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Paper 1

Statistical mechanical models for quantum codes with correlated noise

Christopher T. Chubb, Steven T. Flammia

Year

2018

Journal

arXiv preprint

DOI

arXiv:1809.10704

arXiv

1809.10704

We give a broad generalisation of the mapping, originally due to Dennis, Kitaev, Landahl and Preskill, from quantum error correcting codes to statistical mechanical models. We show how the mapping can be extended to arbitrary stabiliser or subsystem codes subject to correlated Pauli noise models, including models of fault tolerance. This mapping connects the error correction threshold of the quantum code to a phase transition in the statistical mechanical model. Thus, any existing method for finding phase transitions, such as Monte Carlo simulations, can be applied to approximate the threshold of any such code, without having to perform optimal decoding. By way of example, we numerically study the threshold of the surface code under mildly correlated bit-flip noise, showing that noise with bunching correlations causes the threshold to drop to $p_{\textrm{corr}}=10.04(6)\%$, from its known iid value of $p_{\text{iid}}=10.917(3)\%$. Complementing this, we show that the mapping also allows us to utilise any algorithm which can calculate/approximate partition functions of classical statistical mechanical models to perform optimal/approximately optimal decoding. Specifically, for 2D codes subject to locally correlated noise, we give a linear-time tensor network-based algorithm for approximate optimal decoding which extends the MPS decoder of Bravyi, Suchara and Vargo.

Open paper

Paper 2

Conservation of Quantum Correlations in Multimode Systems with $U(1)$ Symmetry

Giuseppe Buonaiuto, David M. Whittaker, Emiliano Cancellieri

Year

2017

Journal

arXiv preprint

DOI

arXiv:1711.00733

arXiv

1711.00733

We present a theoretical investigation of the properties of quantum correlation functions in a dissipative multimode system. We define a total mth order equal-time correlation function, summed over all modes, which is shown to be conserved if the Hamiltonian possesses U(1) symmetry, provided any dissipation processes are linear in the system operators. As examples, we demonstrate this conservation using numerical simulations of a coupled cavity system and the Jaynes-Cummings model.

Open paper