Error Distributions on Large Entangled States with Non-Markovian Dynamics

We investigate the distribution of errors on a computationally useful entangled state generated via the repeated emission from an emitter undergoing strongly non-Markovian evolution. For emitter-environment coupling of pure-dephasing form, we show that the probability that a particular patten of errors occurs has a bound of Markovian form, and thus, accuracy threshold theorems based on Markovian models should be just as effective. Beyond the pure-dephasing assumption, though complicated error structures can arise, they can still be qualitatively bounded by a Markovian error model.

Figure 1

Quantum circuit for the generation of a linear cluster state of five photons and a quantum dot. Non-Markovian decoherence of the dot is modelled as a sequential coupling to an environment. In the ideal case, the unitaries are replaced by $e^{-i{Y}_D\pi /4}\otimes {\mathbb{1}}_E$.

Figure 2

Left panels: all $2^5$ non-Markovian error distribution probabilities for a five-photon state, calculated exactly (blue circles), using the bound given in Eq. (6) (red crosses), and a best fit to a Markovian model of the form $p^h(1-p{)}^{n-h}$ (green circles). The error distributions are ordered along the $x$ axis such that those corresponding to the least number of fundamental errors are to the left. The inset in the lower plot shows a zoom in of the $h(\mathit{\alpha })=2$ band. Right panels: scaling of a typical error distribution probability with increasing environment size.

Figure 3

Top: all $2^8$ exact non-Markovian error probabilities for an eight-photon state in the non-pure-dephasing case. Middle: corresponding polarization $⟨\sum _k{I}_k^y⟩$ of the spin environment: error probabilities falling outside their bands correspond to angular momentum exchanges. Bottom: the qualitative features of the probability distribution can be captured by a simple Markovian model shown in green.