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$\require{cancel} \newcommand{\Ket}[1]{\left|{#1}\right\rangle} \newcommand{\Bra}[1]{\left\langle{#1}\right|} \newcommand{\Braket}[1]{\left\langle{#1}\right\rangle} \newcommand{\Rsr}[1]{\frac{1}{\sqrt{#1}}} \newcommand{\RSR}[1]{1/\sqrt{#1}} \newcommand{\Verti}{\rvert} \newcommand{\HAT}[1]{\hat{\,#1~}} \DeclareMathOperator{\Tr}{Tr}$
First created in December 2018
Quantum computing algorithms, their implementations and implications.
Bernstein-Vazirani Algorithm - How the Bernstein-Vazirani Algorithm can find a secret pattern $s$ presented as a function $f_s(\bullet)$ such that $f_s(x)=s\cdot x$, in a single operation.
Deutsch-Jozsa Algorithm - How the Deutsch-Jozsa Algorithm can determine the input being constant or balance in a single run, with a QISKit demo.
Grover's Algorithm - An explanation of the Grover's Algorithm with proof that the optimal probability can be achieved in $\frac{\pi}{2}\sqrt{N}$ iterations.
Grover's Algorithm Analysis - A discussion on the success rate and the role of the Oracle.
Grover's Algorithm 4-item Trial - A trial of Grover's Algorithm using $n=2$ qubits with $N=2^n=4$ items. This is a special case with $\theta=\pi/6$ (as $\sin\theta=\sqrt{1/N}=1/2$). The result is certain after one iteration (flip down then flip up).
Grover's Algorithm 8-item Trial - A trial of Grover's Algorithm using $n=3$ qubits with $N=2^n=8$ items. The code shows two iterations.
Shor's Algorithm - The basic idea of Shors Algorithm (under development while branching out for Quantum Phase Estimation).
How the quantum computing technology can be used in business context and strategies to commercialize the technology.
Quantum Computing Commercialization - An overview of quantum computing commercialization in terms of business context, strategies, multi-disciplinary effort, technology stack and application potentials.
Probability - Basics - Fundamentals of classical probability.
Probability - Counting - Permutations and Combinations
Probability - Normal Distribution - Probability density function (PDF), mean, variant, error function, cumulative distribution function (CDF) and entropy.
Probability - Discrete Probability Distributions - Discrete random variables, binomial distributions and geometric distributions.
Probability - Continuous Probability Distributions - Continuous random variables, normal distributions and exponential distributions.
Quantum computing related information theories, analysis, qubits and their transformation (rotations on Bloch Sphere).
Bloch Sphere Orthonormality - To illustrate and prove that the two opposite ends of an arbitrary diameter on the Bloch Sphere are orthonormal, and the converse that two orthonormal vectors are on the same diameter.
Bloch Vector - The Bloch Vector composed of the expectation values of $\Ket\psi$ on the three Pauli axes: $\{\Braket{X}_\psi,\Braket{Y}_\psi,\Braket{Z}_\psi\}.$
Eigenstate of Multi-Qubits - To explore the eigenstates of common quantum operations and extend the discussion to multiple qubits, with illustrations.
Kous Delta Operator - Rotation Operator about $\Ket\psi$ is $\boxed{~R_\psi(\theta) =\Braket{X}_\psi R_x(\theta)+\Braket{Y}_\psi R_y(\theta)+\Braket{Z}_\psi R_z(\theta)+\delta(\theta)~},$ where $\Braket{Q}_\psi$ is the expectation value of measurement on Pauli axis $Q$, and $\delta(\theta)=\left(1-\Braket{X}_\psi-\Braket{Y}_\psi-\Braket{Z}_\psi\right)~I\cos\frac{\theta}{2} .~~~~~$ $\delta(\theta)=0$ when $\Ket\psi$ is $X, Y$ or $Z$, or when $\theta=\pi.$
Kous Octant Visualisation - A mind model to imagine the effect of various operators on a qubit by means of rubiks-cube-like rotations on the Bloch Sphere. An example to implement a Hadamard gate using $\sqrt Z\sqrt X\sqrt Z.$
Multi-Qubit Systems - How an $n$-qubit system is in a single state as a whole with $2^n$ dimensions, behaving "like" a $2^{2^n}$ dimensional classical system. The operators $\oplus$ and $\otimes$ and the projector on a multi-qubit system. A study note of "Quantum Computing: A Gentle Introduction" by Eleanor Rieffel, Wolfgang Polak, The MIT Press.
No-Cloning Theorem - The description that a single universal unitary transformation cannot clone a general quantum state, the proof, and why some quantum phenomena are not considered as "cloning".
