#Literature #Error rate #Density Operator

Error rate vs Fidelity (Part 1): Unveil the mathematics behind quantum error rate

In the article of Bernsterin-Vazirani implementation, I’ve addressed one of the most crucial challenges of quantum computing. The field is yet to flourish because quantum gates aren’t constructed to be good enough. One major difference of quantum gates versus classical counterparts is that quantum gates have to be reversible, which means we’re able to retrieve a prior state of a qubit without loss of information. Scientists are now on their way to establish a proper scheme for building decent quantum gates. ...

#Mixed States #Density Operator

Mixed States and General Quantum Operations: A simple view on the framework of Quantum Mechanics

In this text I’ll do some review on the definition of quantum state and quantum measurement, which was introduced here, and reasonably expand the notions into the most general instances of quantum bits. Review Given a state $$|\psi\rangle = \sum_i \alpha_i |\varphi_i \rangle,$$ it’s possible to perform a Von Neumann measurement on the state’s space $\mathscr{H}$ with respect to the orthonormal basis $B = \{|\varphi_i \rangle\}$. The output would be a label $i$ with corresponding probability $|\alpha_i|^2$ and the system left in state $|\varphi_i\rangle$. ...

#project #Bernstein-Vazirani algorithm #Deutsch algorithm

Bernstein-Vazirani algorithm (Part 2): How to run your algorithms on a realistic quantum computer

In the last part of the previous series, Deutsch algorithm, I introduced a manual way to construct the oracle. That oracle must be built from matrix because the oracle function is undeterminate except for the outcome. In Bernsterin-Vazirani problem, the function is, on the other hand, clearly stated: $f(\mathbf{x}) = \mathbf{s}\cdot\mathbf{x}$. So, it’s unnecessary to repeat the previous procedure; instead, we’ll build the oracle without using matrices by understanding what the function really do. ...

#Bernstein-Vazirani algorithm #Deutsch algorithm

Bernstein-Vazirani algorithm (Part 1): A derivation of the Deutsch-Jozsa algorithm

Here we stop by Berstein-Vazirani problem that inherents the structure of its predecessor, Deutsch-Jozsa problem, before we can get started with problems and solutions of practical meaning later. Because this problem is similar to the one stated in the last series, we’ll get straight to it. Berstein-Vazirani problem The access for a black-box function $f: \{0,1\}^n \mapsto \{0,1\}$. It’s guaranteed that $f(\mathbf{x}) = \mathbf{s}\cdot\mathbf{x}$ (mod $2$) for some secret string $\mathbf{s} \in \{0,1\}^n$. ...

#PiMA 2019 #Deep Learning

[TUYỂN SINH] PiMA 2019 - The Mathematics of Deep Learning

Nối tiếp ba mùa trại 2016, 2017 và 2018 đầy thành công, PiMA chính thức quay trở lại với chủ đề “The Mathematics of Deep Learning”. I. Đôi nét về PiMA Chúng mình là ai? Projects in Mathematics and Applications (PiMA) là một dự án phi lợi nhuận được thành lập năm 2016 bởi một nhóm sinh viên có niềm đam mê với Toán ứng dụng từ các trường đại học ở nhiều nơi trên thế giới. ...

#project #Deutsch algorithm

Deutsch Algorithm (Part 3): Implementation of oracle-based algorithms

Because the Deutsch algorithm is just a simple instance of the Deutsch-Jozsa algorithm where $n=2$, we’ll only discuss how to implement the general one. Let’s go. Difficulty of reconstructing gates from matrices This is the first time we encounter a so-called black-box function in an algorithm. It’s called a black box since we don’t how it calculate the output from the input, but we do know the output it returns. ...