T2: 3. Quantum Error Correction

Question 1

What are the three primary challenges of quantum error correction?

Answer 1

• We cannot make copies of arbitrary states
• Errors in a state |ψ⟩ will be continuous and not discrete; the state will evolve into some state
  |ψ⟩+ ϵ|χ⟩ .
• In order to identify an error, we must make a measurement; this will alter the state.

Question 2

What is the main assumption in quantum error correction.

Answer 2

That errors only affect a single bit; the physical realisation is that qubits are assumed to be well separated.

Question 3

How do we protect against single bit flips?

Answer 3

Using logical qubits: map every 0 to 000 and 1 to 111; if a single bit flip occurs, take a majority rule.

Question 4

Define an error syndrome

Answer 4

An operator chosen so different error subspaces are different eigenspaces of the operator with different eigenvalues.

Question 5

What syndromes do we define and how do they identify qubit flips?

Answer 5

Z_0 Z_1 and Z_0 Z_2 (Z_1 Z_2 not independent)

Acting on a state with both will allow you to identify the flipped qubit by +-1 eigenvalues

Question 6

Briefly explain the circuit for detecting and recovering bit flips

Answer 6

Two ancillary qubits set to zero acting CNOTs as targets for the CNOTs representing the syndrome (detection)

Two CNOTs from q_2 and q_1 to a_1 and a_0 respectively then a toffoli from each q with controls on both a.

Question 7

In which case is our bit flip detection and correction not viable? Why?

Answer 7

n = 2. We need n+1 orthogonal two-qubit subspaces for the syndromes; n=2 only gives us a four dimensional space. Hence, only works for 2^n-1>=n+1

Question 8

What is the relation between the Pauli X,Y and Z gates?

Answer 8

iXZ=Y

Question 9

What are the two key properties of the Steane code syndromes M_i and N_i?

Answer 9

M_i ^2 = N_i ^2 = 1

[M_i , M_j] = [N_i , N_j] = 0