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https://github.com/quantumjim/Quantum-Computation-course-Basel.git
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137 lines
6.4 KiB
Plaintext
137 lines
6.4 KiB
Plaintext
{
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"# Exercise 4"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"## 1. Shor Code\n",
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"\n",
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"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",
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"$$\n",
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"|+\\rangle_3 = |+++\\rangle, \\,\\,\\, |-\\rangle_3 = |---\\rangle.\n",
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"$$\n",
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"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",
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"\n",
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"\n",
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"$$\n",
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"|0\\rangle_3 = \\frac{1}{\\sqrt{2}} (|+++\\rangle+|---\\rangle), \\,\\,\\, |1\\rangle_3 = \\frac{1}{\\sqrt{2}} (|+++\\rangle-|---\\rangle)\n",
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"$$\n",
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"\n",
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"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",
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"\n",
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"$$\n",
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"|0\\rangle_9 = |0\\rangle_3 \\otimes |0\\rangle_3 \\otimes |0\\rangle_3, \\,\\,\\, |1\\rangle_9 = |1\\rangle_3 \\otimes |1\\rangle_3 \\otimes |1\\rangle_3.\n",
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"$$\n",
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"\n",
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"The end result is then a code that stores one logical qubit in 9 physical qubits, with stabilizer states\n",
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"\n",
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"$$\n",
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"|0\\rangle_9 = \\left\\lbrack \\frac{1}{\\sqrt{2}} \\left(|+++\\rangle +|---\\rangle \\right) \\right\\rbrack^{\\otimes 3}, \\,\\,\\, |1\\rangle_9 = \\left\\lbrack\\frac{1}{\\sqrt{2}}\\left(|+++\\rangle-|---\\rangle\\right) \\right\\rbrack^{\\otimes 3}\n",
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"$$\n",
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"\n",
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"a) Find operators that act as $X$ and $Z$ on the logical qubit. What are the minimal number of qubits these act on?\n",
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"\n",
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"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",
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"\n",
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"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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]
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},
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"source": [
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"## 2. Concatenated Shor Code\n",
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"\n",
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"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",
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"\n",
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"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",
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"\n",
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"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",
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"\n",
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"b) The standard Shor code has distance $d=3$. What is the distance of a level-$l$ concatenated Shor code?\n",
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"\n",
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"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",
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"$$\n",
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"p_x^{(l)} < 27 \\left( p_x^{(l-1)} \\right)^2.\n",
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"$$\n",
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"\n",
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"d) Show that $p_x^{(l)}$ decays exponentially with $n(l)^\\beta$ when $p_x < 1/27$, and find $\\beta$.\n",
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"\n",
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"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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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 3. The Toric Code\n",
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"\n",
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"The surface code is defined on a plane with rough and smooth boundary conditions.\n",
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"But we could instead wrap the $L \\times L$ square lattice around a torus and have periodic\n",
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"boundary conditions. Then the code would be translationally invariant, and all $A_s$\n",
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"and $B_p$ stabilizers would be four qubit operators.\n",
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"\n",
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"a) The parameter $L$ counts the number of plaquettes along each direction. Show that\n",
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"$n = 2L^2$, where $n$ is the number of qubits.\n",
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"\n",
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"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",
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"\n",
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"c) How many logical qubits, $k$, can be stored in the stabilizer space?\n",
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"\n",
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"d) Define logical $X$ and $Z$ operators for these logical qubits. Note that these are not\n",
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"uniquely defined. However, as with any stabilizer code, you will know that your logical\n",
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"operators are a valid choice if they satisfy the following conditions.\n",
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"\n",
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"1. Logical Pauli operators must commute with all stabilizers.\n",
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"2. Logical Pauli operators for the same logical qubit anticommute: $\\{ X_j, Z_j \\} = 0$.\n",
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"3. Logical Pauli operators for different logical qubits commute: $\\left[ X_j, Z_j \\right] = 0$."
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]
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