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‎doc/pub/week1/html/week1-bs.html‎

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@@ -264,32 +264,17 @@
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None,
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'exercise-set-for-first-week-to-be-discussed-january-28'),
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('Exewrcise 1: Bell states', 2, None, 'exewrcise-1-bell-states'),
268-
('And the next two', 2, None, 'and-the-next-two'),
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('Exercise 2: Entangled state',
267+
('1: Commutator identities', 2, None, '1-commutator-identities'),
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('2: Pauli matrices', 2, None, '2-pauli-matrices'),
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('3: Shared eigenvectors', 2, None, '3-shared-eigenvectors'),
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('4: One-qubit basis and Pauli matrices',
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'exercise-2-entangled-state'),
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('Exercise 3: Commutator identities',
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'4-one-qubit-basis-and-pauli-matrices'),
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('5: Hadamard and Phase gates',
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2,
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None,
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'exercise-3-commutator-identities'),
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('Exercise 4: Pauli matrices',
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2,
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None,
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'exercise-4-pauli-matrices'),
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('Exercise 5: Shared eigenvectors',
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2,
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None,
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'exercise-5-shared-eigenvectors'),
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('Exercise 6: One-qubit basis and Pauli matrices',
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2,
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None,
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'exercise-6-one-qubit-basis-and-pauli-matrices'),
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('Exercise 7: Hadamard and Phase gates',
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2,
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None,
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'exercise-7-hadamard-and-phase-gates')]}
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'5-hadamard-and-phase-gates')]}
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end of tocinfo -->
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<body>
@@ -429,14 +414,11 @@
429414
<!-- navigation toc: --> <li><a href="#explicit-results" style="font-size: 80%;">Explicit results</a></li>
430415
<!-- navigation toc: --> <li><a href="#the-spectral-decomposition" style="font-size: 80%;">The spectral decomposition</a></li>
431416
<!-- navigation toc: --> <li><a href="#exercise-set-for-first-week-to-be-discussed-january-28" style="font-size: 80%;">Exercise set for first week, to be discussed January 28</a></li>
432-
<!-- navigation toc: --> <li><a href="#exewrcise-1-bell-states" style="font-size: 80%;">Exewrcise 1: Bell states</a></li>
433-
<!-- navigation toc: --> <li><a href="#and-the-next-two" style="font-size: 80%;">And the next two</a></li>
434-
<!-- navigation toc: --> <li><a href="#exercise-2-entangled-state" style="font-size: 80%;">Exercise 2: Entangled state</a></li>
435-
<!-- navigation toc: --> <li><a href="#exercise-3-commutator-identities" style="font-size: 80%;">Exercise 3: Commutator identities</a></li>
436-
<!-- navigation toc: --> <li><a href="#exercise-4-pauli-matrices" style="font-size: 80%;">Exercise 4: Pauli matrices</a></li>
437-
<!-- navigation toc: --> <li><a href="#exercise-5-shared-eigenvectors" style="font-size: 80%;">Exercise 5: Shared eigenvectors</a></li>
438-
<!-- navigation toc: --> <li><a href="#exercise-6-one-qubit-basis-and-pauli-matrices" style="font-size: 80%;">Exercise 6: One-qubit basis and Pauli matrices</a></li>
439-
<!-- navigation toc: --> <li><a href="#exercise-7-hadamard-and-phase-gates" style="font-size: 80%;">Exercise 7: Hadamard and Phase gates</a></li>
417+
<!-- navigation toc: --> <li><a href="#1-commutator-identities" style="font-size: 80%;">1: Commutator identities</a></li>
418+
<!-- navigation toc: --> <li><a href="#2-pauli-matrices" style="font-size: 80%;">2: Pauli matrices</a></li>
419+
<!-- navigation toc: --> <li><a href="#3-shared-eigenvectors" style="font-size: 80%;">3: Shared eigenvectors</a></li>
420+
<!-- navigation toc: --> <li><a href="#4-one-qubit-basis-and-pauli-matrices" style="font-size: 80%;">4: One-qubit basis and Pauli matrices</a></li>
421+
<!-- navigation toc: --> <li><a href="#5-hadamard-and-phase-gates" style="font-size: 80%;">5: Hadamard and Phase gates</a></li>
440422

