T-count and T-depth efficient fault-tolerant quantum arithmetic and logic unit

Quantum circuits are one of the best platforms to implement quantum algorithms. Concerning fault-tolerant quantum circuit, the Clifford + T gate set supports quantum circuits against decoherence error. However, they cause physical resource overheads like many qubits and the use of T gates as a high-cost computing element. This work focuses on low T-cost fault tolerant quantum ALU implementation using Clifford + T gate set. Three new different designs of quantum ALU are proposed by introducing a new quantum logic unit, and new low-cost fault tolerant implementations of full adder and subtractor circuits. We present a novel lemma in synthesizing quantum NCV-based circuits to Clifford + T quantum circuits. This lemma shows how an NCV-based structure with less CNOT layer can lead to an improvement in T-count and T-depth criteria in Clifford + T equivalent circuit. We analyze the effect of applying our proposed lemma in implementing low-cost fault tolerant Clifford + T circuits by some examples on adder and subtractors and ALUs. Comparison of the designs shows 50%, 40%, 36%, and 69% superior functionality of our proposed ALU module in terms of T-count, T-depth, number of qubits, and number of calculated operations compared to the existing counterpart, respectively. The proposed lemma can be used as a simplification step in quantum circuit synthesis algorithms and can be extended to use in quantum synthesis tools.