came across those notes for Quantum computation from John Watrous. I am having troubles understanding the last example. We have those two vectors, or if I understood correctly, from now on ...
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82 views

Quantum Teleportation - how to prove the general case?

I've taken a course of quantum information theory and although I can compute a quantum teleportation in an explicit case where I'm given a quantum entanglement shared by Alice and Bob (normally ...
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32 views

Applying an unambiguous quantum state discrimination operator on an entangled qubit.

Given a quantum system $|\psi\rangle=\alpha_0|\psi_0\rangle\otimes |0\rangle+\alpha_1|\psi_1\rangle\otimes |1\rangle$, such that each subsystem $|\psi_i\rangle$ is entangled with a qubit is state ...
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49 views

Quantum algorithm writing

I'm interested in learning to write quantum algorithms, but I don't know where to start. Could someone recommend some resources for me?
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40 views

Coset state for non-abelian hidden subgroup problem

On page 14 of his classic paper, Quantum factoring, discrete logarithms and the hidden subgroup problem, Jozsa introduced a function $f$ for the non-abelian hidden subgroup representation of the graph ...
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62 views

Decomposition of a matrix into a product of two-level unitaries

I am working through example 4.12 and 4.13 from 'Quantum Computing: From Linear Algebra to Physical Realizations' (Pages 83-84 and 405-406) and I can't seem to figure out some what the '*' in figure ...
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3answers
261 views

Prove that the trace of a normal matrix is equal to the sum of the eigenvalues

Prove that the trace (main diagonal sum) of a normal matrix is equal to the sum of the eigenvalues. Note: For a matrix A to be normal we must have AA*=A* A where A* is the Hermitian Conjugate I am ...
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Why a tensor product of $2\times 2$ unitaries cannot implement a $3\times 3$ unitary?

Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows: $$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...
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231 views

Normalize quantum state

There are given two vectors describing quantic state: $$ x= \begin{pmatrix} e^{j^{30^\circ}}\\ 1+2j \end{pmatrix} $$ $$ y= \begin{pmatrix} 3+j\\ e^{j^{60^\circ}} \end{pmatrix} $$ How to normalize ...
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74 views

Toffoli gates can be decomposed into single and two-qubit gates

I'm not sure what the "I" and "-I" gates do. I can't seem to apply them correctly. When I do hadimard I get |00>(Tensor)Hadimard. If I then apply the tensor product to apply the 'i' gate on the last ...
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89 views

How to represent controlled quantum circuits as matrices?

I'm trying to implement (actually, simulate) the quantum algorithm by Harrow, Hassidim and Lloyd - http://arxiv.org/abs/0811.3171 - to solve linear systems of equations using a simple $2\times2$ case ...
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252 views

Notation in formula for tensor product of Hadamard matrix

I'm having trouble understanding the notation used in a linear algebra exercise (it's exercise 2.33 of Nielsen and Chuang's "Quantum Computation and Quantum Information", page 74). The exercise gives ...
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84 views

Understanding the Quantum Fourier Transform

I have a question about the Quantum Fourier Transform. I would like to understand it because I have a re-take for an exam. I have studied the provided / recommended literature extensively. ...
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42 views

Specific system of equations with multiplications

I'm facing a math problem that I thought easy, but I'm stuck with a solution that doesn't seem optimal. The problem is the following : I have "registers" which are the expanded representation of ...
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1answer
46 views

Promise of the hidden subgroup problem for $\mathbb{Z} mod 2$

I am going through the talk, Graph isomorphism, the hidden subgroup problem and identifying quantum states, by Pranab Sen. On slide 3, the promise of the problem is defined as follows. Here is the ...
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1answer
49 views

Hidden subgroup problem for $\mathbb{Z} mod 2$

The definition of the Hidden Subgroup Problem (HSP) is as follows (according to a lecture series by Pranab Sen), Let $G$ be a group, $S$ a set and $f : G \to S$ a function. We are given an ...
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Standard deviation of a quantum walk?

The standard deviation of a classical random walk with $n$ steps is $\sqrt n$ - Standard deviation of a random walk. I have read in many places that the standard deviation of a quantum walk $n$ with a ...
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258 views

In the Stinespring dilation theorem, what is the minimum dimension for which a dilation Hilbert space of this form is guaranteed to exist?

This may look like a problem that could easily be looked up, but it's not quite as easy as it first appears, hence my asking. I'm going to phrase my question in terms of the "Schroedinger picture" ...
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60 views

Quantum: Toffoli gates

How do I prove that a toffoli gate is a controlled CNOT gate. i.e, $G_{Toffoli} = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1|\otimes G_{CNOT}$. I am not sure how to approach this, thoughts? ...
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1answer
201 views

Rotation of the Bloch Sphere

I was reading through the book "Quantum Computation and Quantum Information for Computer Scientists", and I got up to a problem about rotation matrices on the block sphere and I can't figure it out at ...
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59 views

a quantum algorithm with high probability on a 4 to 1 function

Let $f : $ {0,1}$^n \rightarrow $ {0,1}$^n$ be a 4-to-1 function, such that there exist distinct and non-zero $a,b\in $ {0,1}$^n$ such that for all $x\in$ {0,1}$^n$: $f(x) = f(x ⊕ a) = f(x ⊕ b) = ...
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Hermitian - Using the Spectral Theorem

Ok so I am trying to show that if A is normal and has real eigenvalues, then A is hermitian. It was suggested that I try using the spectral theorem. So if we assume A is normal and has real ...
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162 views

Quantum Computing - Conjugate Transpose and Tensor Products

Show that $(A^T)^T = A$, where T is the conjugate transpose of the matrix. $$\left|\psi\right\rangle = \alpha_0\left|0\right\rangle + \alpha_1\left|1\right\rangle\; ...
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73 views

Conveity of quantum Fisher information

Let R be any quantum state and X any observable, then we define F(R,X)=4Tr(L^2 . R) as the quantum Fisher information. Where L is the logarithmic derivative determined by i(R.X - X.R)=(L.R + R.L)/2 I ...
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1answer
49 views

Continous and Discrete basis, Multiplication of Density Matrix and Hamiltonian.

Suppose I have a wave function $\psi(x)$ in position basis. I can make a density function by simply multiplying $\psi(x)$ and its conjugate $\psi^*(x)$. If I operate the density matrix ...
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87 views

Defining conditional quantum probability

My knowledge of quantum mechanics is very limited, but I will try to ask a purely mathematical question here. If there is a text or resource that explains this, I would definitely appreciate any ...