T-count and Qubit Optimized Quantum Circuit Designs of Carry Lookahead Adder

Abstract: Quantum circuits of arithmetic operations such as addition are needed to implement quantum algorithms in hardware. Quantum circuits based on Clifford+T gates are used as they can be made tolerant to noise. The tradeoff of gaining fault tolerance from using Clifford+T gates and error correcting codes is the high implementation overhead of the T gate. As a result, the T-count performance measure has become important in quantum circuit design. Due to noise, the risk for errors in a quantum circuit computation increases as the number of gate layers (or depth) in the circuit increases. As a result, low depth circuits such as quantum carry lookahead adders (QCLA)s have caught the attention of researchers. This work presents two QCLA designs each optimized with emphasis on T-count or qubit cost respectively. In-place and out-of-place versions of each design are shown. The proposed QCLAs are compared against the existing works in terms of T-count. The proposed QCLAs for out-of-place addition achieve average T gate savings of $54.34 \%$ and $37.21 \%$, respectively. The proposed QCLAs for in-place addition achieve average T gate savings of $72.11 \%$ and $35.87 \%$