Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

2
votes
1answer
45 views

Deeper understanding of the adjoint of a linear operator

My undergraduate classes in Q.M describes the adjoint of a linear operator purely as a mathematical formality. At this point, I'd like a deeper and heuristic understanding of it. My questions are ...
1
vote
0answers
13 views

The action of tensor product over N terms on a ket.

Equation (6) of the paper titled, Multi-player and Multi-choice quantum game has left me puzzled-after many hours-as to how it is being derived. My working begins from the generic form seen just after ...
1
vote
1answer
17 views

Configuration of graph for maxcut algorithm

I am attempting to run a maxcut quantum algorithm with qiskit-aqua. I am trying to create a configuration of a graph of n number of nodes. These nodes are random values of a single parameter. For eg. ...
1
vote
0answers
17 views

Prove that the operator of the Grover's algorithm 'inverts about the mean'

I'm trying to solve the following problems that I have found in the book An Introduction to Quantum Computing by Phillip Kaye, but I don't know exactly where should I start. I would appreciate any ...
1
vote
0answers
26 views

Inner Product norm squared

In a book for quantum information, I found the following expression: $$\sum_{x_0\in\{0,1\}^n}4\big\|\langle x_0|\phi^k\rangle|x_0\rangle\big\|^2=1$$ If so, would not it be 4 as a result? Now, I ...
1
vote
2answers
40 views

Triangle inequality squared?

I am in the process of understanding a proof. There, for example, the following is indicated: $$\big\||a\rangle+|b\rangle\big\|^2\leq\big\||a\rangle\big\|^2+2\big\||a\rangle\big\|\big\||b\rangle\big\|...
3
votes
1answer
68 views

What is the braid word for the link L6n1

Please consider the link L6n1 Please note that such link is not the Borromean link L6a4. I am trying to obtain the braid word for L6n1. Using SnapPy with the following code ...
0
votes
1answer
21 views

2 exercises on reversible quantum process and quantum measurements

For question 7, Suppose that there exist such a reversible quantum process such that $$\lvert\varphi^{\perp}\rangle=U\lvert\varphi\rangle$$ where $U$ is a unitary matrix. Then,$$ 0 = \langle\varphi\...
0
votes
0answers
11 views

Tensor product and Unitary matrix

I have already shown parts 1, 2, 3 of the question. However, I am not sure how part 4 is realted to the pervious parts. Can I say that (U⊗U) is also a unitary matrix then the result follows?
4
votes
1answer
33 views

Definition of outer product

I am trying to understand the concept of outer product in quantum mechanics. I read "Quantum Computing explained" of David MacMahon. I can understand the transition in (3.12): $$(|\psi\rangle \...
1
vote
0answers
128 views

Is this link L10a169?

Please consider the following link I am using SnapPy with the following code ...
0
votes
0answers
19 views

Method of approximating arbitrary unitary matrices through a universal quantum set

I have been struggling with problems asking me to construct unitary matrices out of a quantum universal set. The particular universal set I am using is $\{H, T, CNOT\}$. Where $T$ is the matrix ...
1
vote
0answers
22 views

Computation with infinite Weyl algebra

My question here is very computational. My problem is in mathematical physics, so I want to ask the community what kind of software they use to do the following computation if there is any? Let $$L_{...
0
votes
1answer
35 views

A exercise with Dirac notation

I am dealing with the above exercise and not sure about my solution is correct or not, It is possible for me to find another state that the payoff would be larger than 50? Or it is possible for me to ...
6
votes
4answers
122 views

$\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ representation of $B_3$ braid group

I've been trying to find a representation of the braid group $B_3$ acting on $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ but I can't find it anywhere. From what I understand I have to ...
1
vote
0answers
38 views

Black Box optimization of expectation value from probability distribution valued function

So the setup is the following: I have a probability distribution valued function that maps each point in $\mathbb{R}^n$ to a probability density function $f_x$ on $\mathbb{R}$. (actually just a ...
0
votes
1answer
23 views

Going from singular value decomposition to Schmidt decomposition

I'm trying to understand the proof of the Schmidt decomposition found on the Wikipedia page. In particular, the part that I've circled in red in this screenshot: I've tried writing out $U_1, \Sigma, ...
1
vote
1answer
49 views

Inequality relating the probabilities of a quantum state to the euclidean distance of states.

My professor has provided us with the following proposition (without proof). I am trying to prove this. i'm having quite some trouble proving the first inequality, right under the first sentence. ...
0
votes
0answers
38 views

Symmetry preserving generators in Lie Group

In quantum computing, we can use finite types of generator to fill $U(N)$ space of all unitary operations, where $N$ is the number of qubits. e.g. $S=\{X, Z, CNOT\}$, where $X$ and $Z$ are single ...
0
votes
1answer
37 views

Does Raz and Tal 2018 (linked) falsify the Extended Church-Turing Hypothesis?

