From 03accd13c0a6ec4de2d8001edf3ce7553f831160 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Tue, 27 Sep 2022 21:20:05 +0200 Subject: Clean up CSS, minor design changes --- content/_index.md | 11 +++++------ content/blog/2022/email-server-revisited/index.md | 2 +- content/know/concept/bernstein-vazirani-algorithm/index.pdc | 2 +- content/know/concept/bloch-sphere/index.pdc | 2 +- content/know/concept/deutsch-jozsa-algorithm/index.pdc | 4 ++-- content/know/concept/dispersive-broadening/index.pdc | 2 +- content/know/concept/fabry-perot-cavity/index.pdc | 2 +- content/know/concept/feynman-diagram/index.pdc | 10 +++++----- content/know/concept/metacentric-height/index.pdc | 2 +- content/know/concept/modulational-instability/index.pdc | 2 +- content/know/concept/optical-wave-breaking/index.pdc | 6 +++--- content/know/concept/quantum-fourier-transform/index.pdc | 4 ++-- content/know/concept/quantum-gate/index.pdc | 6 +++--- content/know/concept/random-phase-approximation/index.pdc | 8 ++++---- content/know/concept/repetition-code/index.pdc | 10 +++++----- content/know/concept/rutherford-scattering/index.pdc | 4 ++-- content/know/concept/self-energy/index.pdc | 6 +++--- content/know/concept/self-phase-modulation/index.pdc | 2 +- content/know/concept/self-steepening/index.pdc | 2 +- content/know/concept/shors-algorithm/index.pdc | 2 +- content/know/concept/simons-algorithm/index.pdc | 4 ++-- content/know/concept/step-index-fiber/index.pdc | 4 ++-- content/know/concept/toffoli-gate/index.pdc | 12 ++++++------ 23 files changed, 54 insertions(+), 55 deletions(-) (limited to 'content') diff --git a/content/_index.md b/content/_index.md index 833d7b3..47e1c33 100644 --- a/content/_index.md +++ b/content/_index.md @@ -4,14 +4,13 @@ date: 2021-02-22T17:15:50+01:00 draft: false --- -Welcome to my website. + -Once in a blue moon, I'll post something here -related to my areas of interest: -programming, optimization, -mathematics, physics and even linguistics. +Welcome to my website. -You might be interested in my [knowledge base](/know/), +Once in a blue moon, I post something related to my areas of interest: +physics, mathematics, programming and even linguistics. +Perhaps you're interested in my [knowledge base](/know/), where I explain STEM concepts I've learned over the years. This website is made by me using [Hugo](https://gohugo.io), diff --git a/content/blog/2022/email-server-revisited/index.md b/content/blog/2022/email-server-revisited/index.md index 3f45993..838fe24 100644 --- a/content/blog/2022/email-server-revisited/index.md +++ b/content/blog/2022/email-server-revisited/index.md @@ -184,7 +184,7 @@ and OpenSMTPD reported it had been unable to make the delivery, because Microsoft had thrown an error: - + To their credit, they seem to be offering a way out. diff --git a/content/know/concept/bernstein-vazirani-algorithm/index.pdc b/content/know/concept/bernstein-vazirani-algorithm/index.pdc index 22de51a..40f41be 100644 --- a/content/know/concept/bernstein-vazirani-algorithm/index.pdc +++ b/content/know/concept/bernstein-vazirani-algorithm/index.pdc @@ -42,7 +42,7 @@ allows a quantum computer to do it with only a single query. It uses the following circuit: - + Where $U_f$ is a phase oracle, diff --git a/content/know/concept/bloch-sphere/index.pdc b/content/know/concept/bloch-sphere/index.pdc index 27abb54..7f0dfda 100644 --- a/content/know/concept/bloch-sphere/index.pdc +++ b/content/know/concept/bloch-sphere/index.pdc @@ -19,7 +19,7 @@ the **Bloch sphere** is an invaluable tool to visualize qubits. All pure qubit states are represented by a point on the sphere's surface: - + The $x$, $y$ and $z$-axes represent the components of a spin-1/2-alike system, diff --git a/content/know/concept/deutsch-jozsa-algorithm/index.pdc b/content/know/concept/deutsch-jozsa-algorithm/index.pdc index a3acaf4..d8dce8d 100644 --- a/content/know/concept/deutsch-jozsa-algorithm/index.pdc +++ b/content/know/concept/deutsch-jozsa-algorithm/index.pdc @@ -46,7 +46,7 @@ To do this, we use the following quantum