441423
</ul>
442424
</li>
@@ -2366,64 +2348,32 @@ <h2 id="exercise-set-for-first-week-to-be-discussed-january-28" class="anchor">E
23662348
</p>
23672349

23682350
<!-- !split -->
2369-
<h2 id="exewrcise-1-bell-states" class="anchor">Exewrcise 1: Bell states </h2>
2370-
2371-
<p>Show that the so-called Bell states listed here (and to be encountered many times in this course) form an orthogonal basis</p>
2372-
$$
2373-
\vert \Phi^+\rangle = \frac{1}{\sqrt{2}}\left[\vert 00\rangle +\vert 11\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 1\end{bmatrix},
2374-
$$
2375-
2376-
2377-
$$
2378-
\vert \Phi^-\rangle = \frac{1}{\sqrt{2}}\left[\vert 00\rangle -\vert 11\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 0 \\ 0 \\ -1\end{bmatrix},
2379-
$$
2380-
2381-
2382-
<!-- !split -->
2383-
<h2 id="and-the-next-two" class="anchor">And the next two </h2>
2384-
$$
2385-
\vert \Psi^+\rangle = \frac{1}{\sqrt{2}}\left[\vert 10\rangle +\vert 01\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 0 \\ 1 \\ 1 \\ 0\end{bmatrix},
2386-
$$
2387-
2388-
<p>and</p>
2389-
2390-
$$
2391-
\vert \Psi^-\rangle = \frac{1}{\sqrt{2}}\left[\vert 10\rangle -\vert 01\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 0 \\ 1 \\ -1 \\ 0\end{bmatrix}.
2392-
$$
2393-
2394-
2395-
<!-- !split -->
2396-
<h2 id="exercise-2-entangled-state" class="anchor">Exercise 2: Entangled state </h2>
2397-
2398-
<p>Show that the state \( \alpha \vert 00\rangle+\beta\vert 11\rangle \) cannot be written as the product of the tensor product of two states and is thus entangle. The constants \( \alpha \) and \( \beta \) are both nonzero.</p>
2399-
2400-
<!-- !split -->
2401-
<h2 id="exercise-3-commutator-identities" class="anchor">Exercise 3: Commutator identities </h2>
2351+
<h2 id="1-commutator-identities" class="anchor">1: Commutator identities </h2>
24022352
<p>Prove the following commutator relations for different operators (marked with a hat)</p>
24032353
<ol>
24042354
<li> \( [\hat{A}+\hat{B},\hat{C}]= [\hat{A},\hat{C}]+[\hat{B},\hat{C}] \);</li>
24052355
<li> \( [\hat{A},\hat{B}\hat{C}]= [\hat{A},\hat{B}]\hat{C}+\hat{B}[\hat{A},\hat{C}] \); and</li>
24062356
<li> \( [\hat{A},[\hat{B}\hat{C}]]= [\hat{B},[\hat{C},\hat{A}]]+[\hat{C},[\hat{A},\hat{B}]]=0 \) (the so-called Jacobi identity).</li>
24072357
</ol>
24082358
<!-- !split -->
2409-
<h2 id="exercise-4-pauli-matrices" class="anchor">Exercise 4: Pauli matrices </h2>
2359+
<h2 id="2-pauli-matrices" class="anchor">2: Pauli matrices </h2>
24102360
<ol>
24112361
<li> Set up the commutation rules for the Pauli matrices, that is find \( [\sigma_i,\sigma_j] \) where \( i,j=x,y,z \).</li>
24122362
<li> We define \( \boldsymbol{X}=\sigma_x \), \( \boldsymbol{Y}=\sigma_y \) and \( \boldsymbol{Z}=\sigma_z \). Show that \( \boldsymbol{XX}=\boldsymbol{YY}=\boldsymbol{ZZ}=\boldsymbol{I} \).</li>
24132363
<li> Which one of the Pauli matrices has the qubit basis \( \vert 0\rangle \) and \( \vert 1\rangle \) as eigenbasis? What are the eigenvalues?</li>
24142364
</ol>
24152365
<!-- !split -->
2416-
<h2 id="exercise-5-shared-eigenvectors" class="anchor">Exercise 5: Shared eigenvectors </h2>
2366+
<h2 id="3-shared-eigenvectors" class="anchor">3: Shared eigenvectors </h2>
24172367

