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‎doc/pub/week5/html/week5-bs.html‎

Lines changed: 61 additions & 5 deletions
Original file line numberDiff line numberDiff line change
@@ -120,7 +120,9 @@
120120
('CNOT gate', 2, None, 'cnot-gate'),
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('$\\mathrm{CX}$ gate', 2, None, 'mathrm-cx-gate'),
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('Swap gate', 2, None, 'swap-gate'),
123+
('Reminder from last week', 2, None, 'reminder-from-last-week'),
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('Two-qubit Hamiltonian', 2, None, 'two-qubit-hamiltonian'),
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('Definitions', 2, None, 'definitions'),
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('Ex1: Two-qubit Hamiltonian',
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2,
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None,
@@ -206,7 +208,9 @@
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<!-- navigation toc: --> <li><a href="#cnot-gate" style="font-size: 80%;">CNOT gate</a></li>
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<!-- navigation toc: --> <li><a href="#mathrm-cx-gate" style="font-size: 80%;">\( \mathrm{CX} \) gate</a></li>
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<!-- navigation toc: --> <li><a href="#swap-gate" style="font-size: 80%;">Swap gate</a></li>
211+
<!-- navigation toc: --> <li><a href="#reminder-from-last-week" style="font-size: 80%;">Reminder from last week</a></li>
209212
<!-- navigation toc: --> <li><a href="#two-qubit-hamiltonian" style="font-size: 80%;">Two-qubit Hamiltonian</a></li>
213+
<!-- navigation toc: --> <li><a href="#definitions" style="font-size: 80%;">Definitions</a></li>
210214
<!-- navigation toc: --> <li><a href="#ex1-two-qubit-hamiltonian" style="font-size: 80%;">Ex1: Two-qubit Hamiltonian</a></li>
211215
<!-- navigation toc: --> <li><a href="#ex2-two-qubit-hamiltonian-and-entropy" style="font-size: 80%;">Ex2: Two-qubit Hamiltonian and entropy</a></li>
212216

@@ -1164,24 +1168,76 @@ <h2 id="swap-gate" class="anchor">Swap gate </h2>
11641168
$$
11651169

11661170

1171+
<!-- !split -->
1172+
<h2 id="reminder-from-last-week" class="anchor">Reminder from last week </h2>
1173+
1174+
<p>We rewrote the one-qubit Hamiltonian</p>
1175+
$$
1176+
\boldsymbol{H}=\begin{bmatrix} \epsilon_{1}+V_{11} & V_{12}\\
1177+
V_{21} & \epsilon_{2}-V_{22} \end{bmatrix},
1178+
$$
1179+
1180+
<p>as (defining \( V_{12}=V_{21}=V_x \)) </p>
1181+
$$
1182+
\boldsymbol{H}=\frac{\epsilon_{1}+\epsilon_{2}}{2}\boldsymbol{I}+\frac{\epsilon_{1}-\epsilon_{2}}{2}\boldsymbol{Z}+\frac{V_{11}+V_{22}}{2}\boldsymbol{I}+\frac{V_{11}-V_{22}}{2}\boldsymbol{Z}+V_x\boldsymbol{X}.
1183+
$$
1184+
1185+
<p>We will now rewrite our first two-qubit Hamiltonian along the same lines.</p>
1186+
11671187
<!-- !split -->
11681188
<h2 id="two-qubit-hamiltonian" class="anchor">Two-qubit Hamiltonian </h2>
11691189

11701190
<p>We start again with </p>
11711191
$$
1172-
\boldsymbol{H}=\begin{bmatrix} \epsilon_{00}+H_z & 0 & 0 & H_x \\
1173-
0 & \epsilon_{10}-H_z & H_x & 0 \\
1174-
0 & H_x & \epsilon_{01}-H_z & 0 \\
1175-
H_x & 0 & 0 & \epsilon_{11} +H_z \end{bmatrix}.
1192+
\boldsymbol{H}=\begin{bmatrix} \epsilon_{1}+V_z & 0 & 0 & V_x \\
1193+
0 & \epsilon_{2}-V_z & V_x & 0 \\
1194+
0 & H_x & \epsilon_{3}-V_z & 0 \\
1195+
H_x & 0 & 0 & \epsilon_{4} +V_z \end{bmatrix}.
1196+
$$
1197+
1198+
<p>This Hamiltonian, as discussed in the first exercise can be rewritten in terms of various one-qubit matrices.</p>
1199+
1200+
<!-- !split -->
1201+
<h2 id="definitions" class="anchor">Definitions </h2>
1202+
1203+
<p>We define</p>
1204+
$$
1205+
\epsilon_{II}=\frac{\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4}}{4},
1206+
$$
1207+
1208+
$$
1209+
\epsilon_{ZI}=\frac{\epsilon_{1}+\epsilon_{2}-\epsilon_{3}-\epsilon_{4}}{4},
1210+
$$
1211+
1212+
$$
1213+
\epsilon_{IZ}=\frac{\epsilon_{1}-\epsilon_{2}+\epsilon_{3}-\epsilon_{4}}{4},
1214+
$$
1215+
1216+
$$
1217+
\epsilon_{ZZ}=\frac{\epsilon_{1}-\epsilon_{2}-\epsilon_{3}+\epsilon_{4}}{4}.
11761218
$$
11771219

