From e3f5b7d00fd5e4cf0be61ad9e3a703861e6fce59 Mon Sep 17 00:00:00 2001
From: "Kevin J. Sung" <kevjsung@umich.edu>
Date: Mon, 19 Aug 2024 07:36:19 +0800
Subject: [PATCH] run squeaky

---
 docs/tensor-circuit.ipynb | 58446 +-----------------------------------
 1 file changed, 121 insertions(+), 58325 deletions(-)

diff --git a/docs/tensor-circuit.ipynb b/docs/tensor-circuit.ipynb
index f205c3cf..d3fb65eb 100644
--- a/docs/tensor-circuit.ipynb
+++ b/docs/tensor-circuit.ipynb
@@ -3,6 +3,7 @@
   {
    "attachments": {},
    "cell_type": "markdown",
+   "id": "fe4df7ac-b07b-45d7-b8d0-ce2c59a97b39",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -35,6 +36,7 @@
   {
    "cell_type": "code",
    "execution_count": 1,
+   "id": "a2195a10-fecc-429e-bac1-01ad60c068e6",
    "metadata": {},
    "outputs": [
     {
@@ -76,6 +78,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "dc6a9fea-95e3-4160-a204-ea309b061f8b",
    "metadata": {},
    "source": [
     "As mentioned above, various pre-processing steps need to occur (which will happen on the first run if not explicitly called). The results of these are cached such that the more you sample the more the simulation should speed up:"
@@ -84,6 +87,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
+   "id": "460f5032-8085-4f1a-a3d6-af6d494dd3ee",
    "metadata": {},
    "outputs": [
     {
@@ -112,6 +116,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "43d77f06-897a-444f-a404-e13e464fb02f",
    "metadata": {},
    "source": [
     "Collect some statistics:"
@@ -120,6 +125,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
+   "id": "827a1793-4626-493c-8683-80782b84fc12",
    "metadata": {},
    "outputs": [
     {
@@ -152,6 +158,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "afe9431e-c625-40a2-aab8-dcdcd775e11f",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -196,6 +203,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "a8b98fd3-20ad-42a4-9963-f2b348d4b604",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -219,6 +227,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
+   "id": "5ebbe59f-4f11-4209-8ee2-fb2c464af7a8",
    "metadata": {},
    "outputs": [
     {
@@ -268,8342 +277,23 @@
   },
   {
    "cell_type": "markdown",
+   "id": "cb6d2616-0ae0-4562-a091-6413c2821532",
    "metadata": {},
    "source": [
-    "The basic tensor network representing the state is stored in the \n",
+    "The basic tensor network representing the state is stored in the\n",
     "``.psi`` attribute, which we can then visualize:"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 5,
+   "id": "33fdd539-1946-47ba-9d03-392f3ec91ffc",
    "metadata": {},
    "outputs": [
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@@ -8621,8298 +311,26 @@
   },
   {
    "cell_type": "markdown",
+   "id": "e6e0cf85-b8b6-406b-b4fd-852b06f06439",
    "metadata": {},
    "source": [
     "Note by default the CNOT and CZ gates have\n",
     "been split via a rank-2 spatial decomposition into two parts acting on each site seperately\n",
-    "but connected by a new bond. \n",
-    "We can also graph the default (``propagate_tags='register'``) method for \n",
+    "but connected by a new bond.\n",
+    "We can also graph the default (``propagate_tags='register'``) method for\n",
     "adding site tags to the applied operators:"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 6,
+   "id": "eb746b69-eac5-4ec2-82b1-f33bf36026ad",
    "metadata": {},
    "outputs": [
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     "Or since we supplied ``gate_round`` as an keyword (which is optional), the tensors\n",
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@@ -25481,6 +382,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "1d83aeca-abaa-4c94-b276-6ece26adefb0",
