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0answers
11 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>)
0
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0answers
35 views

a matrix metric

Let $U_1,...,U_n$ and $V_1,...,V_n$ be two sets of $n$ 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=\prod_i ...
1
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0answers
26 views

Creating an arbitrary state of the quantum simple harmonic oscillator [migrated]

Suppose $\mathcal{B}=\{\lvert 0\rangle, \lvert 1\rangle, \lvert 2\rangle, ... \}$ is the energy eigen-basis of a quantum simple harmonic oscillator. I want to create the state \begin{equation} ...
-2
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2answers
82 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
65 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
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1answer
88 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
55 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
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1answer
15 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
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0answers
35 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?
0
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0answers
11 views

Is it possible to obtain a Hermitian, positive semidefinite matrix as some sum of non-commuting matrices?

I am working with generalized Pauli matrices given by $X \vert j \rangle = \vert (j+1)mod~p \rangle$, where $p$ is a prime number. $Z = \vert j \rangle = \omega \vert j \rangle$, where $\omega = ...
1
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0answers
34 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
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0answers
18 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
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3answers
74 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
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0answers
284 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) ...
0
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0answers
20 views

How can projection operator be expressed in form $\frac{1}{d} (I + \sum_i r_i \lambda_i)$?

How can projection operator be expressed in form $\frac{1}{d} (I + \sum_i r_i \lambda_i)$? I was reading a papar and found out that the desity matrix in $d$-dimensional Hilbert Space can be expressed ...
-1
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1answer
46 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
33 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
38 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 ...
3
votes
1answer
113 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
72 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
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1answer
41 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
42 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
47 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
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0answers
64 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
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0answers
129 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
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1answer
52 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
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1answer
151 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
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0answers
55 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
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2answers
38 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 ...
0
votes
1answer
116 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\; ...
1
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1answer
55 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 ...
2
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1answer
39 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 ...
5
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1answer
72 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 ...
1
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0answers
52 views

If P = NP can asymmetric key exchanges still exist?

One functions are easy to compute (ie polynomial time checking) but hard to reverse. if P = NP does that mean that asymmetric key exchanges will be reduced from polynomial computation time and ...
0
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1answer
34 views

Special Quantum Gate

Given $\alpha|1\rangle+\beta|0\rangle$, to transform to $|\phi\rangle=\alpha|0\rangle+\beta|1\rangle$ , we use Quantum NOT gate: NOT = $\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$. For ...
0
votes
1answer
45 views

Pauli Spin Operator General Rotation

I would like to calculate the Pauli spin operator rotation $$ U^{\dagger } \overset{\rightharpoonup }{\sigma } U$$ where $$\overset{\rightharpoonup }{\sigma }=\sigma _x \overset{\rightharpoonup ...
1
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1answer
45 views

Probability distribution of two party quantum states

I am going through a blog post written by Thomas Vidick. It states following three assumptions by Bell. Measurement independence (“free will”): the state $\lambda$ is independent of ${x,y}$ (since ...
2
votes
1answer
134 views

Simon's algorithm for n = 3

The Simon's problem is as follows: Suppose we are given a function $f : \{0, 1\}^n \to \{0, 1\}^m$, with $m \ge n$, and we are promised that either $f$ is 1-to-1, or there exists a non-trivial s such ...
1
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1answer
65 views

Fourier transformations in Simon's quantum paper

I am reading this paper by Simon. This is one of the earliest quantum algorithm papers. In the paper he presents a routine starting at the end of page six. The first step runs a Fourier transformation ...
2
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0answers
112 views

Quadratic Diophantine Equations in Polynomial Time

Considering the problem of finding lattice points $(x_1, x_2 ... x_n)$ that satisfy a quadratic law: $F(x_1, x_2... x_n) = 0$ such that $F(x_1, x_2... x_n)$ is a second degree polynomial It is ...
2
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1answer
156 views

Unit Matrices for $M_n(M_n(\mathbb{R}))$

In the case of $M_n(\mathbb{R})$ we can define unit matrices $E_{ij}$ in the following way, $$E_{ij}f_l = \delta_{lj}f_i$$ where the $\{f_i\}$ is a basis for our space. One consequence of this ...
1
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0answers
193 views

How to Diagonalize an Extremely Large Sparse Matrix in SLEPc/PETSc

Dear Friends, Recently I have started with learning SLEPc/PETSc, but I didn't find a way to solve my problem. I have to solve a big sparse matrix which is a two dimensional quantum ...
1
vote
1answer
80 views

Dense in the special unitary group

I was going through the PhD thesis of Daniel Nagaj. In the last paragraph on page 30, there is this following sentence. A universal gate set must be dense in the group $SU(n)\ldots$ My question ...
2
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0answers
317 views

Is quantum game theory reducible to classical game theory? [closed]

Modnote: This question was manually migrated (closed and crossposted) to MathOverflow by request of the OP. Quantum game theory is an extension of classical game theory to the quantum domain. It ...
7
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2answers
584 views

Is a bra the adjoint of a ket?

The instructor in my quantum computation course sometimes uses the equivalence $$(\left|a\right>)^\dagger\equiv\left<a\right|$$ I understand that this is true for the typical matrix ...
2
votes
1answer
457 views

Does bra-ket notation work for all inner product spaces?

My quantum computation instructor keeps referring to the vector space in which he is using Dirac's bra-ket notation as an "inner product space", but doesn't it need additional properties to use that ...
2
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1answer
73 views

Factoring of exponents in Simon's algorithm

In derivations of Simon's algorithm (e.g., p. 4), it's often meant to be apparent that $$(x_0\oplus s)\cdot y=(x_0\cdot y)+(s\cdot y)$$ where $\oplus$ is "direct sum modulo 2", $x_0,s,y$ are all ...
5
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1answer
122 views

Quantum Information: Deutsch-Jozsa Algorithm

There is a step in the construction of this algorithm which I'm not understanding: $\displaystyle \left[\sum_x \frac{| x \rangle}{\sqrt{2^n}}\right]\left[\frac{ | 0 \rangle -| 1 \rangle ...