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"boundary conditions. Then the code would be translationally invariant, and all $A_s$\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", "and $B_p$ stabilizers would be four qubit operators.\n",
"\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", "$n = 2L^2$, 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", "\n",
"* (c) How many logical qubits, $k$, can be stored in the stabilizer space?\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",
"* (d) Define logical $X$ and $Z$ operators for these logical qubits. Note that these are not\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", "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", "operators are a valid choice if they satisfy the following conditions.\n",
" 1. Logical Pauli operators must commute with all stabilizers.\n", "\n",
" 2. Logical Pauli operators for the same logical qubit anticommute: $\\left[ X_j, Z_j \\right] = 0$.\n", "1. Logical Pauli operators must commute with all stabilizers.\n",
" 3. Logical Pauli operators for different logical qubits commute: $\\{ X_j, Z_j \\} = 0$." "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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