    "metadata": {},
    "source": [
     "All of these might be helpful when addressing only certain tensors:"
@@ -25489,6 +391,7 @@
   {
    "cell_type": "code",
    "execution_count": 8,
+   "id": "c3d7734f-7705-497d-8904-0be128f2101e",
    "metadata": {},
    "outputs": [
     {
@@ -25516,6 +419,7 @@
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   {
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+   "id": "07961cd8-8c12-4783-9b53-140472d824b6",
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@@ -25531,6 +435,7 @@
   {
    "cell_type": "code",
    "execution_count": 9,
+   "id": "f9ccc690-8ae9-4121-8431-ab03ad060062",
    "metadata": {},
    "outputs": [
     {
@@ -25595,6 +500,7 @@
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   {
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+   "id": "d7049467-75d5-4cb1-a0dd-fa6306c88cfd",
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@@ -25611,1346 +517,13 @@
   {
    "cell_type": "code",
    "execution_count": 10,
+   "id": "5df00677-5c02-4c29-81a9-67b6768969fe",
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        "<Figure size 600x600 with 1 Axes>"
@@ -26978,6 +551,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "a057d29c-63e5-46f7-b8fe-26abc53aef81",
    "metadata": {},
    "source": [
     "We've used the ``contract=False`` option which doesn't try and split the gate tensor in any way,\n",
@@ -26989,6 +563,7 @@
   {
    "cell_type": "code",
    "execution_count": 11,
+   "id": "4e0e7bc4-e158-4c84-bb4c-c84a7f0453cb",
    "metadata": {},
    "outputs": [
     {
@@ -27022,6 +597,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "f2a7d31e-b7e8-4124-aff9-2435672b1ea6",
    "metadata": {},
    "source": [
     "For most tasks like contraction these are transparently handled like normal tensors:"
@@ -27030,6 +606,7 @@
   {
    "cell_type": "code",
    "execution_count": 12,
+   "id": "d9e61383-7e21-45e1-9078-0767b67ff059",
    "metadata": {},
    "outputs": [
     {
@@ -27049,6 +626,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "da03af62-db40-461c-b7eb-03aa08bffbf5",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -27089,6 +667,7 @@
   {
    "cell_type": "code",
    "execution_count": 13,
+   "id": "c3e79474-81c1-40bd-972f-5a68850b7ce4",
    "metadata": {},
    "outputs": [
     {
@@ -27116,6 +695,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "f7b7624c-7115-4d44-8cb3-7538f0dbd307",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -27146,6 +726,7 @@
   {
    "cell_type": "code",
    "execution_count": 14,
+   "id": "2ebf2bc9-9e84-4ca1-9732-5d9c247f08a5",
    "metadata": {},
    "outputs": [
     {
@@ -27165,6 +746,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "8312fec2-0b30-4668-a583-1f81de036264",
    "metadata": {},
    "source": [
     "You can compute several individual expectations on the same sites by supplying a list (they are computed in a single contraction):"
@@ -27173,6 +755,7 @@
   {
    "cell_type": "code",
    "execution_count": 15,
+   "id": "0b6d4321-af15-4563-8e51-ab56d6870494",
    "metadata": {},
    "outputs": [
     {
@@ -27199,6 +782,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "4c58bd64-a19b-4c48-937e-5090ca65b9c0",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -27225,12 +809,13 @@
     "and so it can be more efficient than repeatedly calling\n",
     "{meth}`~quimb.tensor.circuit.Circuit.local_expectation`.\n",
     "\n",
-    "**Example usage:**\n"
+    "**Example usage:**"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 16,
+   "id": "2e3827fa-935c-42c3-a22a-970880572eba",
    "metadata": {},
    "outputs": [
