  Quantum squaring operation is a useful building block in implementing quantum
algorithms such as linear regression, regularized least squares algorithm,
order-finding algorithm, quantum search algorithm, Newton Raphson division,
Euclidean distance calculation, cryptography, and in finding roots and
reciprocals. 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-gate is very costly to implement. Two qubit gates
(such as the CNOT-gate) are more prone to noise errors than single qubit gates.
Consequently, in order to realize reliable quantum algorithms, the quantum
circuits should have a low T-count and CNOT-count. In this paper, we present a
novel quantum integer squaring architecture optimized for T-count, CNOT-count,
T-depth, CNOT-depth, and $KQ_T$ that produces no garbage outputs. To reduce
costs, we use a novel approach for arranging the generated partial products
that allows us to reduce the number of adders by 50%. We also use the resource
efficient logical-AND gate and uncomputation gate shown in [1] to further save
resources. The proposed quantum squaring circuit sees an asymptotic reduction
of 66.67% in T-count, 50% in T-depth, 29.41% in CNOT-count, 42.86% in
CNOT-depth, and 25% in KQ T with respect to Thapliyal et al. [2]. With respect
to Nagamani et al. [3] the design sees an asymptotic reduction of 77.27% in
T-count, 68.75% in T-depth, 50% in CNOT-count, 61.90% in CNOT-depth, and 6.25%
in the $KQ_T$.