Quantum Fourier Transform - Fouriere Transform in classical and quantum context, analysis of the discrete case in quantum computing, examples of 1, 2 and 3 qubit QFT, the implementation with circuit and code.
Quantum Phase Estimation - How - An implementation of QPE with circuit and code.
Quantum Phase Estimation - Why - The motivation and the fomulation of QPE, the visualisation with examples of 1 and 2 qubit systems and circuit illustration.
Quantum Teleportation - An illustration on quantum teleportation with code implementation for each stage (Bell, Alice and Bob), and some puzzles like what if Alice does not measure or withhold the measurement result from Bob, and more.
Quantum Teleportation Using CReg - An exploration on quantum teleportation using classical registers.
Rotation - Bloch Sphere - A study of bloch sphere rotation from the lecture note of Ian Glendinning of University of Vienna, starting from the density matrix coming up with a universal operator.
Rotation - Matrix Exponential - Definition of Matrix Exponential, its properties, the special condition where if $\displaystyle A^2=I,~~e^{i\theta A}=I\cos\theta+iA\sin\theta$, rotation through a generator $G\equiv\Ket b\Bra a-\Ket{a}\Bra{b}$ and a projector $P\equiv-G^2$, and from that deriving $R_x(\theta)$, $R_y(\theta)$ and $R_z(\theta)$.
Rotation about Pauli Axes - Rotation of $\theta$ about a Pauli Axis on the Bloch Sphere.
Rotation by Phase Shift - The concept of rotation about the $Z$-axis as a phase shift can be generalised to other axes.
Superdense Coding - An illustration on superdense coding with code implementation of four different cases: 00, 01, 10 and 11.
Quantum theories (independent of computing context).
Born Rule - Description of the wave function $\psi(x)\in\mathbb C$ as probability amplitude, of which the norm squared is the probability density. How the wave function is viewed in Hilbert space. The basis vector and the special normalisation to the Dirac Delta function. How an observable relates to its eigenvalues and eigenvectors. A discussion of the antihermitian operator $\displaystyle \HAT D\equiv\frac{\partial}{\partial x}$ and the Hermitian operator $\HAT K=-i\HAT D.$
CHSH Inequality - Derivation of CHSH Inequality, which states that $|S|\le 2$. There is no violation in Hidden Variable theory as the two qubits collapse individually, while with Quantum Entanglement the two qubits collapse at the same time, causing a maximu $|S|=2\sqrt 2$. A QISKit demo shows $|S|=2.82$.
Continuous Measurements - Starting from the discrete case and continuing on continuous case. Discussing the basis, the inner product and dual space of such infinite-dimensional Hilbert Space.
Density Matrix - A Density Matrix representing a mixed state (as different from Transformation and Projector). Measurement of a mixed state by projection and the success rate, hence the expectation value $\Braket{A}_\rho =\sum_i\lambda_i\beta_i =\Tr(\rho A) .$
Dirac Delta Distribution - Dirac Delta Function as Distribution: $\Braket{\delta_\alpha~,~\varphi} \equiv\varphi(\alpha) =\int_\mathbf R\delta_\alpha(x)~\varphi(x)~dx.$ Describing its properties and introducing the Heaviside step function, which differentiate to the delta function.
Fourier Transform as Change of Basis - Illustrating how Fourier transform converts the wave function between the two bases, position $x$ and momentum $k$.
Measurement Operators - Distinction between measurement operators and measurement results, which is in the direct sum of eigenspaces with $k<2^n$ possible outcomes. Density operators on pure state and mixed state. Expectation values.
Mixed States - A mixed state is in a statistical ensemble with non-unique decompositions but if an eigendecomposition is non-degenerate it would be unique. The detectable distinction between an ensemble of pure state and mixed state is illustrated.
Momentum Operator - Deriving the momentum operator $\HAT p=-i\hbar\frac{\partial}{\partial x}$ from $\HAT x=x$ from various perspectives. Discussing the Schrödinger equation and Fourier transform.
QSHO - Quantum Simple Harmonic Oscillator.
Stern-Gerlach Experiment - The Black and White, Hard and Soft illustration of the quantum nature.
Wavefunctions and Relativity - Relationship between wavefuntions and relativity and how they interlinked.
Quantum logic gates and quantum states.
Controlled Gates - The basic idea of Shors Algorithm (under development while branching out for Quantum Phase Estimation).
Controlled Operations as an Eigenvalue - The notation of controlled operation by means of an eigenvalue of phase shift. $c^nR_\phi\Ket x=e^{i\phi\cdot x_1x_2\ldots x_n}\Ket x.$
Error Bars - How to plot error bars with matplotlib.pyplot.
