diff options
Diffstat (limited to 'source')
157 files changed, 151 insertions, 169 deletions
diff --git a/source/_includes/image.html b/source/_includes/image.html new file mode 100644 index 0000000..a9bc3fd --- /dev/null +++ b/source/_includes/image.html @@ -0,0 +1,34 @@ +{% assign name_dot_format = include.file | split: "." %} +{% assign name = name_dot_format | first %} +{% assign format = name_dot_format | last %} + +{% comment %} Let 'suffix' = last 4 chars of 'name', 'prefix' = the rest {% endcomment %} +{% assign suffix = name | slice: -4, 4 %} +{% assign rname = name | split: "" | reverse | join: "" %} +{% assign rsuffix = suffix | split: "" | reverse | join: "" %} +{% assign rprefix = rname | remove_first: rsuffix %} +{% assign prefix = rprefix | split: "" | reverse | join: "" %} + +{% comment %} If 'name' ends in "full", half-size image must exist {% endcomment %} +{% assign name_full = name %} +{% if suffix == "full" %} + {% assign name_half = prefix | append: "half" %} +{% else %} + {% assign name_half = name_full %} +{% endif %} + +{% comment %} Insert the image, linking to the full-size version {% endcomment %} +<p> +<a href="{{ name_full }}.{{ format }}"> + {% assign set_width = "width:" | append: include.width %} + <picture markdown="0"> + <source srcset="{{ name_half }}.avif" type="image/avif"> + {% if name_full == name_half %} + <img src="{{ name_half }}.{{ format }}" style="{{ include.style | default: set_width }}" alt="{{ include.alt }}" title="{{ include.alt }}"> + {% else %} + <source srcset="{{ name_half }}.webp" type="image/webp"> + <img src="{{ name_half }}.jpg" style="{{ include.style | default: set_width }}" alt="{{ include.alt }}" title="{{ include.alt }}"> + {% endif %} + </picture> +</a> +</p> diff --git a/source/know/concept/bernstein-vazirani-algorithm/bernstein-vazirani-circuit.avif b/source/know/concept/bernstein-vazirani-algorithm/bernstein-vazirani-circuit.avif Binary files differnew file mode 100644 index 0000000..7bfd114 --- /dev/null +++ b/source/know/concept/bernstein-vazirani-algorithm/bernstein-vazirani-circuit.avif diff --git a/source/know/concept/bernstein-vazirani-algorithm/index.md b/source/know/concept/bernstein-vazirani-algorithm/index.md index f91c0ba..85017dc 100644 --- a/source/know/concept/bernstein-vazirani-algorithm/index.md +++ b/source/know/concept/bernstein-vazirani-algorithm/index.md @@ -36,9 +36,7 @@ However, the Bernstein-Vazirani algorithm allows a quantum computer to do it with only a single query. It uses the following circuit: -<a href="bernstein-vazirani-circuit.png"> -<img src="bernstein-vazirani-circuit.png" style="width:52%"> -</a> +{% include image.html file="bernstein-vazirani-circuit.png" width="52%" alt="Bernstein-Vazirani circuit" %} Where $$U_f$$ is a phase oracle, whose action is defined as follows, diff --git a/source/know/concept/bloch-sphere/bloch-small.jpg b/source/know/concept/bloch-sphere/bloch-small.jpg Binary files differdeleted file mode 100644 index e99c0e1..0000000 --- a/source/know/concept/bloch-sphere/bloch-small.jpg +++ /dev/null diff --git a/source/know/concept/bloch-sphere/bloch.jpg b/source/know/concept/bloch-sphere/bloch.jpg Binary files differdeleted file mode 100644 index 9515d84..0000000 --- a/source/know/concept/bloch-sphere/bloch.jpg +++ /dev/null diff --git a/source/know/concept/bloch-sphere/index.md b/source/know/concept/bloch-sphere/index.md index 2cb7742..0ca6f1b 100644 --- a/source/know/concept/bloch-sphere/index.md +++ b/source/know/concept/bloch-sphere/index.md @@ -13,9 +13,7 @@ In quantum mechanics, particularly quantum information, the **Bloch sphere** is an invaluable tool to visualize qubits. All pure qubit states are represented by a point on the sphere's surface: -<a href="bloch.jpg"> -<img src="bloch-small.jpg" style="width:60%"> -</a> +{% include image.html file="sketch-full.png" width="67%" alt="Bloch sphere" %} The $$x$$, $$y$$ and $$z$$-axes represent the components of a spin-1/2-alike system, and their extremes are the eigenstates of the Pauli matrices: diff --git a/source/know/concept/bloch-sphere/sketch-full.png b/source/know/concept/bloch-sphere/sketch-full.png Binary files differnew file mode 100644 index 0000000..709aceb --- /dev/null +++ b/source/know/concept/bloch-sphere/sketch-full.png diff --git a/source/know/concept/bloch-sphere/sketch-half.avif b/source/know/concept/bloch-sphere/sketch-half.avif Binary files differnew file mode 100644 index 0000000..1752c9d --- /dev/null +++ b/source/know/concept/bloch-sphere/sketch-half.avif diff --git a/source/know/concept/bloch-sphere/sketch-half.jpg b/source/know/concept/bloch-sphere/sketch-half.jpg Binary files differnew file mode 100644 index 0000000..28b97d7 --- /dev/null +++ b/source/know/concept/bloch-sphere/sketch-half.jpg diff --git a/source/know/concept/bloch-sphere/sketch-half.png b/source/know/concept/bloch-sphere/sketch-half.png Binary files differnew file mode 100644 index 0000000..d48102d --- /dev/null +++ b/source/know/concept/bloch-sphere/sketch-half.png diff --git a/source/know/concept/bloch-sphere/sketch-half.webp b/source/know/concept/bloch-sphere/sketch-half.webp Binary files differnew file mode 100644 index 0000000..257145b --- /dev/null +++ b/source/know/concept/bloch-sphere/sketch-half.webp diff --git a/source/know/concept/deutsch-jozsa-algorithm/deutsch-circuit.avif b/source/know/concept/deutsch-jozsa-algorithm/deutsch-circuit.avif Binary files differnew file mode 100644 index 0000000..c498cd9 --- /dev/null +++ b/source/know/concept/deutsch-jozsa-algorithm/deutsch-circuit.avif diff --git a/source/know/concept/deutsch-jozsa-algorithm/deutsch-jozsa-circuit.avif