In the paper Oracle Separation of BQP and PH, Raz and Tal exhibit an algorithm that is in complexity class BQP but not in PH. Question: Does their proof invalidate the Extended Church-Turing ...
1
vote
1answer
45 views

Partial trace of action on density matrix

This topic is from chapter 11 of Kitaev, Shen and Vyalyi's Classical and Quantum Computation. If $V$ is an isometric (i.e. preserves the inner product) embedding $V: \mathcal{N} \rightarrow \mathcal{N}...
1
vote
1answer
113 views

Schur-Weyl duality for qubits

I am interested in applying Schur-Weyl duality to quantum information theory, specifically "qubits". But I have been stuck for some time on understanding how the Young symmetrizers work in this ...
1
vote
0answers
80 views

Relationship between differential geometry and quantum computation?

I have just started reading about quantum computation (Quantum Computation and Quantum Information, Nielsen and Chuang). In parallel, I have to study differential geometry for a final exam. After the ...
2
votes
1answer
28 views

How to find if an operator is the tensor product of more lower dimensional operators.

In quantum computation and quantum information it is very common to use e.g. the effect of a Hadamard matrix $H$ over $2n$ spins. Using (I think it is called the Kroenecker product in mathematical ...
2
votes
1answer
38 views

Confusion on the Coefficient Formulas for Fourier Series

For Question 12, this formula was used to find the coefficients in the Fourier series, whereas for Question 13, this formula was used. The difference is that for one, the inside of the sin and cos ...
1
vote
1answer
15 views

Qualities of a discontinuous function that can be written as a finite/infinite sum of continuous functions

My question is given a discontinuous function, can it be written as an infinite or finite sum if continuous functions. Here, “continuity” is of course relative to a point in the domain. Also, do the ...
0
votes
0answers
39 views

Is it possible to (naturally) integrate an algebraic curve into quantum computers?

A projective algebraic curve is defined over an projective space as zeros of some homogeneous polynomial. While quantum computers almost canonically holds their states in a "ket," i.e. a vector in the ...
0
votes
0answers
34 views

If two vectors lie on the YZ-plane of a Bloch Sphere, which orientations are required for v1-v2 to be a valid qubit state?

Suppose v1, and v2, are the two vectors on the y-z plane of the Bloch Sphere. What orientation must they be in in order for v1 - v2 to be a valid qubit?
2
votes
2answers
238 views

Linear combination of Pauli matrices and projectors

Premise: this is exercise 2.60 of Quantum Computation and Quantum Information, by Nielsen and Chuang, where I'm currently stuck. Suppose $\vec{v}$ is any real three-dimensional unit vector, and $\...
2
votes
1answer
115 views

Prove that the Schmidt number of a state is equal to the rank of the reduced density matrix

Suppose $|\psi\rangle$ is a pure state of a composite system with components $A$ and $B.$ Prove that the Schmidt number of $|\psi\rangle$ is equal to the rank of the reduced density matrix $\...
3
votes
1answer
63 views

Can I obtain a unitary block matrix from any invertible matrix?

Suppose A is any square invertible complex matrix. Then $$ C = \left[ \begin{array}{c|c} 0 & A \\ \hline A^{\dagger} & 0 \end{array} \right] $$ is a Hermitian matrix. My Question: Is ...
0
votes
0answers
26 views

Lower bounds for classical counting algorithm

This is Exercise 6.14 from Nielsen & Chuang. Prove that any classical counting algorithm with a probability at least $3/4$ for estimating $M$ correctly to within an accuracy $c\sqrt{M}$ for some ...
1
vote
1answer
420 views

Bitwise inner product and orthogonality

I am confused about the definition of bitwise inner product used in quantum algorithms. For example, bitwise inner product of 01111 with itself (in mod2) gives us 0. But they are not orthogonal to ...
3
votes
2answers
56 views

Decomposing the “sign flip” matrix in terms of Pauli matrices

Can the matrix: $A=\left[\begin{matrix}0 & -1 \\ -1 & 0\end{matrix}\right]$ be somehow expressed as a product of the $3$ standard Pauli matrices? I'm being able to diagonalize $A$, but not ...
1
vote
0answers
35 views

coupled non linear different equations in matrix form

Consider following differential equation of the form, $[\frac{d^2}{dx_1^2}+\frac{d^2}{dx_2^2}+x_1^2+x_2^2+gx_1^2x_2^2+x_1^4+x_2^4]u(x_1,x_2)=E u(x_1,x_2),$ where $g$ is the coupling constant. Please ...
2
votes
2answers
265 views

How do unitary matrices preserve the magnitude of unit vectors?

