Daniel Gottesman. Perimeter Institute. The Classical and Quantum Worlds. A general quantum error is a superoperator: Ak Ak†. Examples of single- qubit errors: Bit Flip X: X 0 = 1,. Classical Error- Correcting Codes. The typical classical single- bit error is the bit- flip: 0 ↔ 1. If we assume a simple error model ( the binary symmetric channel) in which bit flips errors occur on each bit independently with probablility p per unit. Classical error correction. The most general classical single- bit error is the bit- flip: 0 ↔ 1.

We will assume a simple error model ( the binary symmetric channel) in which bit flips errors occur on each bit independently with probablility p per unit. Fumiko Yamaguchi edu>. Quantum error correction. ○ Example: ○. Binary symmetric channel. Noise in the channel. = flip the bit with probability p ( 0 ≤ p ≤ 1). To better understand error correction, we look first at classical computers: Classical Error Correction. When transfering one bit of information, an error occurs as a ' bit flip' : A zero bit turns into one bit or vice versa. Classical noise, classical error correction. Classical noise: bit- flip errors. 0 − → 000 and 1 − → 111;. Majorty vote for correction;. New error probability: p − → 3p2 − 2p3 ( probability of two or more. This simple observation is crucial: a phase flip in one basis looks like a bit flip in the other basis.

It is crucial because it immediately shows us how to correct phase flip errors: we need an error correcting code in the new basis! Nielsen & Chuang. Quantum Information and. Quantum Computation, CUP. Bit flip channel. Phase flip channel. Bit- phase flip channel. Amplitude damping channel. Depolarizing channel. To overcome this, a different method, such as the so- called three- qubit bit flip code, has to be used. This technique uses entanglement and syndrome measurements and is comparable in performance with the.

9 Quantum error correction. Preskill: chapter 7. We wish to protect a single qubit against the occurrence of a single error. The error could be a bit flip ( σx), a phase shift ( σz), or the combination of a bit flip and a phase shift. ( σy = iσxσz).