b/source/know/concept/deutsch-jozsa-algorithm/deutsch-jozsa-circuit.avif Binary files differnew file mode 100644 index 0000000..2312ff3 --- /dev/null +++ b/source/know/concept/deutsch-jozsa-algorithm/deutsch-jozsa-circuit.avif diff --git a/source/know/concept/deutsch-jozsa-algorithm/index.md b/source/know/concept/deutsch-jozsa-algorithm/index.md index bbdd58d..5f2f268 100644 --- a/source/know/concept/deutsch-jozsa-algorithm/index.md +++ b/source/know/concept/deutsch-jozsa-algorithm/index.md @@ -27,6 +27,7 @@ while classical computers need up to $$2^{N - 1} + 1$$ queries for an $$N$$-bit $$x$$. + ## Deutsch algorithm The Deutsch algorithm handles the simplest case, @@ -40,9 +41,7 @@ In other words, we only need to determine if $$f(0) = f(1)$$ or $$f(0) \neq f(1) To do this, we use the following quantum circuit, where $$U_f$$ is the oracle we query: -<a href="deutsch-circuit.png"> -<img src="deutsch-circuit.png" style="width:48%"> -</a> +{% include image.html file="deutsch-circuit.png" width="48%" alt="Deutsch circuit" %} Due to unitarity constraints, the action of $$U_f$$ is defined to be as follows, @@ -134,16 +133,15 @@ A classical computer would need to query it twice, once with input $$x = 0$$, and again with $$x = 1$$. -## Full Deutsch-Jozsa algorithm + +## Deutsch-Jozsa algorithm The Deutsch-Jozsa algorithm generalizes the above to $$N$$-bit inputs $$x$$. We are promised that $$f(x)$$ is either constant or balanced; other possibilities are assumed to be impossible. This algorithm is then implemented by the following quantum circuit: -<a href="deutsch-jozsa-circuit.png"> -<img src="deutsch-jozsa-circuit.png" style="width:52%"> -</a> +{% include image.html file="deutsch-jozsa-circuit.png" width="52%" alt="Deutsch-Jozsa circuit" %} There are $$N$$ qubits in initial state $$\Ket{0}$$, and one in $$\Ket{1}$$. For clarity, the oracle $$U_f$$ works like so: diff --git a/source/know/concept/dispersive-broadening/index.md b/source/know/concept/dispersive-broadening/index.md index 4e4cf82..746eb6d 100644 --- a/source/know/concept/dispersive-broadening/index.md +++ b/source/know/concept/dispersive-broadening/index.md @@ -59,9 +59,7 @@ This phenomenon is illustrated below for our example of a Gaussian pulse with parameter values $$T_0 = 1\:\mathrm{ps}$$, $$P_0 = 1\:\mathrm{kW}$$, $$\beta_2 = -10 \:\mathrm{ps}^2/\mathrm{m}$$ and $$\gamma = 0$$: -<a href="pheno-disp.jpg"> -<img src="pheno-disp-small.jpg" style="width:100%"> -</a> +{% include image.html file="simulation-full.png" width="100%" alt="Dispersive broadening simulation results" %} The **instantaneous frequency** $$\omega_\mathrm{GVD}(z, t)$$, which describes the dominant angular frequency at a given point in the time domain, diff --git a/source/know/concept/dispersive-broadening/pheno-disp-small.jpg b/source/know/concept/dispersive-broadening/pheno-disp-small.jpg Binary files differdeleted file mode 100644 index 8c70eac..0000000 --- a/source/know/concept/dispersive-broadening/pheno-disp-small.jpg +++ /dev/null diff --git a/source/know/concept/dispersive-broadening/pheno-disp.jpg b/source/know/concept/dispersive-broadening/pheno-disp.jpg Binary files differdeleted file mode 100644 index a97312b..0000000 --- a/source/know/concept/dispersive-broadening/pheno-disp.jpg +++ /dev/null diff --git a/source/know/concept/dispersive-broadening/simulation-full.png b/source/know/concept/dispersive-broadening/simulation-full.png Binary files differnew file mode 100644 index 0000000..5ff78dc --- /dev/null +++ b/source/know/concept/dispersive-broadening/simulation-full.png diff --git a/source/know/concept/dispersive-broadening/simulation-half.avif b/source/know/concept/dispersive-broadening/simulation-half.avif Binary files differnew file mode 100644 index 0000000..7f5d2ad --- /dev/null +++ b/source/know/concept/dispersive-broadening/simulation-half.avif diff --git a/source/know/concept/dispersive-broadening/simulation-half.jpg b/source/know/concept/dispersive-broadening/simulation-half.jpg Binary files differnew file mode 100644 index 0000000..3d9a53a --- /dev/null +++ b/source/know/concept/dispersive-broadening/simulation-half.jpg diff --git a/source/know/concept/dispersive-broadening/simulation-half.png b/source/know/concept/dispersive-broadening/simulation-half.png Binary files differnew file mode 100644 index 0000000..a7b4aac --- /dev/null +++ b/source/know/concept/dispersive-broadening/simulation-half.png diff --git a/source/know/concept/dispersive-broadening/simulation-half.webp b/source/know/concept/dispersive-broadening/simulation-half.webp Binary files differnew file mode 100644 index 0000000..a781bc4 --- /dev/null +++ b/source/know/concept/dispersive-broadening/simulation-half.webp diff --git a/source/know/concept/fabry-perot-cavity/cavity.png b/source/know/concept/fabry-perot-cavity/cavity.png Binary files differdeleted file mode 100644 index f7b0c2a..0000000 --- a/source/know/concept/fabry-perot-cavity/cavity.png +++ /dev/null diff --git a/source/know/concept/fabry-perot-cavity/index.md b/source/know/concept/fabry-perot-cavity/index.md index 980fa54..6eefc6e 100644 --- a/source/know/concept/fabry-perot-cavity/index.md +++ b/source/know/concept/fabry-perot-cavity/index.md @@ -19,9 +19,8 @@ We divide the $$x$$-axis into three domains: left $$L$$, center $$C$$, and right The cavity $$C$$ has length $$\ell$$ and is centered on $$x = 0$$. Let $$n_L$$, $$n_C$$ and $$n_R$$ be the respective domains' refractive indices: -<a href="cavity.png"> -<img src="cavity.png" style="width:70%"> -</a> +{% include image.html file="sketch-full.png" width="70%" alt="Cavity structure" %} + ## Microscopic cavity @@ -129,6 +128,7 @@ $$A_1$$, $$A_2$$, $$A_3$$ or $$A_4$$ freely, and then the others are determined by $$k_m$$ and the field's continuity. + ## Macroscopic cavity Next, consider a "macroscopic" Fabry-Pérot cavity diff --git a/source/know/concept/fabry-perot-cavity/sketch-full.png b/source/know/concept/fabry-perot-cavity/sketch-full.png Binary files differnew file mode 100644 index 0000000..0568fd2 --- /dev/null +++ b/source/know/concept/fabry-perot-cavity/sketch-full.png diff --git a/source/know/concept/fabry-perot-cavity/sketch-half.avif b/source/know/concept/fabry-perot-cavity/sketch-half.avif Binary files differnew file mode 100644 index 0000000..e980a70 --- /dev/null +++ b/source/know/concept/fabry-perot-cavity/sketch-half.avif diff --git a/source/know/concept/fabry-perot-cavity/sketch-half.jpg b/source/know/concept/fabry-perot-cavity/sketch-half.jpg Binary files differnew file mode 100644 index 0000000..8ca9aad --- /dev/null +++ b/source/know/concept/fabry-perot-cavity/sketch-half.jpg diff --git a/source/know/concept/fabry-perot-cavity/sketch-half.png b/source/know/concept/fabry-perot-cavity/sketch-half.png Binary files differnew file mode 100644 index 0000000..38cace9 --- /dev/null +++ b/source/know/concept/fabry-perot-cavity/sketch-half.png diff --git a/source/know/concept/fabry-perot-cavity/sketch-half.webp b/source/know/concept/fabry-perot-cavity/sketch-half.webp Binary files differnew file mode 100644 index 0000000..d6be4d9 --- /dev/null +++ b/source/know/concept/fabry-perot-cavity/sketch-half.webp diff --git a/source/know/concept/feynman-diagram/boson.avif b/source/know/concept/feynman-diagram/boson.avif Binary files differnew file mode 100644 index 0000000..576e668 --- /dev/null +++ b/source/know/concept/feynman-diagram/boson.avif diff --git a/source/know/concept/feynman-diagram/interaction.png b/source/know/concept/feynman-diagram/boson.png Binary files differindex 3c2d093..3c2d093 100644 --- a/source/know/concept/feynman-diagram/interaction.png +++ b/source/know/concept/feynman-diagram/boson.png diff --git a/source/know/concept/feynman-diagram/example.avif b/source/know/concept/feynman-diagram/example.avif Binary files differnew file mode 100644 index 0000000..0dfa15c --- /dev/null +++ b/source/know/concept/feynman-diagram/example.avif diff --git a/source/know/concept/feynman-diagram/conservation.png b/source/know/concept/feynman-diagram/example.png Binary files differindex 1d19fd0..1d19fd0 100644 --- a/source/know/concept/feynman-diagram/conservation.png +++ b/source/know/concept/feynman-diagram/example.png diff --git a/source/know/concept/feynman-diagram/fermion-heavy.avif b/source/know/concept/feynman-diagram/fermion-heavy.avif Binary files differnew file mode 100644 index 0000000..6618736 --- /dev/null +++ b/source/know/concept/feynman-diagram/fermion-heavy.avif diff --git a/source/know/concept/feynman-diagram/fullgf.png b/source/know/concept/feynman-diagram/fermion-heavy.png Binary files differindex f376ef2..f376ef2 100644 --- a/source/know/concept/feynman-diagram/fullgf.png +++ b/source/know/concept/feynman-diagram/fermion-heavy.png diff --git a/source/know/concept/feynman-diagram/fermion-light.avif b/source/know/concept/feynman-diagram/fermion-light.avif Binary files differnew file mode 100644 index 0000000..196334b --- /dev/null +++ b/source/know/concept/feynman-diagram/fermion-light.avif diff --git a/source/know/concept/feynman-diagram/freegf.png b/source/know/concept/feynman-diagram/fermion-light.png Binary files differindex 632ccaf..632ccaf 100644 --- a/source/know/concept/feynman-diagram/freegf.png +++ b/source/know/concept/feynman-diagram/fermion-light.png diff --git a/source/know/concept/feynman-diagram/impurity.avif b/source/know/concept/feynman-diagram/impurity.avif Binary files differnew file mode 100644 index 0000000..1f78d10 --- /dev/null +++ b/source/know/concept/feynman-diagram/impurity.avif diff --git a/source/know/concept/feynman-diagram/perturbation.png b/source/know/concept/feynman-diagram/impurity.png Binary files differindex 2e8fbf6..2e8fbf6 100644 --- a/source/know/concept/feynman-diagram/perturbation.png +++ b/source/know/concept/feynman-diagram/impurity.png diff --git a/source/know/concept/feynman-diagram/index.md b/source/know/concept/feynman-diagram/index.md index c36e7df..ace8dbc 100644 --- a/source/know/concept/feynman-diagram/index.md +++ b/source/know/concept/feynman-diagram/index.md @@ -25,6 +25,7 @@ Below, we go through the most notable components of Feynman diagrams and how to translate them into a mathematical expression. + ## Real space The most common component is a **fermion line**, which represents @@ -37,9 +38,7 @@ Let the subscript $$I$$ refer to the and $$\mathcal{T}\{\}$$ denote the [time-ordered product](/know/concept/time-ordered-product/): -<a href="freegf.png"> -<img src="freegf.png" style="width:60%"> -</a> +{% include image.html file="fermion-light.png" width="60%" alt="Fermion line diagram" %} $$\begin{aligned} = i \hbar G_{s_2 s_1}^0(\vb{r}_2, t_2; \vb{r}_1, t_1) @@ -59,9 +58,7 @@ Less common is a **heavy fermion line**, representing 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/): -<a href="fullgf.png"> -<img src="fullgf.png" style="width:60%"> -</a> +{% include image.html file="fermion-heavy.png" width="60%" alt="Heavy fermion line diagram" %} $$\begin{aligned} = i \hbar G_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1) @@ -75,9 +72,7 @@ which we assume to be instantaneous, i.e. time-independent hence it starts and ends at the same time, and no arrow is drawn: -<a href="interaction.png"> -<img src="interaction.png" style="width:60%"> -</a> +{% include image.html