Flip Bits - A circuit to flip a qubit without assistance from ancillary qubits. This is useful in constructing the oracle unitary operator for the Grover's Algorithm.
GHZ State - Technical calculation and QISKit demo of a 3-qubit GHZ State $\Rsr2(\Ket{000}+\Ket{111}).$
Hadamard on Multi-Qubits - The unbiased superposition that contains all possible configurations of multi-qubits, useful as input to an algorithm.
IBM Q - Sqrt T - Attempts to meet the challenge from the "IBM Q Experience - Basic Circuit Identities and Larger Circuits" page to construct an approximate controlled-$T$ unitary transformation.
KRT Universal Quantum Gates - A study note of "Quantum Computing: A Gentle Introduction" on KRT Universal Quantum Gates with $K(\sigma), R(\beta)\text{ and }T(\alpha).$
Measurement on Basis Vectors - The mechanism to measure (project) on an arbitary basis vector, by rotating to $+Z$. The difference of the two probabilities $\cos^2(\theta/2)-\sin^2(\theta/2)=\cos\theta$ gives the projection of $\Ket\psi$ on $\Ket k$.
QASM U Gates - How QASM implements quantum gates using the three $U$ Gates, all derivable from $U(\theta,\phi,\lambda):=R_z(\phi)~R_y(\theta)~R_z(\lambda).$ An example of $U(\pi/2,\pi/2,\pi/2)$ is given with visualisation guide.
QASM U Operators Derivation - To prove the three QASM $U$ Gates by deriving their matrix forms.
Quantum Logic Gates - Basics of quantum logic gates and what they do.
Quantum Logic Gates - 2 - Controlled and Swap gates with code implementations.
Square Root of Quantum Gates - Exploration of square root of quantum gates, inspired by the IBM Square Root of $T$-Gate challenge.
T Gate - An illustration of the $T$-Gate, the $\pi/8$ gate, and how it is a shift of $\pi/4$ about the $Z$-axis.
Toffoli Gate - The 3-bit controlled-$X$ gate and its implementation.
The mathematics useful for quantum computing.
It is tempted to dive deep into the mathematics to get to the bottom of it. One must know that the ocean of mathematics has no bottom. Advise to get just enough and turn back.
Block Matrix - Block matrix multiplication.
Eigendecomposition - The decomposition of a matrix into its eigenvectors with their corresponding eigenvalues.
Eigenvalues - Basics of eigenvalues and characteristic polynomial.
Hilber Space Constructions - The operations to construct Hilbert Spaces of higher dimensions from lower ones. The basics of direct sum $\oplus$ and Kronecker product $\otimes$ in the context of qubit operations, as illustrated in the example and notes.
Kronecker Product - Kronecker product with Python.
Matrix Pulled Apart - Elements picking with $\Braket{i\Verti A\Verti j}=[a_{ij}]$ and $A\Ket q\Bra p A^\dagger=[a_{iq}a_{pj}]$, flattening, multiplication with elements and multi-column vectors.
Matrix Types and Properties - The symbols and properties of commonly used matrices such as normal matrix, unitary matrix, Hermitian matrix and many more.
Similarity - The concept of similarity and class of matrics similar to a single diagonal matrix.
Space Mapping - Mapping of two spaces $\left(A\Ket{v_j}=\sum_{i=1}^na_{ij}\Ket{w_i}.\right)$ and Completeness Relation $\left(\sum\Ket i\Bra i=I,~\text{where}\left\{\Ket i\right\}\text{is an orthonormal basis}.\right)$
Sum of Ket-Bra - Outer-product manipulation ($\sum\Ket i\Bra j$) useful in mixed state analysis.
Superoperators - Study note of Superoperator, vectorisation of matrices, the outer product $\Bra{\langle P}=\Ket{P\rangle}^\dagger,$ the "Tall Ket" $\Ket{AP\rangle}=(A\otimes I)\Ket{P\rangle}$ and the "Wide Bra" $\Bra{\langle PA}=\Bra{\langle P}(I\otimes A^T)^\dagger$, arriving at the left and right superoperators $\mathcal{L}(S)[\rho]$ and $\mathcal{R}(S)[\rho]$.
Superoperator Examples - Examples of Superoperator $\mathcal{L}(S)[\rho]$ and $\mathcal{R}(S)[\rho]$ with $\rho =\begin{bmatrix} {\frac{1}{4}}&{\frac{1}{4}}\\ 0&{\frac{1}{2}}\\ \end{bmatrix}$ and $A=\begin{bmatrix}1&0\\0&i\end{bmatrix}.$
Trace - Trace of a square matrix, its properties and proof.
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