Category: Quantum Error Correction

The number of syndromes in a 5-qubit scheme would be 5 × 3 + 1 = 16. We’d thus need 4 stabilizer operators since 2 4 = 16. We’ll simply give the operators (see Mermin [48] or Laflamme et al. [44]): M 0 = Z 1 X 2 X 3 Z 4 , M 2…

One of the main results of the theory of quantum error correction is that any general quantum error can be composed only of discrete errors represented by the Pauli operators X, Z, and Y = −iXZ. Errors are induced on our qubit system due to effects of everything outside this system, which we will call…

Let’s now consider a channel that can produce both bit flips and phase flips. A code that combines bit and phase flip coding should protect against these errors. A simple way to do this is to first encode for phase flips: |0i → |+ + +i; |1i → |− − −i and then encode using…

Bit flips alone are a very limited kind of error a qubit could undergo. Consider phase flips, which have no classical equivalent. A phase-flip quantum channel is defined as one that only allows single phase flips, that is, with probability p, |1i → −|1i. Under the action of this channel, a generic state transforms as…

Let us estimate the probability for the above technique to yield an uncor- rupted state, considering a channel characterized by flipping of a qubit with probability p < 1/2. We list in Table 10.3 the probability of occurrence of various corrupted states in order of decreasing probability. The probability that our procedure corrects errors is…

Why does this scheme work? It is possible to distinguish the syndromes, which are orthogonal states, if we measure a suitable observable of which they are eigenstates. It turns out that the bit-flip syndrome states |ψ i i are eigenstates of the operators Z 1 Z 2 and Z 2 Z 3 with distinct sets…

Classically, the repetition code is the simplest way of introducing redun- dancy to protect information. Assume that noise in the channel is modelled as a bit flip with probability p (and hence 1 − p for not flipping). This is schematised in Figure 10.2. This is known as the binary symmetric channel. 0 1−p //…

discover which error has occurred. Second, the no-cloning theorem prevents us from creating redundant quantum information by cloning a qubit. Also, one cannot think of copying an output before measuring it. Nevertheless, as we will see in this section, it is possible to correct qubits for errors in an intrinsic manner without destroying the information…