file="boson.png" width="60%" alt="Boson/interaction line diagram" %} $$\begin{aligned} = \frac{1}{i \hbar} W_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1) @@ -99,9 +94,7 @@ $$\begin{aligned} One-body (time-dependent) operators $$\hat{V}$$ in $$\hat{H}_1$$ are instead represented by a special vertex: -<a href="perturbation.png"> -<img src="perturbation.png" style="width:35%"> -</a> +{% include image.html file="impurity.png" width="35%" alt="One-body perturbation (e.g. impurity) diagram" %} $$\begin{aligned} = \frac{1}{i \hbar} V_s(\vb{r}, t) @@ -148,6 +141,7 @@ so that a particle with a given spin propagates from vertex to vertex without getting flipped. + ## Fourier space If the system is time-independent and spatially uniform, @@ -177,9 +171,7 @@ Working in Fourier space allows us to simplify calculations. Consider the following diagram and the resulting expression, where $$\tilde{\vb{r}} = (\vb{r}, t)$$, and $$\tilde{\vb{k}} = (\vb{k}, \omega)$$: -<a href="conservation.png"> -<img src="conservation.png" style="width:40%"> -</a> +{% include image.html file="example.png" width="40%" alt="Example: fermion-fermion interaction" %} $$\begin{aligned} &= (i \hbar)^3 \sum_{s s'} \!\!\iint \dd{\tilde{\vb{r}}} \dd{\tilde{\vb{r}}'} @@ -274,6 +266,7 @@ then conservation removes all internal variables, so no integrals would be needed. + ## Imaginary time Feynman diagrams are also useful when working with diff --git a/source/know/concept/metacentric-height/index.md b/source/know/concept/metacentric-height/index.md index 3d0c9a5..3d81d44 100644 --- a/source/know/concept/metacentric-height/index.md +++ b/source/know/concept/metacentric-height/index.md @@ -26,9 +26,7 @@ does not coincide with the origin in general, as is illustrated in the following sketch of our choice of coordinate system: -<a href="sketch.jpg"> -<img src="sketch.jpg" style="width:67%"> -</a> +{% include image.html file="sketch-full.png" width="75%" alt="Boat's coordinate system" %} Here, $$B$$ is the **center of buoyancy**, equal to the center of mass of the volume of water displaced by the boat diff --git a/source/know/concept/metacentric-height/sketch-full.png b/source/know/concept/metacentric-height/sketch-full.png Binary files differnew file mode 100644 index 0000000..f76350b --- /dev/null +++ b/source/know/concept/metacentric-height/sketch-full.png diff --git a/source/know/concept/metacentric-height/sketch-half.avif b/source/know/concept/metacentric-height/sketch-half.avif Binary files differnew file mode 100644 index 0000000..e6be2dc --- /dev/null +++ b/source/know/concept/metacentric-height/sketch-half.avif diff --git a/source/know/concept/metacentric-height/sketch-half.jpg b/source/know/concept/metacentric-height/sketch-half.jpg Binary files differnew file mode 100644 index 0000000..066bc93 --- /dev/null +++ b/source/know/concept/metacentric-height/sketch-half.jpg diff --git a/source/know/concept/metacentric-height/sketch-half.png b/source/know/concept/metacentric-height/sketch-half.png Binary files differnew file mode 100644 index 0000000..6835bbc --- /dev/null +++ b/source/know/concept/metacentric-height/sketch-half.png diff --git a/source/know/concept/metacentric-height/sketch-half.webp b/source/know/concept/metacentric-height/sketch-half.webp Binary files differnew file mode 100644 index 0000000..70af91d --- /dev/null +++ b/source/know/concept/metacentric-height/sketch-half.webp diff --git a/source/know/concept/metacentric-height/sketch.jpg b/source/know/concept/metacentric-height/sketch.jpg Binary files differdeleted file mode 100644 index 6a0544b..0000000 --- a/source/know/concept/metacentric-height/sketch.jpg +++ /dev/null diff --git a/source/know/concept/modulational-instability/index.md b/source/know/concept/modulational-instability/index.md index a01293c..e29b2d5 100644 --- a/source/know/concept/modulational-instability/index.md +++ b/source/know/concept/modulational-instability/index.md @@ -172,9 +172,7 @@ $$\begin{aligned} = \sqrt{P_0} \sech\!\Big(\frac{t}{T_0}\Big) \end{aligned}$$ -<a href="pheno-mi.jpg"> -<img src="pheno-mi-small.jpg" style="width:100%"> -</a> +{% include image.html file="simulation-full.png" width="100%" alt="Modulational instability simulation results" %} Where $$L_\mathrm{NL} = 1/(\gamma P_0)$$ is the characteristic length of nonlinear effects. 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To demonstrate this, spectrograms of the same simulation @@ -53,9 +51,7 @@ After OWB, a train of small waves falls off the edges, which eventually melt together, leading to a trapezoid shape in the $$t$$-domain. Dispersive broadening then continues normally: -<a href="pheno-break-sgram.jpg"> -<img src="pheno-break-sgram-small.jpg" style="width:80%"> -</a> +{% include image.html file="spectrograms-full.png" width="100%" alt="Spectrograms of pulse shape evolution" %} We call the distance at which the wave breaks $$L_\mathrm{WB}$$, and would like to analytically predict it. @@ -183,9 +179,7 @@ $$\begin{aligned} This prediction for $$L_\mathrm{WB}$$ appears to agree well with the OWB observed in the simulation: -<a href="pheno-break.jpg"> -<img src="pheno-break-small.jpg" style="width:100%"> -</a> +{% include image.html file="simulation-full.png" width="100%" alt="Optical wave breaking simulation results" %} Because all spectral broadening up to $$L_\mathrm{WB}$$ is caused by SPM, whose frequency behaviour is known, it is in fact possible to draw diff --git a/source/know/concept/optical-wave-breaking/pheno-break-inst-small.jpg b/source/know/concept/optical-wave-breaking/pheno-break-inst-small.jpg 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is illustrated