     {
@@ -27253,6 +838,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "233fb601-bf86-49fd-83cb-61ae4f6d38d0",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -27284,12 +870,13 @@
     "The result is a $2^{|\\bar{q}|}$ dimensional tensor containing the probabilites for each\n",
     "bit-string $x_{\\bar{q}}$, given that we have already 'measured' $x_{\\bar{f}}$.\n",
     "\n",
-    "**Example usage:**\n"
+    "**Example usage:**"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 17,
+   "id": "23e532cd-059f-4238-9893-6daf6f4e97d8",
    "metadata": {},
    "outputs": [
     {
@@ -27312,6 +899,7 @@
   {
    "cell_type": "code",
    "execution_count": 18,
+   "id": "a5a9bcc3-0b3f-48af-848e-aae43d07ced0",
    "metadata": {},
    "outputs": [
     {
@@ -27331,6 +919,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "8a53a99a-1e79-4f86-abb4-f6b14641539f",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -27378,7 +967,7 @@
     "\n",
     ":::{note}\n",
     "You can set the group size to be that of the whole sytem, which is equivalent to sampling from the full wavefunction,\n",
-    "if you want to do this, it would be more efficient to call \n",
+    "if you want to do this, it would be more efficient to call\n",
     "[`Circuit.simulate_counts`](quimb.tensor.circuit.Circuit.simulate_counts), which\n",
     "doesn't draw the samples individually.\n",
     ":::\n",
@@ -27403,6 +992,7 @@
   {
    "cell_type": "code",
    "execution_count": 19,
+   "id": "20516146-84d4-404d-9395-1c0d43cc9fe0",
    "metadata": {},
    "outputs": [
     {
@@ -27429,6 +1019,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "15f0df48-dbfe-4c59-883f-bc11da3360e2",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -27464,12 +1055,13 @@
     "superconducting processor\"](https://www.nature.com/articles/s41586-019-1666-5).\n",
     ":::\n",
     "\n",
-    "**Example usage:**\n"
+    "**Example usage:**"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 20,
+   "id": "3b37c799-138a-41d6-af55-27604b9dff41",
    "metadata": {},
    "outputs": [
     {
@@ -27496,6 +1088,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "3a2a4c3b-5415-4c6a-9ec0-583cc6da03bd",
    "metadata": {},
    "source": [
     "Five of these qubits will now be sampled completely randomly."
@@ -27503,6 +1096,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "da48c0a7-2fc8-4213-b1ad-28a080a7cf81",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -27533,12 +1127,13 @@
     "the circuit geometry once.\n",
     ":::\n",
     "\n",
-    "**Example usage:**\n"
+    "**Example usage:**"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 21,
+   "id": "577bce82-85e6-4c69-b25d-17f616750d5e",
    "metadata": {},
    "outputs": [
     {
@@ -27564,6 +1159,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "b109ca39-074b-40c1-9022-03c65a0340c6",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -27585,12 +1181,13 @@
     "These each return a `dict` with the tensor network that *would* be\n",
     "contracted in the main part of the computation (with the key `'tn'`),\n",
     "and the {class}`cotengra.ContractionTree` object describing the contraction\n",
-    "path found for that tensor network (with the key `'tree'`). For example:\n"
+    "path found for that tensor network (with the key `'tree'`). For example:"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 22,
+   "id": "4539b749-4e02-49f9-9a75-eb538d7a351a",
    "metadata": {},
    "outputs": [
     {
@@ -27614,6 +1211,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "278204ba-a9f7-4772-aa54-8ec22bac5c90",
    "metadata": {},
    "source": [
     "Upper twenties is the limit for standard (~10GB) amounts of RAM."