11781220

11791221
<!-- !split -->
11801222
<h2 id="ex1-two-qubit-hamiltonian" class="anchor">Ex1: Two-qubit Hamiltonian </h2>
11811223

11821224
<p>This exercise is also part of the first project.
1183-
Rewrite the Hamiltonian in term of various Pauli matrices.
1225+
Rewrite the Hamiltonian in term of various Pauli matrices and show that you can rwrite the Hamiltonian as
11841226
</p>
1227+
$$
1228+
\boldsymbol{H}=\boldsymbol{H}_0+\boldsymbol{H}_I,
1229+
$$
1230+
1231+
<p>with </p>
1232+
$$
1233+
\boldsymbol{H}_0=\epsilon_{II}\boldsymbol{I}\otimes\boldsymbol{I}
1234+
$$
1235+
1236+
<p>and </p>
1237+
$$
1238+
\boldsymbol{H}_I=V_x\boldsymbol{X}\otimes\boldsymbol{X}+V_z\boldsymbol{Z}\otimes\boldsymbol{Z}
1239+
$$
1240+
11851241

11861242
<!-- !split -->
11871243
<h2 id="ex2-two-qubit-hamiltonian-and-entropy" class="anchor">Ex2: Two-qubit Hamiltonian and entropy </h2>

‎doc/pub/week5/html/week5-reveal.html‎

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@@ -1285,16 +1285,68 @@ <h2 id="swap-gate">Swap gate </h2>
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<p>&nbsp;<br>
12861286
</section>
12871287

1288+
<section>
1289+
<h2 id="reminder-from-last-week">Reminder from last week </h2>
1290+
1291+
<p>We rewrote the one-qubit Hamiltonian</p>
1292+
<p>&nbsp;<br>
1293+
$$
1294+
\boldsymbol{H}=\begin{bmatrix} \epsilon_{1}+V_{11} & V_{12}\\
1295+
V_{21} & \epsilon_{2}-V_{22} \end{bmatrix},
1296+
$$
1297+
<p>&nbsp;<br>
1298+
1299+
<p>as (defining \( V_{12}=V_{21}=V_x \)) </p>
1300+
<p>&nbsp;<br>
1301+
$$
1302+
\boldsymbol{H}=\frac{\epsilon_{1}+\epsilon_{2}}{2}\boldsymbol{I}+\frac{\epsilon_{1}-\epsilon_{2}}{2}\boldsymbol{Z}+\frac{V_{11}+V_{22}}{2}\boldsymbol{I}+\frac{V_{11}-V_{22}}{2}\boldsymbol{Z}+V_x\boldsymbol{X}.
1303+
$$
1304+
<p>&nbsp;<br>
1305+
1306+
<p>We will now rewrite our first two-qubit Hamiltonian along the same lines.</p>
1307+
</section>
1308+
12881309
<section>
12891310
<h2 id="two-qubit-hamiltonian">Two-qubit Hamiltonian </h2>
12901311

