  In the framework of quantum computational tensor network [D. Gross and J.
Eisert, Phys. Rev. Lett. {\bf98}, 220503 (2007)], which is a general framework
of measurement-based quantum computation, the resource many-body state is
represented in a tensor-network form, and universal quantum computation is
performed in a virtual linear space, which is called a correlation space, where
tensors live. Since any unitary operation, state preparation, and the
projection measurement in the computational basis can be simulated in a
correlation space, it is natural to expect that fault-tolerant quantum circuits
can also be simulated in a correlation space. However, we point out that not
all physical errors on physical qudits appear as linear completely-positive
trace-preserving errors in a correlation space. Since the theories of
fault-tolerant quantum circuits known so far assume such noises, this means
that the simulation of fault-tolerant quantum circuits in a correlation space
is not so straightforward for general resource states.

Quantum Physics