below, excluding the swapping part to get the right order. Here, $$R_m$$ means $$R_\phi$$ with $$\phi = 2 \pi / 2^m$$: -<a href="qft-circuit-noswap.png"> -<img src="qft-circuit-noswap.png" style="width:100%"> -</a> +{% include image.html file="qft-circuit-noswap.png" width="100%" alt="QFT circuit, without final swap" %} Again, note how the inputs $$\Ket{x_j}$$ and outputs $$\Ket{k_j}$$ are in the opposite order. The complete circuit, including the swapping at the end, therefore looks like this: -<a href="qft-circuit-swap.png"> -<img src="qft-circuit-swap.png" style="width:85%"> -</a> +{% include image.html file="qft-circuit-swap.png" width="85%" alt="QFT circuit, including final swap" %} For each of the $$n$$ qubits, $$\mathcal{O}(n)$$ gates are applied, so overall the QFT algorithm is $$\mathcal{O}(n^2)$$. diff --git a/source/know/concept/quantum-fourier-transform/qft-circuit-noswap.avif b/source/know/concept/quantum-fourier-transform/qft-circuit-noswap.avif Binary files differnew file mode 100644 index 0000000..bee92cd --- /dev/null +++ b/source/know/concept/quantum-fourier-transform/qft-circuit-noswap.avif diff --git a/source/know/concept/quantum-fourier-transform/qft-circuit-swap.avif b/source/know/concept/quantum-fourier-transform/qft-circuit-swap.avif Binary files differnew file mode 100644 index 0000000..6f41319 --- /dev/null +++ b/source/know/concept/quantum-fourier-transform/qft-circuit-swap.avif diff --git a/source/know/concept/quantum-gate/cnot.avif b/source/know/concept/quantum-gate/cnot.avif Binary files differnew file mode 100644 index 0000000..75e36ae --- /dev/null +++ b/source/know/concept/quantum-gate/cnot.avif diff --git a/source/know/concept/quantum-gate/cu.avif b/source/know/concept/quantum-gate/cu.avif Binary files differnew file mode 100644 index 0000000..e58c66a --- /dev/null +++ b/source/know/concept/quantum-gate/cu.avif diff --git a/source/know/concept/quantum-gate/index.md b/source/know/concept/quantum-gate/index.md index 8c251be..e8ff579 100644 --- a/source/know/concept/quantum-gate/index.md +++ b/source/know/concept/quantum-gate/index.md @@ -14,6 +14,7 @@ the number of possible quantum gates is uncountably infinite, so we only consider the most important examples here. + ## One-qubit gates As an example, consider the following must general single-qubit state $$\Ket{\psi}$$: @@ -165,6 +166,7 @@ This is the definition of universality: any state can be approximated. + ## Two-qubit gates As an example, let us consider @@ -202,9 +204,7 @@ but not always in the basis of $$\Ket{0}_1$$, $$\Ket{1}_1$$, $$\Ket{0}_2$$ and $ With that said, the first two-qubit gate is $$\mathrm{SWAP}$$, which simply swaps $$\Ket{\psi_1}$$ and $$\Ket{\psi_2}$$: -<a href="swap.png"> -<img src="swap.png" style="width:22%"> -</a> +{% include image.html file="swap.png" width="22%" alt="SWAP gate diagram" %} $$\begin{aligned} \boxed{ @@ -231,9 +231,7 @@ $$\begin{aligned} Next, there is the **controlled NOT gate** $$\mathrm{CNOT}$$, which "flips" (applies $$X$$ to) $$\Ket{\psi_2}$$ if $$\Ket{\psi_1}$$ is true: -<a href="cnot.png"> -<img src="cnot.png" style="width:22%"> -</a> +{% include image.html file="cnot.png" width="22%" alt="CNOT gate diagram" %} $$\begin{aligned} \boxed{ @@ -258,9 +256,7 @@ More generally, from every one-qubit gate $$U$$, we can define a two-qubit **controlled U gate** $$\mathrm{CU}$$, which applies $$U$$ to $$\Ket{\psi_2}$$ if $$\Ket{\psi_1}$$ is true: -<a href="cu.png"> -<img src="cu.png" style="width:22%"> -</a> +{% include image.html file="cu.png" width="22%" alt="CU gate diagram" %} $$\begin{aligned} \boxed{ diff --git a/source/know/concept/quantum-gate/swap.avif b/source/know/concept/quantum-gate/swap.avif Binary files differnew file mode 100644 index 0000000..a70c43c --- /dev/null +++ b/source/know/concept/quantum-gate/swap.avif diff --git a/source/know/concept/random-phase-approximation/dyson.avif b/source/know/concept/random-phase-approximation/dyson.avif Binary files differnew file mode 100644 index 0000000..dc80856 --- /dev/null +++ b/source/know/concept/random-phase-approximation/dyson.avif diff --git a/source/know/concept/random-phase-approximation/index.md b/source/know/concept/random-phase-approximation/index.md index 0f53136..698e1e7 100644 --- a/source/know/concept/random-phase-approximation/index.md +++ b/source/know/concept/random-phase-approximation/index.md @@ -71,23 +71,17 @@ leaving only the single most divergent one at each order $$n$$, i.e. the ones where all $$n$$ interaction lines carry the same momentum and energy: -<a href="rpasigma.png"> -<img src="rpasigma.png" style="width:92%"> -</a> +{% include image.html file="self-energy.png" width="92%" alt="RPA self-energy definition" %} Where we have defined the **screened interaction** $$W^\mathrm{RPA}$$, denoted by a double wavy line: -<a href="screened.png"> -<img src="screened.png" style="width:95%"> -</a> +{% include image.html file="interaction.png" width="95%" alt="RPA screened interaction definition" %} Rearranging the above sequence of diagrams quickly leads to the following [Dyson equation](/know/concept/dyson-equation/): -<a href="dyson.png"> -<img src="dyson.png" style="width:55%"> -</a> +{% include image.html file="dyson.png" width="55%" alt="Dyson equation for screened interaction" %} In Fourier space, this equation's linear shape means it is algebraic, so we can write it out: @@ -104,9 +98,7 @@ with an internal wavevector $$\vb{q}$$, fermionic frequency $$i \omega_m^F$$, an Abbreviating $$\tilde{\vb{k}} \equiv (\vb{k}, i \omega_n^B)$$ and $$\tilde{\vb{q}} \equiv (\vb{q}, i \omega_n^F)$$: -<a href="pairbubble.png"> -<img