Systematic Design and Optimization of Quantum Circuits for Stabilizer
  Codes
Quantum computing is an emerging technology that has the potential to achieve
exponential speedups over their classical counterparts. To achieve quantum
advantage, quantum principles are being applied to fields such as
communications, information processing, and artificial intelligence. However,
quantum computers face a fundamental issue since quantum bits are extremely
noisy and prone to decoherence. Keeping qubits error free is one of the most
important steps towards reliable quantum computing. Different stabilizer codes
for quantum error correction have been proposed in past decades and several
methods have been proposed to import classical error correcting codes to the
quantum domain. However, formal approaches towards the design and optimization
of circuits for these quantum encoders and decoders have so far not been
proposed. In this paper, we propose a formal algorithm for systematic
construction of encoding circuits for general stabilizer codes. This algorithm
is used to design encoding and decoding circuits for an eight-qubit code. Next,
we propose a systematic method for the optimization of the encoder circuit thus
designed. Using the proposed method, we optimize the encoding circuit in terms
of the number of 2-qubit gates used. The proposed optimized eight-qubit encoder
uses 18 CNOT gates and 4 Hadamard gates, as compared to 14 single qubit gates,
33 2-qubit gates, and 6 CCNOT gates in a prior work. The encoder and decoder
circuits are verified using IBM Qiskit. We also present optimized encoder
circuits for Steane code and a 13-qubit code in terms of the number of gates
used.
Arijit Mondal Keshab K. Parhi