  The demonstration of quantum error correction (QEC) is one of the most
important milestones in the realization of fully-fledged quantum computers.
Toward this, QEC experiments using the surface codes have recently been
actively conducted. However, it has not yet been realized to protect logical
quantum information beyond the physical coherence time. In this work, we
performed a full simulation of QEC for the rotated surface codes with a code
distance 5, which employs 49 qubits and is within reach of the current
state-of-the-art quantum computers. In particular, we evaluate the logical
error probability in a realistic noise model that incorporates not only
stochastic Pauli errors but also coherent errors due to a systematic control
error or unintended interactions. While a straightforward simulation of 49
qubits is not tractable within a reasonable computational time, we reduced the
number of qubits required to 26 qubits by delaying the syndrome measurement in
simulation. This and a fast quantum computer simulator, Qulacs, implemented on
GPU allows us to simulate full QEC with an arbitrary local noise within
reasonable simulation time. Based on the numerical results, we also construct
and verify an effective model to incorporate the effect of the coherent error
into a stochastic noise model. This allows us to understand what the effect
coherent error has on the logical error probability on a large scale without
full simulation based on the detailed full simulation of a small scale. The
present simulation framework and effective model, which can handle arbitrary
local noise, will play a vital role in clarifying the physical parameters that
future experimental QEC should target.