src="pairbubble.png" style="width:45%"> -</a> +{% include image.html file="pairbubble.png" width="45%" alt="Internal variables of pair-bubble diagram" %} We isolate the Dyson equation for $$W^\mathrm{RPA}$$, which reveals its physical interpretation as a *screened* interaction: diff --git a/source/know/concept/random-phase-approximation/interaction.avif b/source/know/concept/random-phase-approximation/interaction.avif Binary files differnew file mode 100644 index 0000000..f3f6cbe --- /dev/null +++ b/source/know/concept/random-phase-approximation/interaction.avif diff --git a/source/know/concept/random-phase-approximation/screened.png b/source/know/concept/random-phase-approximation/interaction.png Binary files differindex bb5ccdd..bb5ccdd 100644 --- a/source/know/concept/random-phase-approximation/screened.png +++ b/source/know/concept/random-phase-approximation/interaction.png diff --git a/source/know/concept/random-phase-approximation/pairbubble.avif b/source/know/concept/random-phase-approximation/pairbubble.avif Binary files differnew file mode 100644 index 0000000..bca964e --- /dev/null +++ b/source/know/concept/random-phase-approximation/pairbubble.avif diff --git a/source/know/concept/random-phase-approximation/self-energy.avif b/source/know/concept/random-phase-approximation/self-energy.avif Binary files differnew file mode 100644 index 0000000..3ddfed7 --- /dev/null +++ b/source/know/concept/random-phase-approximation/self-energy.avif diff --git a/source/know/concept/random-phase-approximation/rpasigma.png b/source/know/concept/random-phase-approximation/self-energy.png Binary files differindex fedc9f5..fedc9f5 100644 --- a/source/know/concept/random-phase-approximation/rpasigma.png +++ b/source/know/concept/random-phase-approximation/self-energy.png diff --git a/source/know/concept/repetition-code/bit-flip-detect.avif b/source/know/concept/repetition-code/bit-flip-detect.avif Binary files differnew file mode 100644 index 0000000..83d509a --- /dev/null +++ b/source/know/concept/repetition-code/bit-flip-detect.avif diff --git a/source/know/concept/repetition-code/bit-flip-encode.avif b/source/know/concept/repetition-code/bit-flip-encode.avif Binary files differnew file mode 100644 index 0000000..37bc495 --- /dev/null +++ b/source/know/concept/repetition-code/bit-flip-encode.avif diff --git a/source/know/concept/repetition-code/index.md b/source/know/concept/repetition-code/index.md index 99ac630..678211e 100644 --- a/source/know/concept/repetition-code/index.md +++ b/source/know/concept/repetition-code/index.md @@ -27,6 +27,7 @@ albeit with some complications, as discussed below. + ## Bit flip code Suppose that we want to detect errors in @@ -76,9 +77,7 @@ $$\begin{aligned} Such a transformation is easy to achieve with the following sequence of [quantum gates](/know/concept/quantum-gate/): -<a href="bit-flip-encode.png"> -<img src="bit-flip-encode.png" style="width:32%"> -</a> +{% include image.html file="bit-flip-encode.png" width="32%" alt="Bit flip code encoder" %} So, a little while after encoding the state $$\Ket{\psi}$$ like that, a bit flip occurs on the 2nd qubit: @@ -166,31 +165,61 @@ But by using both, we know exactly which qubit was flipped thanks to the eigenvalues: <table style="width:30%;margin:auto;text-align:center;"> - <tr> - <th>Error</th> - <th>$$ZZI$$</th> - <th>$$IZZ$$</th> - </tr> - <tr> - <td>$$I$$</td> - <td>$$+1$$</td> - <td>$$+1$$</td> - </tr> - <tr> - <td>$$X_1$$</td> - <td>$$-1$$</td> - <td>$$+1$$</td> - </tr> - <tr> - <td>$$X_2$$</td> - <td>$$-1$$</td> - <td>$$-1$$</td> - </tr> - <tr> - <td>$$X_1$$</td> - <td>$$+1$$</td> - <td>$$-1$$</td> - </tr> +<tr> + <th> + Error + </th> + <th markdown="1"> + $$ZZI$$ + </th> + <th markdown="1"> + $$IZZ$$ + </th> +</tr> +<tr> + <td markdown="1"> + $$I$$ + </td> + <td markdown="1"> + $$+1$$ + </td> + <td markdown="1"> + $$+1$$ + </td> +</tr> +<tr> + <td markdown="1"> + $$X_1$$ + </td> + <td markdown="1"> + $$-1$$ + </td> + <td markdown="1"> + $$+1$$ + </td> +</tr> +<tr> + <td markdown="1"> + $$X_2$$ + </td> + <td markdown="1"> + $$-1$$ + </td> + <td markdown="1"> + $$-1$$ + </td> +</tr> +<tr> + <td markdown="1"> + $$X_1$$ + </td> + <td markdown="1"> + $$+1$$ + </td> + <td markdown="1"> + $$-1$$ + </td> +</tr> </table> Where e.g. $$X_3$$ denotes that the 3rd qubit was flipped. @@ -202,9 +231,7 @@ without affecting $$\ket{\overline{\psi}}$$ itself, by applying $$\mathrm{CNOT}$$s to some ancillary qubits and then measuring those: -<a href="bit-flip-detect.png"> -<img src="bit-flip-detect.png" style="width:62%"> -</a> +{% include image.html file="bit-flip-detect.png" width="62%" alt="Bit flip code decoder" %} The two measurements, respectively representing $$ZZI$$ and $$IZZ$$, yield $$\Ket{1}$$ if a bit flip definitely occurred, @@ -213,6 +240,7 @@ There is no entanglement, so the input is untouched. + ## Phase flip code The above system protects us against all single-qubit bit flips. @@ -254,23 +282,20 @@ $$\begin{aligned} = \alpha \Ket{+\!+\!+} + \beta \Ket{-\!-\!