cleanup

exercises/Exercise3.ipynb

@@ -15,11 +15,11 @@
    "source": [
     "## 1\n",
     "\n",
-    "Given an abitrary single qubit state $a|0\\rangle +b|1\\rangle$:\n",
+    "Given an arbitrary single qubit state $a|0\\rangle +b|1\\rangle$:\n",
     "\n",
-    "(a) Write down the correponding density matrix.\n",
+    "(a) Write down the corresponding density matrix.\n",
     "\n",
-    "(b) Write down the density matrix representing the effect of applying an $X$ with probability $q_x$, $Y$ with probability $q_y$ and $Z$ with probability $q_z$.\n",
+    "(b) Write down the density matrix representing the effect of applying an $X$ with probability $q_x$, $Y$ with probability $q_y$, $Z$ with probability $q_z$ (and doing nothing with probability $1-q_x-q_y-q_z$).\n",
     "\n",
     "(c) Write down the density matrix for representing the effect of replacing the state with $I/2$ with probability $p$.\n",
     "\n",

exercises/Exercise4.ipynb

@@ -7,7 +7,7 @@
     "id": "pmm5uV8cQQN6"
    },
    "source": [
-    "# Exercise 10"
+    "# Exercise 4"
    ]
   },
   {
@@ -74,11 +74,27 @@
    ]
   },
   {
-   "cell_type": "code",
-   "execution_count": null,
+   "cell_type": "markdown",
    "metadata": {},
-   "outputs": [],
-   "source": []
+   "source": [
+    "## 3. The Toric Code\n",
+    "\n",
+    "The surface code is defined on a plane with rough and smooth boundary conditions.\n",
+    "But we could instead wrap the $L \\times L$ square lattice around a torus and have periodic\n",
+    "boundary conditions. Then the code would be translationally invariant, and all $A_s$\n",
+    "and $B_p$ stabilizers would be four qubit operators.\n",
+    "\n",
+    "* (a) The parameter $L$ counts the number of plaquettes along each direction. Show that\n",
+    "$n = 2L^2$, where $n$ is the number of qubits.\n",
+    "* (b) Show that the number of plaquette operators is $L^2$, but that the number of independent plaquette operators is $L^2-1$. Show the same thing for the vertex operators.\n",
+    "* (c) How many logical qubits, $k$, can be stored in the stabilizer space?\n",
+    "* (d) Define logical $X$ and $Z$ operators for these logical qubits. Note that these are not\n",
+    "uniquely defined. However, as with any stabilizer code, you will know that your logical\n",
+    "operators are a valid choice if they satisfy the following conditions.\n",
+    "    1. Logical Pauli operators must commute with all stabilizers.\n",
+    "    2. Logical Pauli operators for the same logical qubit anticommute: $\\left[ X_j, Z_j \\right] = 0$.\n",
+    "    3. Logical Pauli operators for different logical qubits commute: $\\{ X_j, Z_j \\} = 0$."
+   ]
   }
  ],
  "metadata": {