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+   "source": [
+    "# Exercise 10"
+   ]
+  },
+  {
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+   "source": [
+    "## 1 Shor Code\n",
+    "\n",
+    "The Shor (or 9 Qubit) code is a method for quantum error correction based on the classical repetition code. Here we will consider the version where a logical qubit is first stored in three physical qubits using the encoding\n",
+    "$$\n",
+    "|+\\rangle_3 = |+++\\rangle, \\,\\,\\, |-\\rangle_3 = |---\\rangle.\n",
+    "$$\n",
+    "This protects against errors that try to flip $|+\\rangle_3$ to $|-\\rangle_3$, and vice-versa. But the corresponding errors for the $Z$ basis states\n",
+    "\n",
+    "\n",
+    "$$\n",
+    "|0\\rangle_3 = \\frac{1}{\\sqrt{2}} (|+++\\rangle+|---\\rangle), \\,\\,\\, |1\\rangle_3 = \\frac{1}{\\sqrt{2}} (|+++\\rangle-|---\\rangle)\n",
+    "$$\n",
+    "\n",
+    "become more likely. To deal with this we take three of these logical qubits and use them (like the original physical qubits) to encode a single logical qubit. This uses the encoding:\n",
+    "\n",
+    "$$\n",
+    "|0\\rangle_9 = |0\\rangle_3 \\otimes |0\\rangle_3 \\otimes |0\\rangle_3, \\,\\,\\, |1\\rangle_9 = |1\\rangle_3 \\otimes |1\\rangle_3 \\otimes |1\\rangle_3.\n",
+    "$$\n",
+    "\n",
+    "The end result is then a code that stores one logical qubit in 9 physical qubits, with stabilizer states\n",
+    "\n",
+    "$$\n",
+    "|0\\rangle_9 = \\left\\lbrack \\frac{1}{\\sqrt{2}} \\left(|+++\\rangle  +|---\\rangle \\right) \\right\\rbrack^{\\otimes 3}, \\,\\,\\, |1\\rangle_9 = \\left\\lbrack\\frac{1}{\\sqrt{2}}\\left(|+++\\rangle-|---\\rangle\\right) \\right\\rbrack^{\\otimes 3}\n",
+    "$$\n",
+    "\n",
+    "a) Find operators that act as $X$ and $Z$ on the logical qubit. What are the minimal number of qubits these act on?\n",
+    "\n",
+    "b) Suppose $\\sigma_x$ errors occur independently on each qubit with probability $p_x$. What is the probability $P_x$ that a logical $X$ occurs after syndrome measurement and error correction? For simplicity you can determine this only up to lowest order in $p_x$.\n",
+    "\n",
+    "c) Similarly, what is the probability $P_z$ of $Z$ errors, given that $\\sigma_z$ errors occur with probability $p_z$?  For simplicity you can determine this only up to lowest order in $p_z$."
+   ]
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+   "source": [
+    "## 2  Concatenated Shor Code\n",
+    "\n",
+    "In order to increase the performance of a code we can use the concept of concatenation. We will now consider this process for the Shor code.\n",
+    "\n",
+    "Let us describe physical qubits as level-0 qubits, and suppose we have $n$ of them. We can use these to encode $n/9$ logical qubits, which we call level-1 qubits. These will have lower probabilities for noise than the level-0 qubits, but maybe not as low as we require. We can then use the level-1 qubits as if they were physical qubits, using them to encode $n/9^2$ level-2 qubits. This procedure can then be continued as many times as required, with the level-$(l-1)$ qubits always used as the physical qubits of the Shor codes that encode the level-$l$ qubits.\n",
+    "\n",
+    "a) In order to encode a single level-$l$ qubit, for arbitrary $l$, what is the number $n(l)$ of level-0 qubits required?\n",
+    "\n",
+    "b) The standard Shor code has distance $d=3$. What is the distance of a level-$l$ concatenated Shor code?\n",
+    "\n",
+    "c) We will use $p_x^{(l)}$ to denote the bit flip error probility at level-$l$. The level-0 error rate is that of the phyical qubit errors: $p_x^{(0)}=p_x$. The level-1 error rate is that of the logical qubits of the 9 qubit code: $p_x^{(1)}=P_x$. Use your result from question 1(b) to show that $p_x^{(l-1)}$ and $p_x^{(l)}$ are related by\n",
+    "$$\n",
+    "p_x^{(l)} < 27 \\left( p_x^{(l-1)} \\right)^2.\n",
+    "$$\n",
+    "\n",
+    "d) Show that $p_x^{(l)}$ decays exponentially with $n(l)^\\beta$ when $p_x < 1/27$, and find $\\beta$.\n",
+    "\n",
+    "This is a proof of the `threshold theorem' for this code and error model. As long as the physical noise rate $p_x$ is below the threshold value of $1/27$ ( and $p_z$ is below its threshold of $1/9$), concatenation of the Shor code can achieve arbitrarily low error rates.\n"
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