Design and Logic Synthesis of a Scalable, Efficient Quantum Number Theoretic Transform

The advent of quantum computing has engendered a widespread proliferation of efforts utilizing qubits for optimizing classical computational algorithms. Number Theoretic Transform (NTT) is one such popular algorithm that accelerates polynomial multiplication significantly and is consequently, the core arithmetic operation in most homomorphic encryption algorithms. Hence, fast and efficient execution of NTT is highly imperative for practical implementation of homomorphic encryption schemes in different computing paradigms. In this paper, we, for the first time, propose an efficient and scalable Quantum Number Theoretic Transform (QNTT) circuit using quantum gates. We introduce a novel exponential unit for modular exponential operation, which furnishes an algorithmic complexity of O(n). Our proposed methodology performs further optimization and logic synthesis of QNTT, that is significantly fast and facilitates efficient implementations on IBM’s quantum computers. The optimized QNTT achieves a gate-level complexity reduction from power of two to one with respect to bit length. Our methodology utilizes 44.2% fewer gates, thereby minimizing the circuit depth and a corresponding reduction in overhead and error probability, for a 4-point QNTT compared to its unoptimized counterpart.