Quantum error correction is used in Quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise but also faulty quantum gates, faulty quantum preparation and faulty measurements.
Decoherence: with passage of time qubit gets destroyed due to noise or the environment
- No cloning theorem
- Measurment will destroy the information
- Errors are continuous, errors can be bit flip error, phase error or a linear combination of both
Classical computing can see certain errors such as 0 being received as 1 and 1 as 0. It employs redundancy, 0 could be sent over the classical channel as 000 (logical 0) and 1 as 111 (logical 1)
| Classical channel | Receiving end | No. of 0s, 1s | Parity (no.of 1s) | Majority bit | Output |
|---|---|---|---|---|---|
| 000 | 001 | 2, 1 | 1 (odd) | 0 | 0 |
| 000 | 010 | 2, 1 | 1 (odd) | 0 | 0 |
| 000 | 100 | 2, 1 | 1 (odd) | 0 | 0 |
| 111 | 110 | 1, 2 | 2 (even) | 1 | 1 |
| 111 | 101 | 1, 2 | 2 (even) | 1 | 1 |
| 111 | 011 | 1, 2 | 2 (even) | 1 | 1 |
Output generated by majority rule
Consider single bit error for inputs 00 and 11 over the classical channels
| Input | Output |
|---|---|
| 00 | 01 |
| 00 | 10 |
| 11 | 10 |
| 11 | 01 |
There is no majority bit and output cannot be obtained by majority rule
Consider a binary symmetric channel where a bit is sent as an input and has a certain probability of generating an output
| Input | Output | Probability |
|---|---|---|
| 0 | 0 | P ------ (1) |
| 0 | 1 | 1-P |
| 1 | 0 | P ------ (2) |
| 1 | 1 | 1-P |
(1) and (2) comprise bit flip error.
Create an encoding by entanglement and send it to the channel to generate certain outputs
Outputs generated:
- P0 has no error
- In P1 flip the first qubit, there would be no error
- In P2 flip the second qubit, there would be no error
- In P3 flip the third qubit, there would be no error
There is a possibility that while being transmitted in a channel the relative sign between |0> and |1> can become inverted.
A qubit in the state |-⟩ = (|0⟩ - |1⟩)/√2 can be flipped to |+⟩ = (|0⟩ + |1⟩)/√2
In orignal state of the qubit, a|+⟩ + b|-⟩ will be changed to a|+++⟩ + b|---⟩
Ephase is a quantum channel that can cause at most one phase flip. Then the bit flip code from above can recover |ψ⟩ by transforming into the Hadamard basis before and after transmission through Ephase
Shor code can correct both the bit flip and phase flip errors. In fact, it corrects arbitrary single-qubit errors
Let E be a quantum channel that can arbitrarily corrupt a single qubit. The 1st, 4th and 7th qubits are for the sign flip code, while the three group of qubits (1,2,3), (4,5,6), and (7,8,9) are designed for the bit flip code.
A qubit in the state, |ψ⟩ = a0|0s⟩ + a1|1s⟩, where
If a bit flip error happens to a qubit, the syndrome analysis will be performed on each set of states (1,2,3), (4,5,6), and (7,8,9), then correct the error. The Shor code also can correct any arbitrary errors to a single qubit. If an error is modeled by a unitary transform U, which will act on a qubit |ψ⟩, then U can be described in the form
where c0, c1, c2 and c3 are complex constants, I is identity and Pauli matrices are given by
- U = I, no error
, bit flip error occurs
, phase flip error occurs
, both the errors occur
