Error correction with stabilizer codes #1365
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| In this demo, we will introduce the stabilizer formalism using bottom-up approach. We construct and | ||
| some well-known codes using the quantum circuit formalism and then derive their **stabilizer generators,** from which | ||
| the code can be reconstructed. This enables the construction of a wide range of error correction codes | ||
| directly from their stabilizer generators. |
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| In this demo, we will introduce the stabilizer formalism using bottom-up approach. We construct and | |
| some well-known codes using the quantum circuit formalism and then derive their **stabilizer generators,** from which | |
| the code can be reconstructed. This enables the construction of a wide range of error correction codes | |
| directly from their stabilizer generators. | |
| In this demo, we will introduce the stabilizer formalism using bottom-up approach. We will start by constructing some well-known codes using the quantum circuit formalism. From these circuits, we'll derive their **stabilizer generators,** from which | |
| the code can be reconstructed. This enables the construction of a wide range of error correction codes | |
| directly from their stabilizer generators. |
| ~~~~~~~~~~~~~~~ | ||
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| The first step in an error correction code is **encoding** one abstract or **logical qubit** into a set of many on-device **physical qubits.** | ||
| The rationale is that, if some external factor changes the state of one of the qubits, the remaining qubits still provide information about the original logical qubit. |
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| The rationale is that, if some external factor changes the state of one of the qubits, the remaining qubits still provide information about the original logical qubit. | |
| The rationale is that if an external factor changes the state of one of the qubits, the remaining qubits still provide information about the original logical qubit. |
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| Let's code this below and verify the output |
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Let's or let us? One is used above, the other here. Probably just change this one because it seems that 'let us' is the most used in the demo.
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| Let's code this below and verify the output |
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| Let's code this below and verify the output | |
| Let's code this below and verify the output. |
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| - Qubit encoding | ||
| - Error detection | ||
| - Error correction |
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do you want any punctuation here or it's fine as is?
| ############################################################################## | ||
| # | ||
| # .. note:: | ||
| # In the literature, you may have come across an error correction code being called an ":math:`[n,k]`-stabilizer code." In this notation, the number :math:`n` represents |
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| # In the literature, you may have come across an error correction code being called an ":math:`[n,k]`-stabilizer code." In this notation, the number :math:`n` represents | |
| # In the literature, you may have come across an error correction code called an ":math:`[n,k]`-stabilizer code." In this notation, the number :math:`n` represents |
| # .. math:: | ||
| # | ||
| # \begin{align*} | ||
| # \vert \bar{0}\rangle = &\frac{1}{4}\left(\vert 00000 \rangle \vert 10010 \rangle + \vert 01001 \rangle + \vert 10100 \rangle + \vert 01010 \rangle - \vert 11011 \rangle - \vert 00110 \rangle \right.\\ |
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there is a missing sign here between the first two states
| # S_2 = X_0 I_1 X_2 Z_3 Z_4,\\ | ||
| # S_3 = Z_0 X_1 I_2 Z_3 X_4. | ||
| # | ||
| # The calculations are a bit cumbersome, but with some patience we can find the common :math:`+1`-eigenspace of the stabilizer generators, |
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| # The calculations are a bit cumbersome, but with some patience we can find the common :math:`+1`-eigenspace of the stabilizer generators, | |
| # The calculations are a bit cumbersome, but with patience, we can determine the common :math:`+1`-eigenspace of the stabilizer generators |
| # | ||
| # \vert \bar{1}\rangle = X\otimes X \otimes X \otimes X \otimes X \vert \bar{0} \rangle. | ||
| # | ||
| # The logical operators bit-flip and phase-flip are for this code are :math:`\bar{X}= X^{\otimes 5}` and :math:`\bar{Z}=Z^{\otimes 5}.` With these |
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| # The logical operators bit-flip and phase-flip are for this code are :math:`\bar{X}= X^{\otimes 5}` and :math:`\bar{Z}=Z^{\otimes 5}.` With these | |
| # The logical bit-flip and phase-flip operators for this code are :math:`\bar{X}= X^{\otimes 5}` and :math:`\bar{Z}=Z^{\otimes 5}.` With these |
| print(f"{error} {wire}", five_qubit_code(1/2, np.sqrt(3)/2, error, wire)) | ||
| ############################################################################## | ||
| # | ||
| # The syndrome table is printed, and with we can apply the necessary operators to fix the corresponding Pauli errors. The script above is straightforward |
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| # The syndrome table is printed, and with we can apply the necessary operators to fix the corresponding Pauli errors. The script above is straightforward | |
| # The syndrome table is printed and with it, we can apply the necessary operators to fix the corresponding Pauli errors. The script above is straightforward |
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