12911312
<p>We start again with </p>
12921313
<p>&nbsp;<br>
12931314
$$
1294-
\boldsymbol{H}=\begin{bmatrix} \epsilon_{00}+H_z & 0 & 0 & H_x \\
1295-
0 & \epsilon_{10}-H_z & H_x & 0 \\
1296-
0 & H_x & \epsilon_{01}-H_z & 0 \\
1297-
H_x & 0 & 0 & \epsilon_{11} +H_z \end{bmatrix}.
1315+
\boldsymbol{H}=\begin{bmatrix} \epsilon_{1}+V_z & 0 & 0 & V_x \\
1316+
0 & \epsilon_{2}-V_z & V_x & 0 \\
1317+
0 & H_x & \epsilon_{3}-V_z & 0 \\
1318+
H_x & 0 & 0 & \epsilon_{4} +V_z \end{bmatrix}.
1319+
$$
1320+
<p>&nbsp;<br>
1321+
1322+
<p>This Hamiltonian, as discussed in the first exercise can be rewritten in terms of various one-qubit matrices.</p>
1323+
</section>
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1325+
<section>
1326+
<h2 id="definitions">Definitions </h2>
1327+
1328+
<p>We define</p>
1329+
<p>&nbsp;<br>
1330+
$$
1331+
\epsilon_{II}=\frac{\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4}}{4},
1332+
$$
1333+
<p>&nbsp;<br>
1334+
1335+
<p>&nbsp;<br>
1336+
$$
1337+
\epsilon_{ZI}=\frac{\epsilon_{1}+\epsilon_{2}-\epsilon_{3}-\epsilon_{4}}{4},
1338+
$$
1339+
<p>&nbsp;<br>
1340+
1341+
<p>&nbsp;<br>
1342+
$$
1343+
\epsilon_{IZ}=\frac{\epsilon_{1}-\epsilon_{2}+\epsilon_{3}-\epsilon_{4}}{4},
1344+
$$
1345+
<p>&nbsp;<br>
1346+
1347+
<p>&nbsp;<br>
1348+
$$
1349+
\epsilon_{ZZ}=\frac{\epsilon_{1}-\epsilon_{2}-\epsilon_{3}+\epsilon_{4}}{4}.
12981350
$$
12991351
<p>&nbsp;<br>
13001352
</section>
@@ -1303,8 +1355,27 @@ <h2 id="two-qubit-hamiltonian">Two-qubit Hamiltonian </h2>
13031355
<h2 id="ex1-two-qubit-hamiltonian">Ex1: Two-qubit Hamiltonian </h2>
13041356

13051357
<p>This exercise is also part of the first project.
1306-
Rewrite the Hamiltonian in term of various Pauli matrices.
1358+
Rewrite the Hamiltonian in term of various Pauli matrices and show that you can rwrite the Hamiltonian as
13071359
</p>
1360+
<p>&nbsp;<br>
1361+
$$
1362+
\boldsymbol{H}=\boldsymbol{H}_0+\boldsymbol{H}_I,
1363+
$$
1364+
<p>&nbsp;<br>
1365+
1366+
<p>with </p>
1367+
<p>&nbsp;<br>
1368+
$$
1369+
\boldsymbol{H}_0=\epsilon_{II}\boldsymbol{I}\otimes\boldsymbol{I}
1370+
$$
1371+
<p>&nbsp;<br>
1372+
1373+
<p>and </p>
1374+
<p>&nbsp;<br>
1375+
$$
1376+
\boldsymbol{H}_I=V_x\boldsymbol{X}\otimes\boldsymbol{X}+V_z\boldsymbol{Z}\otimes\boldsymbol{Z}
1377+
$$
1378+
<p>&nbsp;<br>
13081379
</section>
13091380

13101381
<section>

‎doc/pub/week5/html/week5-solarized.html‎

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@@ -147,7 +147,9 @@
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('CNOT gate', 2, None, 'cnot-gate'),
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('$\\mathrm{CX}$ gate', 2, None, 'mathrm-cx-gate'),
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('Swap gate', 2, None, 'swap-gate'),
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('Reminder from last week', 2, None, 'reminder-from-last-week'),
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('Two-qubit Hamiltonian', 2, None, 'two-qubit-hamiltonian'),
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('Definitions', 2, None, 'definitions'),
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('Ex1: Two-qubit Hamiltonian',
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2,
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None,
@@ -1116,24 +1118,76 @@ <h2 id="swap-gate">Swap gate </h2>
11161118
$$
11171119

