- STABILIZER CODES AND QUANTUM ERROR CORRECTION UPDATE
- STABILIZER CODES AND QUANTUM ERROR CORRECTION CODE
STABILIZER CODES AND QUANTUM ERROR CORRECTION CODE
How can we correct continuous errors and decoherence? Ĭorrecting Just Phase Errors Hadamard transform H exchanges bit flip and phase errors: H (0 + 1) = + + - X+ = +, X- = -- (acts like phase flip) Z+ = -, Z- = + (acts like bit flip) Repetition code corrects a bit flip error Repetition code in Hadamard basis corrects a phase error. Must correct multiple types of errors (e.g., bit flip and phase errors).
STABILIZER CODES AND QUANTUM ERROR CORRECTION UPDATE
Update on the Problems Measurement of error destroys superpositions. Redundancy, Not Repetition This encoding does not violate the no-cloning theorem: 0 + 1 000 + 111 (0 + 1)3 We have repeated the state only in the computational basis superposition states are spread out (redundant encoding), but not repeated (which would violate no-cloning). We have learned about the error without learning about the data, so superpositions are preserved! Note that the syndrome does not depend on and . Correct it with a X operation on the second qubit. Measure the Error, Not the Data With the information from the error syndrome, we can determine whether there is an error and where it is: E.g., 010 + 101 has syndrome 11, which means the second bit is different. 2nd bit of error syndrome says whether the second two bits of the state are the same or different. Measure the Error, Not the Data Use this circuit: Encoded state Ancilla qubits 0 Error syndrome 0 1st bit of error syndrome says whether the first two bits of the state are the same or different. We wish to measure that it is different without finding its actual value. Measurement Destroys Superpositions? Let us apply the classical repetition code to a quantum state to try to correct a single bit flip error: 0 + 1 000 + 111 Bit flip error (X) on 2nd qubit: 010 + 101 2nd qubit is now different from 1st and 3rd.
How can we correct continuous errors and decoherence?
No-Cloning Theorem There is no device that will copy an unknown quantum state: 0 00, 1 11 By linearity, 0 + 1 00 + 11 (0 + 1)(0 + 1) (Related to Heisenberg Uncertainty Principle)īarriers to Quantum Error Correction Measurement of error destroys superpositions. When errors are rare, one error is more likely than two. Quantum Errors A general quantum error is a superoperator: Ak Ak† Examples of single-qubit errors: Bit Flip X: X0 = 1, X1 = 0 Phase Flip Z: Z0 = 0, Z1 = -1 Complete dephasing: ( + ZZ†)/2 (decoherence) Rotation: R0 = 0, R1 = ei1Ĭlassical Repetition Code To correct a single bit-flip error for classical data, we can use the repetition code: 0 000 1 111 If there is a single bit flip error, we can correct the state by choosing the majority of the three bits, e.g. Quantum Error Correction Daniel Gottesman Perimeter Institute
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