@@ -27622,6 +1220,7 @@
   {
    "cell_type": "code",
    "execution_count": 23,
+   "id": "5c895eb6-eb54-493d-a2d9-5bcc524a2e68",
    "metadata": {},
    "outputs": [
     {
@@ -27647,6 +1246,7 @@
   {
    "cell_type": "code",
    "execution_count": 24,
+   "id": "ff947e35-5dc8-4638-91fa-f05f8db9f11d",
    "metadata": {},
    "outputs": [
     {
@@ -27667,6 +1267,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "03caaef4-d506-4929-bb2e-9453b53958aa",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -27682,6 +1283,7 @@
   {
    "cell_type": "code",
    "execution_count": 25,
+   "id": "b0930b08-4341-453e-8d3e-3cd3c4ee3a61",
    "metadata": {},
    "outputs": [
     {
@@ -27703,6 +1305,7 @@
   {
    "cell_type": "code",
    "execution_count": 26,
+   "id": "a28294de-9cca-4ad3-98ed-1352ec5fbf7b",
    "metadata": {},
    "outputs": [
     {
@@ -27722,6 +1325,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "3e256cb8-706a-4639-b93f-c7d6f2b86de6",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -27747,12 +1351,13 @@
     "Whilst such simplifications can be found by the local simplifications (see below) its\n",
     "easier and quicker to drop these explicitly.\n",
     "\n",
-    "You can see which gate tags are in the reverse lightcone of which regions of qubits by calling:\n"
+    "You can see which gate tags are in the reverse lightcone of which regions of qubits by calling:"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 27,
+   "id": "b1337c61-d109-405c-81de-a58f068356c7",
    "metadata": {},
    "outputs": [
     {
@@ -27784,7992 +1389,13 @@
   {
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    "execution_count": 28,
+   "id": "24522528-d0e6-46f6-be7d-7c0cf528d02f",
    "metadata": {},
    "outputs": [
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@@ -35787,6 +1413,7 @@
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   {
    "cell_type": "markdown",
+   "id": "2ae0add9-562a-4dc3-b49b-e90567e2c77e",
    "metadata": {},
    "source": [
     "We can plot the effect this has as selecting only these, $| \\psi \\rangle \\rightarrow | \\psi_{\\bar{q}} \\rangle$, on the norm with the following:"
@@ -35795,11052 +1422,13 @@
   {
    "cell_type": "code",
    "execution_count": 29,
+   "id": "66450825-eb07-4231-90a5-f21040f02e7c",
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@@ -46862,6 +1450,7 @@
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@@ -46902,3214 +1492,13 @@
   {
    "cell_type": "code",
    "execution_count": 30,
+   "id": "386f3831-abc2-4a13-b232-72aff53e8d98",
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@@ -50133,3902 +1522,23 @@
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+   "id": "757503e5-212b-4b45-8332-8f0af5d8fb58",
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    "source": [
-    "You can see that only 3+ dimensional tensors remain. Now if we turn on all the \n",
+    "You can see that only 3+ dimensional tensors remain. Now if we turn on all the\n",
     "simplification methods we get an even smaller tensor network:"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 31,
+   "id": "0edcf125-b9c5-43b2-9524-c79c7ec4dcc3",
    "metadata": {},
    "outputs": [
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       "text/plain": [
        "<Figure size 600x600 with 1 Axes>"
@@ -54052,6 +1562,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "a38035e4-6a6f-438d-81e9-1312e4d3b0af",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -54072,6 +1583,7 @@
   {
    "cell_type": "code",
    "execution_count": 32,
+   "id": "09f22ca6-d3a4-4be2-85d0-f7e01c94b2cc",
    "metadata": {},
    "outputs": [
     {
@@ -54306,6 +1818,7 @@
   {
    "cell_type": "code",
    "execution_count": 33,
+   "id": "eb16a0d5-e568-4a54-8bb9-db0762b11a97",
    "metadata": {},
    "outputs": [
     {
@@ -54566,6 +2079,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "da1fc24a-3ba3-4af8-9913-ac700fef353f",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -54578,6 +2092,7 @@