In quantum mechanics and quantum computing, quantum particles evolve in a unitary manner. That is to say at any point in time the particle/system (represented as a vector) has a magnitude of 1, ...
1
vote
0answers
30 views

Evaluating $|\frac{1}{2}(|a\rangle \otimes|b\rangle+|b\rangle\otimes |a\rangle) |^2$

I have a quantum state of 3-qubits : $|\psi\rangle = \frac{1}{2}|0\rangle (|a\rangle \otimes|b\rangle+|b\rangle\otimes |a\rangle) + \frac{1}{2}|1\rangle (|a\rangle \otimes|b\rangle - |b\rangle\otimes |...
0
votes
0answers
57 views

What geometry or topology best embodies the nonlocality of quantum entanglement?

I am a Princeton physics major. What geometry or topology best embodies the nonlocality of quantum entanglement? https://en.wikipedia.org/wiki/Quantum_entanglement: "Each particle cannot be ...
1
vote
0answers
59 views

Quantum cirucit with two hadamard gates - unitary matrix & eigenvalues

Let $X = \{|0\rangle,|1\rangle\}$ base of Hilbert space $H$ and $X' = \{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}$ base $H^{\otimes2}$ I have quantum circuit with two hadamard gates working ...
0
votes
1answer
73 views

Understanding of matrix XOR product

I'm new to the topics of quantum computing in the theory of computation. However, I'm quite lost trying to understanding the mechanism of how to XOR 2 simple matrices. $A \oplus B$ given $A = \begin{...
1
vote
0answers
29 views

Reduced density operators of a pure Bipartite state?

In (Bellucci, 2010; pg89) it is said (my wording): A pure bipartite state is separable if and only if the two reduced density matrices are pure. Proving the "only if" part is easy but I am ...
1
vote
3answers
49 views

Why are n-1 linearly independent equations sufficient to solve for the secret string in Simon's algorithm?

Professor Vazirani mentions in this lecture on Simon's algorithm that $n-1$ linearly independent equations are sufficient to solve for $s_1,s_2,...,s_n$ from the following system: $y_1^{(1)}s_1+...+...
1
vote
0answers
72 views

Proving unitary similarity between two Hermitian operators with the same eigenvalues

I've been working on a quantum computation problem set at Uni and came across the following problem. I managed to arrive at a proof, but I don't know if I have solved it to its fullest generality. ...
1
vote
0answers
130 views

How to find the projection matrix to find the projection along one of the basis matrices?

How to write the matrix operator for finding projection of a matrix along one of the basis matrices? For example I have a matrix $\mathcal{M}$ which can be written in terms of basis matrices like: $...
1
vote
0answers
14 views

In the Bloch representation why pre and post multiply by rotation operator?

In these notes to see the effect of the rotation operator $R$, on slide 11, the calculation $\rho' = R\rho R^{\dagger}$ is performed. Why isn't the operator only applied to the state AS $\rho' = R\rho$...
0
votes
0answers
119 views

Show that R is a rotation of the Bloch sphere

I've shown that $$R=e^{-i\frac{\theta}{2}X} = \cos\left(\frac{\theta}{2}\right)I - i\sin\left(\frac{\theta}{2}\right)X$$ Where $X$ is the matrix $\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$, ...
0
votes
1answer
40 views

An $N$-qubit system keeps $2^N$ pieces of information or should we say $2^{N+1}$?

Say my quantum register is of size $3$ qubits. This means I'll need $2^3$ complex numbers to describe all of its possible arrangements. But each complex number requires $2$ real numbers, so maybe I ...
0
votes
1answer
79 views

How to read Mermin's use of subscripts in his Cbit operators?

$\newcommand{\qr}[1]{|#1\rangle}$ I'm reading "Quantum computer science, an introduction," N. David Mermin, Cambridge University Press, 2007. I don't understand Mermin's use of the subscripts in his ...
0
votes
1answer
66 views

Derivative of fidelity with respect to time

Consider the quantum fidelity between two states defined as $$ F(\rho(t),\sigma(t)):=\text{Tr}\left(\sqrt{\sqrt{\rho(t)}\sigma(t)\sqrt{\rho(t)}}\right)^2 $$ Does $dF/dt$ have a closed form equation?
0
votes
1answer
577 views

Why does this solution to the Bernstein-Vazirani problem use $ (-1)^{f(x)} $

I am reading a solution to the Bernstein-Vazirani problem. For those unaware, the issue is to find a randomly selected $ 0 \leq a \lt 2^n $ given only a function $ f(x) = a_0x_0 \oplus a_1x_1 \oplus .....