-} \end{aligned}$$ -<a href="phase-flip-encode.png"> -<img src="phase-flip-encode.png" style="width:40%"> -</a> +{% include image.html file="phase-flip-encode.png" width="40%" alt="Phase flip code encoder" %} A phase flip along the $$Z$$-axis corresponds to a bit flip along the $$X$$-axis $$\Ket{+} \to \Ket{-}$$. In this case, the stabilizers are $$XXI$$ and $$IXX$$, and the error detection circuit is as follows: -<a href="phase-flip-detect.png"> -<img src="phase-flip-detect.png" style="width:70%"> -</a> +{% include image.html file="phase-flip-detect.png" width="70%" alt="Phase flip code decoder" %} This system protects us against all single-qubit phase flips, but not against bit flips. + ## Shor code What kind of repetition code would we need @@ -307,9 +332,7 @@ This encoding is achieved by the following quantum circuit, which simply consists of the phase flip encoder, followed by 3 copies of the bit flip encoder: -<a href="shor-code-encode.png"> -<img src="shor-code-encode.png" style="width:55%"> -</a> +{% include image.html file="shor-code-encode.png" width="55%" alt="Shor code encoder" %} We thus use 9 physical qubits to store 1 logical qubit. Fortunately, more efficient schemes exist. diff --git a/source/know/concept/repetition-code/phase-flip-detect.avif b/source/know/concept/repetition-code/phase-flip-detect.avif Binary files differnew file mode 100644 index 0000000..b69044b --- /dev/null +++ b/source/know/concept/repetition-code/phase-flip-detect.avif diff --git a/source/know/concept/repetition-code/phase-flip-encode.avif b/source/know/concept/repetition-code/phase-flip-encode.avif Binary files differnew file mode 100644 index 0000000..bdefddd --- /dev/null +++ b/source/know/concept/repetition-code/phase-flip-encode.avif diff --git a/source/know/concept/repetition-code/shor-code-encode.avif b/source/know/concept/repetition-code/shor-code-encode.avif Binary files differnew file mode 100644 index 0000000..37be14a --- /dev/null +++ b/source/know/concept/repetition-code/shor-code-encode.avif diff --git a/source/know/concept/rutherford-scattering/index.md b/source/know/concept/rutherford-scattering/index.md index a7375d5..6f5a21f 100644 --- a/source/know/concept/rutherford-scattering/index.md +++ b/source/know/concept/rutherford-scattering/index.md @@ -19,9 +19,7 @@ Let 2 be initially at rest, and 1 approach it with velocity $$\vb{v}_1$$. Coulomb repulsion causes 1 to deflect by an angle $$\theta$$, and pushes 2 away in the process: -<a href="two-body.png"> -<img src="two-body.png" style="width:50%"> -</a> +{% include image.html file="two-body-full.png" width="50%" alt="Two-body repulsive 'collision'" %} Here, $$b$$ is called the **impact parameter**. 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100644 --- a/source/know/concept/self-energy/index.md +++ b/source/know/concept/self-energy/index.md @@ -204,9 +204,7 @@ that exactly $$2^m m!$$ diagrams at each order are topologically equivalent, so we are left with non-equivalent diagrams only. Let $$G(b,a) = G_{ba}$$: -<a href="fullgf.png"> -<img src="fullgf.png" style="width:90%"> -</a> +{% include image.html file="expansion.png" width="90%" alt="Full expansion of G in Feynman diagrams" %} A **reducible diagram** is a Feynman diagram that can be cut in two valid diagrams @@ -217,9 +215,7 @@ At last, we define the **self-energy** $$\Sigma(y,x)$$ as the sum of all irreducible terms in $$G(b,a)$$, after removing the two external lines from/to $$a$$ and $$b$$: -<a href="selfenergy.png"> -<img src="selfenergy.png" style="width:90%"> -</a> +{% include image.html file="definition.png" width="90%" alt="Definition of self-energy" %} Despite its appearance, the self-energy has the semantics of a line, so it has two endpoints over which to integrate if necessary. @@ -238,9 +234,7 @@ Thanks to this recursive structure, you can convince youself that $$G(b,a)$$ obeys a [Dyson equation](/know/concept/dyson-equation/) involving $$\Sigma(y, x)$$: -<a href="dyson.png"> -<img src="dyson.png" style="width:95%"> -</a> +{% include image.html file="dyson.png" width="95%" alt="Dyson equation in Feynman diagrams" %} This makes sense: in the "normal" Dyson equation we have a one-body perturbation instead of $$\Sigma$$, diff --git a/source/know/concept/self-phase-modulation/index.md b/source/know/concept/self-phase-modulation/index.md index f13ad2f..48ea20b 100644 --- a/source/know/concept/self-phase-modulation/index.md +++ b/source/know/concept/self-phase-modulation/index.md @@ -66,9 +66,7 @@ $$\begin{aligned} A(z, t) = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big) \exp\!\bigg( i \gamma z P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) \end{aligned}$$ -<a href="pheno-spm.jpg"> -<img src="pheno-spm-small.jpg" style="width:100%"> -</a> +{% include image.html file="simulation-full.png" width="100%" alt="Self-phase modulation simulation results" %} The **instantaneous frequency** $$\omega_\mathrm{SPM}(z, t)$$, which describes the dominant angular frequency at a given point in 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diff --git a/source/know/concept/self-steepening/index.md b/source/know/concept/self-steepening/index.md index a8bc3c2..f934ab7 100644 --- a/source/know/concept/self-steepening/index.md +++ b/source/know/concept/self-steepening/index.md @@ -113,9 +113,7 @@ $$L_\mathrm{shock} = 0.847\,\mathrm{m}$$, which turns out to be accurate, although the simulation breaks down due to insufficient resolution: -<a href="pheno-steep.jpg"> -<img src="pheno-steep-small.jpg" style="width:100%"> -</a> +{% include image.html file="simulation-full.png" width="100%" alt="Self-steepening simulation results" %} Unfortunately, self-steepening cannot be simulated perfectly: as the pulse approaches $$L_\mathrm{shock}$$, its spectrum broadens to infinite diff --git a/source/know/concept/self-steepening/pheno-steep-small.jpg b/source/know/concept/self-steepening/pheno-steep-small.jpg Binary files differdeleted file mode 100644 index bb2a158..0000000 --- a/source/know/concept/self-steepening/pheno-steep-small.jpg 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algorithm was designed to factorize an integer $$N$$, @@ -46,9 +47,7 @@ The period $$s$$ is the smallest integer satisfying $$f(x) = f(x+s)$$. To do this, the following $$2q$$-qubit