11181120

1121+
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
1122+
<h2 id="reminder-from-last-week">Reminder from last week </h2>
1123+
1124+
<p>We rewrote the one-qubit Hamiltonian</p>
1125+
$$
1126+
\boldsymbol{H}=\begin{bmatrix} \epsilon_{1}+V_{11} & V_{12}\\
1127+
V_{21} & \epsilon_{2}-V_{22} \end{bmatrix},
1128+
$$
1129+
1130+
<p>as (defining \( V_{12}=V_{21}=V_x \)) </p>
1131+
$$
1132+
\boldsymbol{H}=\frac{\epsilon_{1}+\epsilon_{2}}{2}\boldsymbol{I}+\frac{\epsilon_{1}-\epsilon_{2}}{2}\boldsymbol{Z}+\frac{V_{11}+V_{22}}{2}\boldsymbol{I}+\frac{V_{11}-V_{22}}{2}\boldsymbol{Z}+V_x\boldsymbol{X}.
1133+
$$
1134+
1135+
<p>We will now rewrite our first two-qubit Hamiltonian along the same lines.</p>
1136+
11191137
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
11201138
<h2 id="two-qubit-hamiltonian">Two-qubit Hamiltonian </h2>
11211139

11221140
<p>We start again with </p>
11231141
$$
1124-
\boldsymbol{H}=\begin{bmatrix} \epsilon_{00}+H_z & 0 & 0 & H_x \\
1125-
0 & \epsilon_{10}-H_z & H_x & 0 \\
1126-
0 & H_x & \epsilon_{01}-H_z & 0 \\
1127-
H_x & 0 & 0 & \epsilon_{11} +H_z \end{bmatrix}.
1142+
\boldsymbol{H}=\begin{bmatrix} \epsilon_{1}+V_z & 0 & 0 & V_x \\
1143+
0 & \epsilon_{2}-V_z & V_x & 0 \\
1144+
0 & H_x & \epsilon_{3}-V_z & 0 \\
1145+
H_x & 0 & 0 & \epsilon_{4} +V_z \end{bmatrix}.
1146+
$$
1147+
1148+
<p>This Hamiltonian, as discussed in the first exercise can be rewritten in terms of various one-qubit matrices.</p>
1149+
1150+
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
1151+
<h2 id="definitions">Definitions </h2>
1152+
1153+
<p>We define</p>
1154+
$$
1155+
\epsilon_{II}=\frac{\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4}}{4},
1156+
$$
1157+
1158+
$$
1159+
\epsilon_{ZI}=\frac{\epsilon_{1}+\epsilon_{2}-\epsilon_{3}-\epsilon_{4}}{4},
1160+
$$
1161+
1162+
$$
1163+
\epsilon_{IZ}=\frac{\epsilon_{1}-\epsilon_{2}+\epsilon_{3}-\epsilon_{4}}{4},
1164+
$$
1165+
1166+
$$
1167+
\epsilon_{ZZ}=\frac{\epsilon_{1}-\epsilon_{2}-\epsilon_{3}+\epsilon_{4}}{4}.
11281168
$$
11291169

11301170

11311171
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
11321172
<h2 id="ex1-two-qubit-hamiltonian">Ex1: Two-qubit Hamiltonian </h2>
11331173

11341174
<p>This exercise is also part of the first project.
1135-
Rewrite the Hamiltonian in term of various Pauli matrices.
1175+
Rewrite the Hamiltonian in term of various Pauli matrices and show that you can rwrite the Hamiltonian as
11361176
</p>
1177+
$$
1178+
\boldsymbol{H}=\boldsymbol{H}_0+\boldsymbol{H}_I,
1179+
$$
1180+
1181+
<p>with </p>
1182+
$$
1183+
\boldsymbol{H}_0=\epsilon_{II}\boldsymbol{I}\otimes\boldsymbol{I}
1184+
$$
1185+
1186+
<p>and </p>
1187+
$$
1188+
\boldsymbol{H}_I=V_x\boldsymbol{X}\otimes\boldsymbol{X}+V_z\boldsymbol{Z}\otimes\boldsymbol{Z}
1189+
$$
1190+
11371191

11381192
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
11391193
<h2 id="ex2-two-qubit-hamiltonian-and-entropy">Ex2: Two-qubit Hamiltonian and entropy </h2>

‎doc/pub/week5/html/week5.html‎

Lines changed: 59 additions & 5 deletions
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@@ -224,7 +224,9 @@
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('CNOT gate', 2, None, 'cnot-gate'),
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('$\\mathrm{CX}$ gate', 2, None, 'mathrm-cx-gate'),
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('Swap gate', 2, None, 'swap-gate'),
227+
('Reminder from last week', 2, None, 'reminder-from-last-week'),
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('Two-qubit Hamiltonian', 2, None, 'two-qubit-hamiltonian'),
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('Definitions', 2, None, 'definitions'),
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('Ex1: Two-qubit Hamiltonian',
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2,
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None,
@@ -1193,24 +1195,76 @@ <h2 id="swap-gate">Swap gate </h2>
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$$
11941196

