Qubit and T-count Optimized Quantum Circuit Design for Fixed Precision Square Root

Quantum square root operation is a useful building block in implementing important quantum algorithms such as the Harrow–Hassidim–Lloyd (HHL) algorithm, algorithms in physics and quantum chemistry, and also used in computing roots of quadratic equation, evaluating quadratic congruence, solving Hamiltonian and Poisson equations, trigonometric functions, computing logarithm and fractional exponent, and in quantum chemistry [1]–[5]. Quantum circuits could be made fault-tolerant by using error correcting codes and fault-tolerant quantum gates (such as the Clifford + T-gates). However, The T-gates are expensive to implement than other fault-tolerant gates (such as Clifford+T gates) [6] [7]. Therefore, quantum circuits with low T-count and qubit cost is preferred. In this paper, we present a novel quantum square root architecture optimized for T-count and Qubit cost that produces no garbage outputs. The proposed square root circuit accepts integer and fractional inputs. We use the Babylonian method to calculate the square root. To reduce the number of iteration, the initial guess required in the Babylonian method is calculated from the integer square root circuit in [8] which reduces the cost measures. To further save the cost, we use the resource efficient non-restoring division circuit shown in [9] and quantum adder shown in [6]. The proposed quantum square root circuit sees an asymptotic reduction of 95.84% in T-count and 88.89% in Qubit cost with respect to Bhaskar et al. [1]. With respect to Dutta et al. [10] the design sees an asymptotic reduction of 96% in T-count, and 88.89% in Qubit cost.