circuit, where $U_f$ is the oracle we query: - + Due to unitarity constraints, @@ -147,7 +147,7 @@ other possibilities are assumed to be impossible. This algorithm is then implemented by the following quantum circuit: - + There are $N$ qubits in initial state $\ket{0}$, and one in $\ket{1}$. diff --git a/content/know/concept/dispersive-broadening/index.pdc b/content/know/concept/dispersive-broadening/index.pdc index cae856d..6efda9f 100644 --- a/content/know/concept/dispersive-broadening/index.pdc +++ b/content/know/concept/dispersive-broadening/index.pdc @@ -65,7 +65,7 @@ with parameter values $T_0 = 1\:\mathrm{ps}$, $P_0 = 1\:\mathrm{kW}$, $\beta_2 = -10 \:\mathrm{ps}^2/\mathrm{m}$ and $\gamma = 0$: - + The **instantaneous frequency** $\omega_\mathrm{GVD}(z, t)$, diff --git a/content/know/concept/fabry-perot-cavity/index.pdc b/content/know/concept/fabry-perot-cavity/index.pdc index e4195d0..066749c 100644 --- a/content/know/concept/fabry-perot-cavity/index.pdc +++ b/content/know/concept/fabry-perot-cavity/index.pdc @@ -25,7 +25,7 @@ The cavity $C$ has length $\ell$ and is centered on $x = 0$. Let $n_L$, $n_C$ and $n_R$ be the respective domains' refractive indices: - + diff --git a/content/know/concept/feynman-diagram/index.pdc b/content/know/concept/feynman-diagram/index.pdc index 600be61..98ed668 100644 --- a/content/know/concept/feynman-diagram/index.pdc +++ b/content/know/concept/feynman-diagram/index.pdc @@ -43,7 +43,7 @@ and $\mathcal{T}\{\}$ denote the [time-ordered product](/know/concept/time-ordered-product/): - + $$\begin{aligned} = i \hbar G_{s_2 s_1}^0(\vb{r}_2, t_2; \vb{r}_1, t_1) @@ -64,7 +64,7 @@ a causal Green's function $G$ for the entire Hamiltonian $\hat{H}$, where the subscript $H$ refers to the [Heisenberg picture](/know/concept/heisenberg-picture/): - + $$\begin{aligned} = i \hbar G_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1) @@ -79,7 +79,7 @@ hence it starts and ends at the same time, and no arrow is drawn: - + $$\begin{aligned} = \frac{1}{i \hbar} W_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1) @@ -102,7 +102,7 @@ One-body (time-dependent) operators $\hat{V}$ in $\hat{H}_1$ are instead represented by a special vertex: - + $$\begin{aligned} = \frac{1}{i \hbar} V_s(\vb{r}, t) @@ -179,7 +179,7 @@ Consider the following diagram and the resulting expression, where $\tilde{\vb{r}} = (\vb{r}, t)$, and $\tilde{\vb{k}} = (\vb{k}, \omega)$: - + $$\begin{aligned} &= (i \hbar)^3 \sum_{s s'} \!\!\iint \dd{\tilde{\vb{r}}} \dd{\tilde{\vb{r}}'} diff --git a/content/know/concept/metacentric-height/index.pdc b/content/know/concept/metacentric-height/index.pdc index 1fc6aca..1668cc0 100644 --- a/content/know/concept/metacentric-height/index.pdc +++ b/content/know/concept/metacentric-height/index.pdc @@ -32,7 +32,7 @@ as is illustrated in the following sketch of our choice of coordinate system: - + Here, $B$ is the **center of buoyancy**, equal to diff --git a/content/know/concept/modulational-instability/index.pdc b/content/know/concept/modulational-instability/index.pdc index 993dec9..a0c2c91 100644 --- a/content/know/concept/modulational-instability/index.pdc +++ b/content/know/concept/modulational-instability/index.pdc @@ -178,7 +178,7 @@ $$\begin{aligned} \end{aligned}$$ - + Where $L_\mathrm{NL} = 1/(\gamma P_0)$ is the characteristic length of nonlinear effects. diff --git a/content/know/concept/optical-wave-breaking/index.pdc b/content/know/concept/optical-wave-breaking/index.pdc index 30305f5..ecd5a4f 100644 --- a/content/know/concept/optical-wave-breaking/index.pdc +++ b/content/know/concept/optical-wave-breaking/index.pdc @@ -40,7 +40,7 @@ small waves start "falling off" the edge of the pulse, hence the name *wave breaking*: - + Several interesting things happen around this moment. @@ -59,7 +59,7 @@ which eventually melt together, leading to a trapezoid shape in the $t$-domain. Dispersive broadening then continues normally: - + We call the distance at which the wave breaks $L_\mathrm{WB}$, @@ -189,7 +189,7 @@ This prediction for $L_\mathrm{WB}$ appears to agree well with the OWB observed in the simulation: - + Because all spectral broadening up to $L_\mathrm{WB}$ is caused by SPM, diff --git a/content/know/concept/quantum-fourier-transform/index.pdc b/content/know/concept/quantum-fourier-transform/index.pdc index 5a3de7b..cfdbc07 100644 --- a/content/know/concept/quantum-fourier-transform/index.pdc +++ b/content/know/concept/quantum-fourier-transform/index.pdc @@ -178,7 +178,7 @@ excluding the swapping part to get the right order. Here, $R_m$ means $R_\phi$ with $\phi = 2 \pi / 2^m$: - + Again, note how the inputs $\ket{x_j}$ and outputs $\ket{k_j}$ are in the opposite order. @@ -186,7 +186,7 @@ The complete circuit, including the swapping at the end, therefore looks like this: - + For each of the $n$ qubits, $\mathcal{O}(n)$ gates are applied, diff --git a/content/know/concept/quantum-gate/index.pdc b/content/know/concept/quantum-gate/index.pdc index 189145f..b9682ec 100644 --- a/content/know/concept/quantum-gate/index.pdc +++ b/content/know/concept/quantum-gate/index.pdc @@ -209,7 +209,7 @@ With that said, the first two-qubit gate is $\mathrm{SWAP}$, which simply swaps $\ket{\psi_1}$ and $\ket{\psi_2}$: - + $$\begin{aligned} @@ -238,7 +238,7 @@ Next, there is the **controlled NOT gate** $\mathrm{CNOT}$, which "flips" (applies $X$ to) $\ket{\psi_2}$ if $\ket{\psi_1}$ is true: - + $$\begin{aligned} @@ -265,7 +265,7 @@ we can define a two-qubit **controlled U gate** $\mathrm{CU}$, which applies $U$ to $\ket{\psi_2}$ if $\ket{\psi_1}$ is true: - + $$\begin{aligned} diff --git a/content/know/concept/random-phase-approximation/index.pdc b/content/know/concept/random-phase-approximation/index.pdc index ed85106..0089dff 100644 --- a/content/know/concept/random-phase-approximation/index.pdc +++ b/content/know/concept/random-phase-approximation/index.pdc @@ -77,21 +77,21 @@ i.e. the ones where all $n$ interaction lines carry the same momentum and energy: - + Where we have defined the **screened interaction** $W^\mathrm{RPA}$, denoted by a double wavy line: - + Rearranging the above sequence of diagrams quickly leads to the following [Dyson equation](/know/concept/dyson-equation/): - + In Fourier space, this equation's linear shape @@ -110,7 +110,7 @@ Abbreviating $\tilde{\vb{k}} \equiv (\vb{k}, i \omega_n^B)$ and $\tilde{\vb{q}} \equiv (\vb{q}, i \omega_n^F)$: - + We isolate the Dyson equation for $W^\mathrm{RPA}$, diff --git a/content/know/concept/repetition-code/index.pdc b/content/know/concept/repetition-code/index.pdc index 7245cbc..d9eec2c 100644 --- a/content/know/concept/repetition-code/index.pdc +++ b/content/know/concept/repetition-code/index.pdc @@ -82,7 +82,7 @@ Such a transformation is easy to achieve with the following sequence of [quantum gates](/know/concept/quantum-gate/): - + So, a little while after encoding the state $\ket{\psi}$ like that, @@ -208,7 +208,7 @@ by applying $\mathrm{CNOT}$s to some ancillary qubits and then measuring those: - + The two measurements, respectively representing $ZZI$ and $IZZ$, @@ -260,7 +260,7 @@ $$\begin{aligned} \end{aligned}$$ - + A phase flip along the $Z$-axis @@ -269,7 +269,7 @@ In this case, the stabilizers are $XXI$ and $IXX$, and the error detection circuit is as follows: - + This system protects us against all single-qubit phase flips, @@ -313,7 +313,7 @@ which simply consists of the phase flip encoder, followed by 3 copies of the bit flip encoder: - + We thus use 9 physical qubits to store 1 logical qubit. diff --git a/content/know/concept/rutherford-scattering/index.pdc b/content/know/concept/rutherford-scattering/index.pdc index c89b477..d91ae40 100644 --- a/content/know/concept/rutherford-scattering/index.pdc +++ b/content/know/concept/rutherford-scattering/index.pdc @@ -25,7 +25,7 @@ Coulomb repulsion causes 1 to deflect by an angle $\theta$, and pushes 2 away in the process: - + Here, $b$ is called the **impact parameter**. @@ -75,7 +75,7 @@ we can see that $v_x$ is unchanged for any given $\pm t$, while $v_y$ simply changes sign: - + From our expression for $\vb{r}$, diff --git a/content/know/concept/self-energy/index.pdc b/content/know/concept/self-energy/index.pdc index 935cca8..c6aa0c5 100644 --- a/content/know/concept/self-energy/index.pdc +++ b/content/know/concept/self-energy/index.pdc @@ -210,7 +210,7 @@ so we are left with non-equivalent diagrams only. Let $G(b,a) = G_{ba}$: - + A **reducible diagram** is a Feynman diagram @@ -223,7 +223,7 @@ as the sum of all irreducible terms in $G(b,a)$, after removing the two external lines from/to $a$ and $b$: - + Despite its appearance, the self-energy has the semantics of a line, @@ -244,7 +244,7 @@ you can convince youself that $G(b,a)$ obeys a [Dyson equation](/know/concept/dyson-equation/) involving $\Sigma(y, x)$: - + This makes sense: in the "normal" Dyson equation diff --git a/content/know/concept/self-phase-modulation/index.pdc b/content/know/concept/self-phase-modulation/index.pdc index 3d33746..64f68d4 100644 --- a/content/know/concept/self-phase-modulation/index.pdc +++ b/content/know/concept/self-phase-modulation/index.pdc @@ -72,7 +72,7 @@ $$\begin{aligned} \end{aligned}$$ - + The **instantaneous frequency** $\omega_\mathrm{SPM}(z, t)$, diff --git a/content/know/concept/self-steepening/index.pdc b/content/know/concept/self-steepening/index.pdc index 7349854..808240b 100644 --- a/content/know/concept/self-steepening/index.pdc +++ b/content/know/concept/self-steepening/index.pdc @@ -119,7 +119,7 @@ which turns out to be accurate, although the simulation breaks down due to insufficient resolution: - + Unfortunately, self-steepening cannot be simulated perfectly: as the diff --git a/content/know/concept/shors-algorithm/index.pdc b/content/know/concept/shors-algorithm/index.pdc index 643337c..8fc528e 100644 --- a/content/know/concept/shors-algorithm/index.pdc +++ b/content/know/concept/shors-algorithm/index.pdc @@ -52,7 +52,7 @@ To do this, the following $2q$-qubit quantum circuit is used, with $q$ chosen so that $N^2 \le 2^q < 2 N^2$: - + Here, $\mathrm{QFT}_q$ refers to the $q$-qubit diff --git a/content/know/concept/simons-algorithm/index.pdc b/content/know/concept/simons-algorithm/index.pdc index f6b7d69..a8c5eb1 100644 --- a/content/know/concept/simons-algorithm/index.pdc +++ b/content/know/concept/simons-algorithm/index.pdc @@ -57,8 +57,8 @@ A quantum computer needs to query $f$ only $\mathcal{O}(n)$ times, although the exact number varies due to the algorithm's probabilistic nature. It uses the following circuit: - - + + The XOR oracle $U_f$ implements $f$, diff --git a/content/know/concept/step-index-fiber/index.pdc b/content/know/concept/step-index-fiber/index.pdc index 8847fff..4ca2ade 100644 --- a/content/know/concept/step-index-fiber/index.pdc +++ b/content/know/concept/step-index-fiber/index.pdc @@ -246,7 +246,7 @@ $$\begin{aligned} \end{aligned}$$ - + Looking at these solutions with our constraints for $R_o$ in mind, @@ -400,7 +400,7 @@ is illustrated below for a fiber with $V = 5$, where red and blue respectively denote the left and right-hand side: - + This shows that each $\mathrm{LP}_{\ell m}$ has an associated cut-off $V_{\ell m}$, diff --git a/content/know/concept/toffoli-gate/index.pdc b/content/know/concept/toffoli-gate/index.pdc index f3ab0ba..f0b39ad 100644 --- a/content/know/concept/toffoli-gate/index.pdc +++ b/content/know/concept/toffoli-gate/index.pdc @@ -22,7 +22,7 @@ and flips $C$ if both $A$ and $B$ are true. In circuit diagrams, its representation is: - + This gate is reversible, because $A$ and $B$ are preserved, @@ -31,7 +31,7 @@ Moreover, this gate is universal, because we can make a NAND gate from it: - + A NAND is enough to implement every conceivable circuit. @@ -39,15 +39,15 @@ That said, we can efficiently implement NOT, AND, and XOR using a single Toffoli Note that NOT is a special case of NAND: - + - + - + Using these, we can, as an example, make an OR gate @@ -56,7 +56,7 @@ thanks to the fact that $A \lor B = \neg (\neg A \land \neg B)$, i.e. OR is NAND of NOT $A$ and NOT $B$: - + Thanks to its reversibility and universality, -- cgit v1.2.3