t b a b a

of the model can be written as:
H = vb
2
� +∞
dx[∂xφ(x)]
−∞
2 + [Π(x)] 2 +
�
π �
λ ∂xφ(n)σ
2 z n.
with φ and Π = ∂xθ canonic conjugate variables and σ z n the Pauli matrices acting in the
Hilbert space of the qubits. vb is the velocity of the bosonic excitations and the units are
such that � = kB = 1. The exact time evolution between gates of a qubit, can be expressed
as the product of two vertex operators of the free bosonic theory:
�
π
Un(t, 0) = exp[i
2 λ(θ(n, t) − θ(n, 0))σz n]
with λ the qubits bosonic coupling strength. Using this model the authors analyze the
three qubit Steane code. They calculate the probability of having errors in quantum error
correction cycles starting at times t1 and t2 and show that the probability of errors consists
of two terms; the first is the uncorrelated probability and the second is the contribution due
to correlation between errors in different cycles (∆ is the error correcting cycle):
P ≈ ( ɛ
2 )2 + λ4 ∆ 4
8(t1 − t2) 4
We see that correlations in the quantum system decay algebraically in time, and the latest
error will re-influence the system with a much higher probability than others.
Only phase-flip errors are discussed in detail in [NB06]. The approach introduced by the
authors is very general and can be extended to include other types of errors. Nevertheless,
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n