quantum circuit is used, with $$q$$ chosen so that $$N^2 \le 2^q < 2 N^2$$: -<a href="shors-circuit.png"> -<img src="shors-circuit.png" style="width:70%"> -</a> +{% include image.html file="shors-circuit.png" width="70%" alt="Shor's circuit" %} Here, $$\mathrm{QFT}_q$$ refers to the $$q$$-qubit [quantum Fourier transform](/know/concept/quantum-fourier-transform/), diff --git a/source/know/concept/shors-algorithm/shors-circuit.avif b/source/know/concept/shors-algorithm/shors-circuit.avif Binary files differnew file mode 100644 index 0000000..94b3925 --- /dev/null +++ b/source/know/concept/shors-algorithm/shors-circuit.avif diff --git a/source/know/concept/simons-algorithm/index.md b/source/know/concept/simons-algorithm/index.md index 5502837..294912b 100644 --- a/source/know/concept/simons-algorithm/index.md +++ b/source/know/concept/simons-algorithm/index.md @@ -52,9 +52,7 @@ A quantum computer needs to query $$f$$ only $$\mathcal{O}(n)$$ times, although the exact number varies due to the algorithm's probabilistic nature. It uses the following circuit: -<a href="simons-circuit.png"> -<img src="simons-circuit.png" style="width:52%"> -</a> +{% include image.html file="simons-circuit.png" width="52%" alt="Simon's circuit" %} The XOR oracle $$U_f$$ implements $$f$$, and has the following action for $$n$$-bit $$a$$ and $$b$$: @@ -98,7 +96,6 @@ $$\begin{aligned} &\frac{1}{2^n} \sum_{x = 0}^{2^n - 1} \bigg( \sum_{y = 0}^{2^n - 1} (-1)^{x \cdot y} \Ket{y} \bigg) \Ket{f(x)} \end{aligned}$$ - Next, we measure all qubits. 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$$I_\ell(\rho)$$ do not decay at all, +Of the remaining candidates, $$\ln(r)$$, $$r^\ell$$ and $$I_\ell(\rho)$$ do not decay at all, leading to the following $$R_o$$: $$\begin{aligned} @@ -394,9 +392,7 @@ An example graphical solution of the transcendental equation is illustrated below for a fiber with $$V = 5$$, where red and blue respectively denote the left and right-hand side: -<a href="modes.jpg"> -<img src="modes-small.jpg" style="width:100%"> -</a> +{% include image.html file="transcendental-full.png" width="100%" alt="Graphical solution of transcendental equation" %} This shows that each $$\mathrm{LP}_{\ell m}$$ has an associated cut-off $$V_{\ell m}$$, so that if $$V > V_{\ell m}$$ then $$\mathrm{LP}_{lm}$$ exists, diff --git a/source/know/concept/step-index-fiber/modes-small.jpg b/source/know/concept/step-index-fiber/modes-small.jpg Binary files differdeleted file mode 100644 index cd9c2a2..0000000 --- a/source/know/concept/step-index-fiber/modes-small.jpg +++ /dev/null diff 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7de7600..ba9d3ba 100644 --- a/source/know/concept/toffoli-gate/index.md +++ b/source/know/concept/toffoli-gate/index.md @@ -16,43 +16,31 @@ of which it returns $$A$$ and $$B$$ unchanged, and flips $$C$$ if both $$A$$ and $$B$$ are true. In circuit diagrams, its representation is: -<a href="toffoli.png"> -<img src="toffoli.png" style="width:19%"> -</a> +{% include image.html file="toffoli.png" width="19%" alt="Toffoli gate symbol" %} This gate is reversible, because $$A$$ and $$B$$ are preserved, and are all you need to reconstruct to $$C$$. Moreover, this gate is universal, because we can make a NAND gate from it: -<a href="nand.png"> -<img src="nand.png" style="width:38%"> -</a> +{% include image.html file="nand.png" width="38%" alt="NAND gate made of Toffoli gate" %} A NAND is enough to implement every conceivable circuit. That said, we can efficiently implement NOT, AND, and XOR using a single Toffoli gate too. Note that NOT is a special case of NAND: -<a href="not.png"> -<img src="not.png" style="width:32%"> -</a> +{% include image.html file="not.png" width="32%" alt="NOT gate made of Toffoli gate" %} -<a href="and.png"> -<img src="and.png" style="width:35%"> -</a> +{% include image.html file="and.png" width="35%" alt="AND gate made of Toffoli gate" %} -<a href="xor.png"> -<img src="xor.png" style="width:35%"> -</a> +{% include image.html file="xor.png" width="35%" alt="XOR gate made of Toffoli gate" %} Using these, we can, as an example, make an OR gate from three Toffoli gates, thanks to the fact that $$A \lor B = \neg (\neg A \land \neg B)$$, i.e. OR is NAND of NOT $$A$$ and NOT $$B$$: -<a href="or.png"> -<img src="or.png" style="width:50%"> -</a> +{% include image.html file="or.png" width="50%" alt="OR gate made of Toffoli gates" %} Thanks to its reversibility and universality, the Toffoli gate is interesting for quantum computing. diff --git a/source/know/concept/toffoli-gate/nand.avif b/source/know/concept/toffoli-gate/nand.avif Binary files differnew file mode 100644 index 0000000..53a5d65 --- /dev/null +++ b/source/know/concept/toffoli-gate/nand.avif diff --git a/source/know/concept/toffoli-gate/not.avif b/source/know/concept/toffoli-gate/not.avif Binary files differnew file mode 100644 index 0000000..b277a83 --- /dev/null +++ b/source/know/concept/toffoli-gate/not.avif diff --git a/source/know/concept/toffoli-gate/or.avif b/source/know/concept/toffoli-gate/or.avif Binary files differnew file mode 100644 index 0000000..82dd391 --- /dev/null +++ b/source/know/concept/toffoli-gate/or.avif diff --git a/source/know/concept/toffoli-gate/toffoli.avif b/source/know/concept/toffoli-gate/toffoli.avif Binary files differnew file mode 100644 index 0000000..9c34431 --- /dev/null +++ b/source/know/concept/toffoli-gate/toffoli.avif diff --git a/source/know/concept/toffoli-gate/xor.avif b/source/know/concept/toffoli-gate/xor.avif Binary files differnew file mode 100644 index 0000000..8ed742d --- /dev/null +++ b/source/know/concept/toffoli-gate/xor.avif |
