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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Unitary transform of a partially entangled 3-qubit state

Suppose we have a partially separable 3-qubit state $$φ = \left(a_0\left|0\right\rangle + a_1\left|1\right \rangle\right) \otimes \left(b_{00}\left|00\right \rangle + b_{01}\left|01\right \rangle + ...
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Entanglement of 3-qubit states

Given a separable 3-qubit state φ = φ0 ⊗ φ1 ⊗ φ2 with φi= ai0|0> + ai1|1>, |0>, |1> being the computational base. φ thus can be written as φ = b000|000> + b001|001> + b010|010> + b011|011> + ...
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Gradient of piece wise constant quantum control problem to steer system evolution to a target state

I'm looking for an exact gradient for the piece wise constant control of a quantum system to steer it towards a desired state at time T. It is worth mentioning, the Hamiltonians have been expanded ...
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57 views

Topology of non-degenerate $n\times n$ Hermitian matrices

.. where I guess by topology I mean its homology / homotopy groups. Here "degenerate" means having repeated eigenvalues. This is interesting because it defines the space that can be explored via ...
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39 views

Does Bennett's copy-uncompute trick always work for quantum computing?

In the proof to the theorem that $BQP \subseteq PP$, it is assumed that the quantum circuits C starts with $x0^{m-n}$, and ends with $10^{m-1}$ or $00^{m-1}$. So we can get the acceptance probability ...
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22 views

exercise 10.34 in Nielsen & Chuang <Quantum Computation and Quantum Information>

Let $S=\langle g_1 , g_2, \cdots , g_l \rangle $ be a subgroup of a $n-$fold Pauli group $P_n$ generated by $g_1, g_2, \cdots, g_l$. Exercise 10.34 in Nielsen & Chuang requires to show that $S$ ...
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39 views

Function that grows faster then BB(n) using quantum computation?

Is it possible to define a function in terms of quantum computers that grows faster then any that is defined in terms of turing machines?
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14 views

Using hadamard transform after a CNOT transform

I saw that mathematica already had other quantum computer questions here, so I think this is the best stack-exchange to ask it. Lets say you have two qubits. The first is hadamard transformed and the ...
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33 views

Linear algebra textbook for quantum computing?

I'm looking for an recommendation for a linear algebra textbook specifically to give me the background for learning about quantum computing, and quantum mechanics more generally. In particular, none ...
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15 views

Considering $Res^G_{H_\rho}$ instead of $G$ in quantum Fourier sampling

I am going through the proof of theorem 4 in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. Here, they are trying to calculate the probability of measuring the ...
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38 views

Can anyone answer this equation or tell me what it is? [closed]

I have been sent an image, and have no idea what it means. Think this is the best site to try. The image is of the equation $$W_T=\sum_{i=1}^nW_i\;.$$
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22 views

What is the lovasz theta number of complement of Grötzsch graph?

Lovasz theta number: (It is said to be polynomial, but I do not know how to computer it) https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Grötzsch graph: https://en.wikipedia.org/wiki/Gr%C3%...
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1answer
37 views

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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1answer
20 views

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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23 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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20 views

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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1answer
83 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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26 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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57 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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75 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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27 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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1answer
39 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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39 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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1answer
59 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
81 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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23 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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1answer
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}, -1/\...
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80 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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1answer
71 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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77 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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1answer
111 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 $V=\...
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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 ...
2
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106 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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1answer
117 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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86 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 1/root(...
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1answer
34 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 $|i\...
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50 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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41 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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69 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
292 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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1answer
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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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117 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 2 ...
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1answer
100 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 ...
6
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1answer
287 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 ...
3
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87 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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1answer
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 ...