24182368
<p>Prove that if two operators \( \hat{A} \) and \( \hat{B} \) commute they will share a basis of eigenstates</p>
24192369

24202370
<!-- !split -->
2421-
<h2 id="exercise-6-one-qubit-basis-and-pauli-matrices" class="anchor">Exercise 6: One-qubit basis and Pauli matrices </h2>
2371+
<h2 id="4-one-qubit-basis-and-pauli-matrices" class="anchor">4: One-qubit basis and Pauli matrices </h2>
24222372

24232373
<p>Write a function (in Python for example) which sets up a one-qubit basis and apply the various Pauli matrices to these basis states.</p>
24242374

24252375
<!-- !split -->
2426-
<h2 id="exercise-7-hadamard-and-phase-gates" class="anchor">Exercise 7: Hadamard and Phase gates </h2>
2376+
<h2 id="5-hadamard-and-phase-gates" class="anchor">5: Hadamard and Phase gates </h2>
24272377

24282378
<p>Apply the Hadamard and Phase matrices (or gates) to the same one-qubit basis states and study their actions on these states.
24292379
Write a code which applies these matrices to the same one-qubit basis.

‎doc/pub/week1/html/week1-reveal.html‎

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@@ -2321,47 +2321,7 @@ <h2 id="exercise-set-for-first-week-to-be-discussed-january-28">Exercise set for
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</section>
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<section>
2324-
<h2 id="exewrcise-1-bell-states">Exewrcise 1: Bell states </h2>
2325-
2326-
<p>Show that the so-called Bell states listed here (and to be encountered many times in this course) form an orthogonal basis</p>
2327-
<p>&nbsp;<br>
2328-
$$
2329-
\vert \Phi^+\rangle = \frac{1}{\sqrt{2}}\left[\vert 00\rangle +\vert 11\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 1\end{bmatrix},
2330-
$$
2331-
<p>&nbsp;<br>
2332-
2333-
<p>&nbsp;<br>
2334-
$$
2335-
\vert \Phi^-\rangle = \frac{1}{\sqrt{2}}\left[\vert 00\rangle -\vert 11\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 0 \\ 0 \\ -1\end{bmatrix},
2336-
$$
2337-
<p>&nbsp;<br>
2338-
</section>
2339-
2340-
<section>
2341-
<h2 id="and-the-next-two">And the next two </h2>
2342-
<p>&nbsp;<br>
2343-
$$
2344-
\vert \Psi^+\rangle = \frac{1}{\sqrt{2}}\left[\vert 10\rangle +\vert 01\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 0 \\ 1 \\ 1 \\ 0\end{bmatrix},
2345-
$$
2346-
<p>&nbsp;<br>
2347-
2348-
<p>and</p>
2349-
2350-
<p>&nbsp;<br>
2351-
$$
2352-
\vert \Psi^-\rangle = \frac{1}{\sqrt{2}}\left[\vert 10\rangle -\vert 01\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 0 \\ 1 \\ -1 \\ 0\end{bmatrix}.
2353-
$$
2354-
<p>&nbsp;<br>
2355-
</section>
2356-
2357-
<section>
2358-
<h2 id="exercise-2-entangled-state">Exercise 2: Entangled state </h2>
2359-
2360-
<p>Show that the state \( \alpha \vert 00\rangle+\beta\vert 11\rangle \) cannot be written as the product of the tensor product of two states and is thus entangle. The constants \( \alpha \) and \( \beta \) are both nonzero.</p>
2361-
</section>
2362-
2363-
<section>
2364-
<h2 id="exercise-3-commutator-identities">Exercise 3: Commutator identities </h2>
2324+
<h2 id="1-commutator-identities">1: Commutator identities </h2>
23652325
<p>Prove the following commutator relations for different operators (marked with a hat)</p>
23662326
<ol>
23672327
<p><li> \( [\hat{A}+\hat{B},\hat{C}]= [\hat{A},\hat{C}]+[\hat{B},\hat{C}] \);</li>
@@ -2371,7 +2331,7 @@ <h2 id="exercise-3-commutator-identities">Exercise 3: Commutator identities </h
23712331
</section>
23722332