   {
    "cell_type": "code",
    "execution_count": 34,
+   "id": "5447fd05-4c19-469b-b28a-c978a8e78986",
    "metadata": {},
    "outputs": [
     {
@@ -54598,6 +2113,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "3ba34d2e-84bd-4116-8d16-86c217f5761c",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -54627,12 +2143,13 @@
     "\n",
     "$$\n",
     "\\langle \\psi_{3, 4} | Z_3 \\otimes Z_4 | \\psi_{3, 4} \\rangle\n",
-    "$$\n"
+    "$$"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 35,
+   "id": "557a18cd-ec4b-4b86-9748-68efa614151a",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -54643,6 +2160,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "c6f4a6fa-9851-46f3-92f7-ab7f1e65c6de",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -54655,6 +2173,7 @@
   {
    "cell_type": "code",
    "execution_count": 36,
+   "id": "89a2360f-9dfc-40dc-9a55-1627591c295e",
    "metadata": {},
    "outputs": [
     {
@@ -54676,6 +2195,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "df7ee8d9-31dd-44a3-a6a0-fb68b5e59863",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -54689,6 +2209,7 @@
   {
    "cell_type": "code",
    "execution_count": 37,
+   "id": "1da577f6-832c-4c53-81c6-bbd57abe7c7f",
    "metadata": {},
    "outputs": [
     {
@@ -54706,6 +2227,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "201e70c9-005c-4b62-b11c-505cf8156fc3",
    "metadata": {},
    "source": [
     "We can compare this to just performing the main contraction:"
@@ -54714,6 +2236,7 @@
   {
    "cell_type": "code",
    "execution_count": 38,
+   "id": "ae92a4af-ea2b-4701-b819-b30c610070ee",
    "metadata": {},
    "outputs": [
     {
@@ -54731,6 +2254,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "813190f1-41ca-4045-894e-4a3f58a31a54",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -54742,12 +2266,13 @@
     "You can also supply a customized `PathOptimizer` instance here, an example of which\n",
     "is the `opt_einsum.RandomGreedy`\n",
     "[optimizer](https://optimized-einsum.readthedocs.io/en/stable/random_greedy_path.html)\n",
-    "(which is itself called by `'auto-hq'` in fact).\n"
+    "(which is itself called by `'auto-hq'` in fact)."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 39,
+   "id": "1402e3ad-8637-49dc-9418-258c9a207bd5",
    "metadata": {},
    "outputs": [
     {
@@ -54774,6 +2299,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "81acf90b-dd87-4573-b0ec-22f3fad4cf4a",
    "metadata": {},
    "source": [
     "We see it has found a much better path than `'greedy'`, which is not so surprising.\n",
@@ -54785,6 +2311,7 @@
   {
    "cell_type": "code",
    "execution_count": 40,
+   "id": "589582e8-a0cf-42c4-822f-0ac5b125c5e1",
    "metadata": {},
    "outputs": [
     {
@@ -54805,6 +2332,7 @@
   {
    "cell_type": "code",
    "execution_count": 41,
+   "id": "b74225ff-5cc2-4e4c-932d-5f3e1fd8a8bd",
    "metadata": {},
    "outputs": [
     {
@@ -54825,6 +2353,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "ed2977b1-ec79-46dd-9796-e470068b2060",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -54858,6 +2387,7 @@
   {
    "cell_type": "code",
    "execution_count": 42,
+   "id": "b31213c1-c579-406a-b58c-b695abc55b50",
    "metadata": {},
    "outputs": [
     {
@@ -54897,6 +2427,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "751de381-51f8-48c0-87cf-9a8f084302fd",
    "metadata": {},
    "source": [
     "We can see even for this small contraction it has improved on the `RandomGreedy` path cost.\n",
@@ -54908,6 +2439,7 @@
   {
    "cell_type": "code",
    "execution_count": 43,
+   "id": "998779f6-f62a-4adf-a17e-de776cce5d06",
    "metadata": {},
    "outputs": [
     {
@@ -54925,6 +2457,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "83ddc8be-d0f1-4272-b2eb-cb130f373c33",
    "metadata": {},
    "source": [
     "Again, since the main contraction is very small, we don't see any real improvement."
