Realizing a class of stabilizer quantum error correction codes using a
  single ancilla and circular connectivity
We describe a class of "neighboring-blocks" stabilizer quantum error
correction codes and demonstrate that such class of codes can be implemented in
a resource-efficient manner using a single ancilla and circular near-neighbor
qubit connectivity. We propose an implementation for syndrome-measurement
circuits for codes from the class and illustrate its workings for cases of
3-qubit repetition code, Laflamme's 5-qubit code, and Shor's 9-qubit code. For
3-qubit repetition code and Laflamme's 5-qubit code suggested scheme has the
property that it uses only native two-qubit CNS gates, which potentially
reduces the amount of non-correctable errors due to the shorter gate time.
Elements of the scheme can be used to implement surface code with
near-neighbour connectivity using single ancilla, as demonstrated in an
example. We developed efficient decoding procedures for repetition codes and
the Laflamme's 5-qubit code using a minimum weight-perfect matching approach to
account for the specific order of measurements in our scheme. The analysis of
noise levels for which the scheme could show improvements in the fidelity of a
stored logical qubit in the 3-qubit repetition code and Laflamme's 5-qubit code
cases is provided. We complement our results by realizing the developed scheme
for a 3-qubit code using an IBM quantum processor and the Laflamme's 5-qubit
code using the state-vector simulator.
A. V. Antipov E. O. Kiktenko A. K. Fedorov