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exercises/Exercise4.ipynb
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"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",
"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",
"\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",
"\n",
"c) How many logical qubits, $k$, can be stored in the stabilizer space?\n",
"\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: $\\{ X_j, Z_j \\} = 0$."
"\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: $\\{ X_j, Z_j \\} = 0$."
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