@@ -54932,6 +2465,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "8f533952-d814-47fb-a23b-8aca61bd6c47",
    "metadata": {},
    "source": [
     "`cotengra` also has a `ContractionTree` object for manipulating and visualizing\n",
@@ -54941,5758 +2475,13 @@
   {
    "cell_type": "code",
    "execution_count": 44,
+   "id": "6a9f618f-d90a-4e0f-b4e0-44d52393bb01",
    "metadata": {},
    "outputs": [
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       ],
       "text/plain": [
        "<Figure size 500x500 with 3 Axes>"
@@ -60720,6 +2509,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "34bd9aa7-607f-4a72-ac1b-1ca753a0ec40",
    "metadata": {},
    "source": [
     "Here the, grey network at the bottom is the TN to be contracted, and the tree\n",
@@ -60729,6 +2519,7 @@
   },
   {
    "cell_type": "markdown",
+   "id": "f9e5a557-066f-476f-b7cc-6906e6a863ca",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -60742,6 +2533,7 @@
   {
    "cell_type": "code",
    "execution_count": 45,
+   "id": "3be67053-25e3-4c0d-8eef-9778b8125e0d",
    "metadata": {},
    "outputs": [
     {
@@ -60761,12 +2553,13 @@
   },
   {
    "cell_type": "markdown",
+   "id": "9977e99d-b884-4d3c-b418-6cf9a97934b9",
    "metadata": {},
    "source": [
     "When it comes to the actual contraction however, where large tensors will appear, it can be advantageous to\n",
     "use a different library to perform the contractions. If you specify a ``backend`` kwarg, before contraction,\n",
     "the arrays will converted to the ``backend``, then the contraction performed, and the result converted back\n",
-    "to ``numpy``. \n",
+    "to ``numpy``.\n",
     "[The list of available backends is here](https://optimized-einsum.readthedocs.io/en/stable/backends.html#special-gpu-backends-for-numpy-arrays), including:\n",
     "\n",
     "\n",
@@ -60778,16 +2571,18 @@
   },
   {
    "cell_type": "markdown",
+   "id": "276f5861-2c97-4bfe-88a2-63ed0d4a68a9",
    "metadata": {},
    "source": [
     "Sampling is an excellent candidate for GPU acceleration as the same geometry TNs\n",
-    "are contracted over and over again and since sampling is inherently a low \n",
+    "are contracted over and over again and since sampling is inherently a low\n",
     "precision task, single precision arrays are a natural fit."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 46,
+   "id": "ffa17477-8130-4b0b-9685-bc6e1bddcb34",
    "metadata": {},
    "outputs": [
     {
@@ -60809,11 +2604,12 @@
    ],
    "source": [
     "for b in circ.sample(10, backend='jax', dtype='complex64'):\n",
-    "    print(b)\n"
+    "    print(b)"
    ]
   },
   {
    "cell_type": "markdown",
+   "id": "2073a6e3-cd2f-455b-b078-9e960e616409",
    "metadata": {
     "raw_mimetype": "text/restructuredtext"
    },
@@ -60835,13 +2631,13 @@
     "- How are you applying the gates? For example, `gate_opts={'contract': False}` (no decomposition) can be better for diagonal 2-qubit gates.\n",
     "- How are you grouping the qubits? For sampling, there is a sweet spot for `group_size` usually. For chaotic sampling, you might try the *last* \\$M\\$ marginal qubits rather than the *first*, for example.\n",
     "- What local simplifications are you using, and in what order? `simplify_sequence='SADCR'` is also good sometimes.\n",
-    "- If the computation is contraction dominated, can you run it on a GPU?\n"
+    "- If the computation is contraction dominated, can you run it on a GPU?"
    ]
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3 (ipykernel)",
+   "display_name": "Python 3",
    "language": "python",
    "name": "python3"
   },
@@ -60855,7 +2651,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.11.8"
+   "version": "3"
   }
  },
  "nbformat": 4,