23732333
<section>
2374-
<h2 id="exercise-4-pauli-matrices">Exercise 4: Pauli matrices </h2>
2334+
<h2 id="2-pauli-matrices">2: Pauli matrices </h2>
23752335
<ol>
23762336
<p><li> Set up the commutation rules for the Pauli matrices, that is find \( [\sigma_i,\sigma_j] \) where \( i,j=x,y,z \).</li>
23772337
<p><li> We define \( \boldsymbol{X}=\sigma_x \), \( \boldsymbol{Y}=\sigma_y \) and \( \boldsymbol{Z}=\sigma_z \). Show that \( \boldsymbol{XX}=\boldsymbol{YY}=\boldsymbol{ZZ}=\boldsymbol{I} \).</li>
@@ -2380,19 +2340,19 @@ <h2 id="exercise-4-pauli-matrices">Exercise 4: Pauli matrices </h2>
23802340
</section>
23812341

23822342
<section>
2383-
<h2 id="exercise-5-shared-eigenvectors">Exercise 5: Shared eigenvectors </h2>
2343+
<h2 id="3-shared-eigenvectors">3: Shared eigenvectors </h2>
23842344

23852345
<p>Prove that if two operators \( \hat{A} \) and \( \hat{B} \) commute they will share a basis of eigenstates</p>
23862346
</section>
23872347

23882348
<section>
2389-
<h2 id="exercise-6-one-qubit-basis-and-pauli-matrices">Exercise 6: One-qubit basis and Pauli matrices </h2>
2349+
<h2 id="4-one-qubit-basis-and-pauli-matrices">4: One-qubit basis and Pauli matrices </h2>
23902350

23912351
<p>Write a function (in Python for example) which sets up a one-qubit basis and apply the various Pauli matrices to these basis states.</p>
23922352
</section>
23932353

23942354
<section>
2395-
<h2 id="exercise-7-hadamard-and-phase-gates">Exercise 7: Hadamard and Phase gates </h2>
2355+
<h2 id="5-hadamard-and-phase-gates">5: Hadamard and Phase gates </h2>
23962356

23972357
<p>Apply the Hadamard and Phase matrices (or gates) to the same one-qubit basis states and study their actions on these states.
23982358
Write a code which applies these matrices to the same one-qubit basis.