11951197

1198+
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
1199+
<h2 id="reminder-from-last-week">Reminder from last week </h2>
1200+
1201+
<p>We rewrote the one-qubit Hamiltonian</p>
1202+
$$
1203+
\boldsymbol{H}=\begin{bmatrix} \epsilon_{1}+V_{11} & V_{12}\\
1204+
V_{21} & \epsilon_{2}-V_{22} \end{bmatrix},
1205+
$$
1206+
1207+
<p>as (defining \( V_{12}=V_{21}=V_x \)) </p>
1208+
$$
1209+
\boldsymbol{H}=\frac{\epsilon_{1}+\epsilon_{2}}{2}\boldsymbol{I}+\frac{\epsilon_{1}-\epsilon_{2}}{2}\boldsymbol{Z}+\frac{V_{11}+V_{22}}{2}\boldsymbol{I}+\frac{V_{11}-V_{22}}{2}\boldsymbol{Z}+V_x\boldsymbol{X}.
1210+
$$
1211+
1212+
<p>We will now rewrite our first two-qubit Hamiltonian along the same lines.</p>
1213+
11961214
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
11971215
<h2 id="two-qubit-hamiltonian">Two-qubit Hamiltonian </h2>
11981216

11991217
<p>We start again with </p>
12001218
$$
1201-
\boldsymbol{H}=\begin{bmatrix} \epsilon_{00}+H_z & 0 & 0 & H_x \\
1202-
0 & \epsilon_{10}-H_z & H_x & 0 \\
1203-
0 & H_x & \epsilon_{01}-H_z & 0 \\
1204-
H_x & 0 & 0 & \epsilon_{11} +H_z \end{bmatrix}.
1219+
\boldsymbol{H}=\begin{bmatrix} \epsilon_{1}+V_z & 0 & 0 & V_x \\
1220+
0 & \epsilon_{2}-V_z & V_x & 0 \\
1221+
0 & H_x & \epsilon_{3}-V_z & 0 \\
1222+
H_x & 0 & 0 & \epsilon_{4} +V_z \end{bmatrix}.
1223+
$$
1224+
1225+
<p>This Hamiltonian, as discussed in the first exercise can be rewritten in terms of various one-qubit matrices.</p>
1226+
1227+
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
1228+
<h2 id="definitions">Definitions </h2>
1229+
1230+
<p>We define</p>
1231+
$$
1232+
\epsilon_{II}=\frac{\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4}}{4},
1233+
$$
1234+
1235+
$$
1236+
\epsilon_{ZI}=\frac{\epsilon_{1}+\epsilon_{2}-\epsilon_{3}-\epsilon_{4}}{4},
1237+
$$
1238+
1239+
$$
1240+
\epsilon_{IZ}=\frac{\epsilon_{1}-\epsilon_{2}+\epsilon_{3}-\epsilon_{4}}{4},
1241+
$$
1242+
1243+
$$
1244+
\epsilon_{ZZ}=\frac{\epsilon_{1}-\epsilon_{2}-\epsilon_{3}+\epsilon_{4}}{4}.
12051245
$$
12061246

12071247

12081248
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
12091249
<h2 id="ex1-two-qubit-hamiltonian">Ex1: Two-qubit Hamiltonian </h2>
12101250

12111251
<p>This exercise is also part of the first project.
1212-
Rewrite the Hamiltonian in term of various Pauli matrices.
1252+
Rewrite the Hamiltonian in term of various Pauli matrices and show that you can rwrite the Hamiltonian as
12131253
</p>
1254+
$$
1255+
\boldsymbol{H}=\boldsymbol{H}_0+\boldsymbol{H}_I,
1256+
$$
1257+
1258+
<p>with </p>
1259+
$$
1260+
\boldsymbol{H}_0=\epsilon_{II}\boldsymbol{I}\otimes\boldsymbol{I}
1261+
$$
1262+
1263+
<p>and </p>
1264+
$$
1265+
\boldsymbol{H}_I=V_x\boldsymbol{X}\otimes\boldsymbol{X}+V_z\boldsymbol{Z}\otimes\boldsymbol{Z}
1266+
$$
1267+
12141268

12151269
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
12161270
<h2 id="ex2-two-qubit-hamiltonian-and-entropy">Ex2: Two-qubit Hamiltonian and entropy </h2>

‎doc/pub/week5/ipynb/ipynb-week5-src.tar.gz‎

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