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 $KQ_T$.

翻译：量子平方运算是在实现线性回归、正则化最小二乘算法、阶寻找算法、量子搜索算法、牛顿-拉夫森除法、欧几里得距离计算、密码学以及求根与求倒数等多种量子算法中有用的构建模块。通过使用纠错码和容错量子门（如Clifford+T门集），量子电路可以实现容错。然而，T门的实现成本非常高。双量子比特门（如CNOT门）比单量子比特门更容易受到噪声误差的影响。因此，为了实现可靠的量子算法，量子电路应具有较低的T门计数和CNOT门计数。本文提出了一种新颖的量子整数平方架构，该架构针对T门计数、CNOT门计数、T门深度、CNOT门深度以及$KQ_T$进行了优化，且不产生垃圾输出。为降低成本，我们采用了一种新颖的生成部分积排列方法，使加法器数量减少了50%。我们还使用了文献[1]中展示的资源高效逻辑与门和反计算门以进一步节省资源。与Thapliyal等人[2]的方案相比，所提出的量子平方电路在T门计数上实现了66.67%的渐进式减少，T门深度减少50%，CNOT门计数减少29.41%，CNOT门深度减少42.86%，$KQ_T$减少25%。与Nagamani等人[3]的方案相比，该设计在T门计数上实现了77.27%的渐进式减少，T门深度减少68.75%，CNOT门计数减少50%，CNOT门深度减少61.90%，$KQ_T$减少6.25%。