Efficient recursive encoders for quantum Reed-Muller codes towards Fault tolerance

Transversal gates are logical gate operations on encoded quantum information that are efficient in gate count and depth, and are designed to minimize error propagation. Efficient encoding circuits for quantum codes that admit transversal gates are thus crucial to reduce noise and realize useful quantum computers. The class of punctured Quantum Reed-Muller codes admit transversal gates. We construct resource efficient recursive encoders for the class of quantum codes constructed from Reed-Muller and punctured Reed-Muller codes. These encoders on $n$ qubits have circuit depth of $O(\log n)$ and lower gate counts compared to previous works. The number of CNOT gates in the encoder across bi-partitions of the qubits is found to be equal to the entanglement entropy across these partitions, demonstrating that the encoder is optimal in terms of CNOT gates across these partitions. Finally, connecting these ideas, we explicitly show that entanglement can be extracted from QRM codewords.