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 ...

learn more… | top users | synonyms

0
votes
0answers
26 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 ...
0
votes
0answers
12 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 ...
-4
votes
1answer
36 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 $$W_T=\sum_{i=1}^nW_i\;.$$
1
vote
0answers
21 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: ...
1
vote
1answer
32 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 ...
1
vote
1answer
19 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 ...
0
votes
0answers
22 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 ...
2
votes
0answers
19 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 ...
2
votes
1answer
78 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 ...
0
votes
0answers
17 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 = ...
2
votes
0answers
54 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 ...
1
vote
0answers
53 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 $\mathcal{H}_{d_1} \otimes \mathcal{H}_{d_1}$. ...
1
vote
0answers
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 ...
3
votes
1answer
33 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 ...
0
votes
0answers
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 ...
1
vote
0answers
37 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 ...
2
votes
1answer
58 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 ...
2
votes
1answer
78 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 ...
0
votes
0answers
22 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 ...
0
votes
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
vote
1answer
64 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 ...
0
votes
1answer
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 $U^ \dagger AU=B$?
2
votes
1answer
66 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 ...
1
vote
1answer
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 ...
1
vote
0answers
65 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>)
3
votes
1answer
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 ...
-2
votes
2answers
158 views

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
votes
0answers
103 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 ...
1
vote
1answer
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 ...
2
votes
0answers
85 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 ...
0
votes
1answer
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 ...
1
vote
0answers
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?
1
vote
0answers
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 ...
0
votes
0answers
64 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 ...
1
vote
3answers
271 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 ...
4
votes
0answers
337 views

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) ...
-1
votes
1answer
240 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 ...
0
votes
1answer
90 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 ...
0
votes
1answer
93 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
votes
1answer
271 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
votes
0answers
85 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. ...
1
vote
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 ...
3
votes
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 ...
3
votes
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 ...
5
votes
0answers
96 views

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 ...
1
vote
0answers
270 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" ...
1
vote
1answer
61 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? ...
2
votes
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 ...
1
vote
0answers
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) = ...
1
vote
2answers
40 views

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 ...