‎doc/pub/week1/html/week1-solarized.html‎

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('5: Hadamard and Phase gates',
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('Exercise 6: One-qubit basis and Pauli matrices',
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end of tocinfo -->
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<body>
@@ -2195,64 +2180,32 @@ <h2 id="exercise-set-for-first-week-to-be-discussed-january-28">Exercise set for
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</p>
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21972182
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
2198-
<h2 id="exewrcise-1-bell-states">Exewrcise 1: Bell states </h2>
2199-
2200-
<p>Show that the so-called Bell states listed here (and to be encountered many times in this course) form an orthogonal basis</p>
2201-
$$
2202-
\vert \Phi^+\rangle = \frac{1}{\sqrt{2}}\left[\vert 00\rangle +\vert 11\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 1\end{bmatrix},
2203-
$$
2204-
2205-
2206-
$$
2207-
\vert \Phi^-\rangle = \frac{1}{\sqrt{2}}\left[\vert 00\rangle -\vert 11\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 0 \\ 0 \\ -1\end{bmatrix},
2208-
$$
2209-
2210-
2211-
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
2212-
<h2 id="and-the-next-two">And the next two </h2>
2213-
$$
2214-
\vert \Psi^+\rangle = \frac{1}{\sqrt{2}}\left[\vert 10\rangle +\vert 01\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 0 \\ 1 \\ 1 \\ 0\end{bmatrix},
2215-
$$
2216-
2217-
<p>and</p>
2218-
2219-
$$
2220-
\vert \Psi^-\rangle = \frac{1}{\sqrt{2}}\left[\vert 10\rangle -\vert 01\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 0 \\ 1 \\ -1 \\ 0\end{bmatrix}.
2221-
$$
2222-
2223-
2224-
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
2225-
<h2 id="exercise-2-entangled-state">Exercise 2: Entangled state </h2>
2226-
2227-
<p>Show that the state \( \alpha \vert 00\rangle+\beta\vert 11\rangle \) cannot be written as the product of the tensor product of two states and is thus entangle. The constants \( \alpha \) and \( \beta \) are both nonzero.</p>
2228-
2229-
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
2230-
<h2 id="exercise-3-commutator-identities">Exercise 3: Commutator identities </h2>
2183+
<h2 id="1-commutator-identities">1: Commutator identities </h2>
22312184
<p>Prove the following commutator relations for different operators (marked with a hat)</p>
22322185
<ol>
22332186
<li> \( [\hat{A}+\hat{B},\hat{C}]= [\hat{A},\hat{C}]+[\hat{B},\hat{C}] \);</li>
22342187
<li> \( [\hat{A},\hat{B}\hat{C}]= [\hat{A},\hat{B}]\hat{C}+\hat{B}[\hat{A},\hat{C}] \); and</li>
22352188
<li> \( [\hat{A},[\hat{B}\hat{C}]]= [\hat{B},[\hat{C},\hat{A}]]+[\hat{C},[\hat{A},\hat{B}]]=0 \) (the so-called Jacobi identity).</li>
22362189
</ol>
22372190
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
2238-
<h2 id="exercise-4-pauli-matrices">Exercise 4: Pauli matrices </h2>
2191+
<h2 id="2-pauli-matrices">2: Pauli matrices </h2>
22392192
<ol>
22402193
<li> Set up the commutation rules for the Pauli matrices, that is find \( [\sigma_i,\sigma_j] \) where \( i,j=x,y,z \).</li>
22412194
<li> We define \( \boldsymbol{X}=\sigma_x \), \( \boldsymbol{Y}=\sigma_y \) and \( \boldsymbol{Z}=\sigma_z \). Show that \( \boldsymbol{XX}=\boldsymbol{YY}=\boldsymbol{ZZ}=\boldsymbol{I} \).</li>
22422195
<li> Which one of the Pauli matrices has the qubit basis \( \vert 0\rangle \) and \( \vert 1\rangle \) as eigenbasis? What are the eigenvalues?</li>
22432196
</ol>
22442197
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
2245-
<h2 id="exercise-5-shared-eigenvectors">Exercise 5: Shared eigenvectors </h2>
2198+
<h2 id="3-shared-eigenvectors">3: Shared eigenvectors </h2>
22462199

22472200
<p>Prove that if two operators \( \hat{A} \) and \( \hat{B} \) commute they will share a basis of eigenstates</p>
22482201

22492202
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
2250-
<h2 id="exercise-6-one-qubit-basis-and-pauli-matrices">Exercise 6: One-qubit basis and Pauli matrices </h2>
2203+
<h2 id="4-one-qubit-basis-and-pauli-matrices">4: One-qubit basis and Pauli matrices </h2>
22512204

22522205
<p>Write a function (in Python for example) which sets up a one-qubit basis and apply the various Pauli matrices to these basis states.</p>
22532206

22542207
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
2255-
<h2 id="exercise-7-hadamard-and-phase-gates">Exercise 7: Hadamard and Phase gates </h2>
2208+
<h2 id="5-hadamard-and-phase-gates">5: Hadamard and Phase gates </h2>
22562209

22572210
<p>Apply the Hadamard and Phase matrices (or gates) to the same one-qubit basis states and study their actions on these states.
22582211
Write a code which applies these matrices to the same one-qubit basis.

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