Robustness and measurement-induced percolation of the surface code

Journal of Physics A Mathematical and Theoretical, vol. 58, pp. 205304

Abstract

Abstract We examine the robustness of a logical qubit in the planar surface code subject to ‘measurement-errors’, i.e. to local Pauli measurements at known positions. This yields a measurement-only dynamics, which is driven by the competition between local Pauli measurements and stabilizer measurements. The setup differs from the conventional surface code for which errors are caused by decoherence and their positions are unknown. Yet, our setting allows us to examine the dynamics of the encoded logical qubit by using a combination of analytical arguments based on percolation theory and numerical simulations. Firstly, we show that for a single round of Pauli measurements only, the threshold corresponding to the irreversible loss of the logical qubit depends only on the rate of X ^ - and Z ^ -measurements, and that this loss of logical quantum information is equivalent to the bond percolation transition in a 2D square lattice. Local Y ^ -measurements, which affect both X and Z stabilizers, erase the logical qubit only if all physical qubits along one of the diagonals are measured, and are thus negligible at large code distance. Secondly, we examine the dynamics in the code by considering the interplay between rounds of Pauli measurements and rounds of stabilizer measurements. Focusing on the lifetime of the logical qubit, we obtain a rich phase diagram featuring different dynamical regimes. We argue that the loss of the logical qubit in this setting can still be understood by percolation theory and underpin our arguments with numerical simulations.

Authors 4

1. Thomas Botzung Aachen Institute for Quantum Information

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany

   Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany

2. Michael Buchhold

   University of Cologne

   Affiliation as printed

   Institut für Theoretische Physik, Universität zu Köln, D-50937 Cologne, Germany

3. Sebastian Diehl

   University of Cologne

   Affiliation as printed

   Institut für Theoretische Physik, Universität zu Köln, D-50937 Cologne, Germany

4. Markus Michael Müller Aachen Institute for Quantum Information

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany

   Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany