Efficient quantum circuits for arithmetic operations are vital for quantum algorithms. A fault-tolerant circuit is required for a robust quantum computing in the presence of noise. Quantum circuits based on Clifford+T gates are easily rendered fault-tolerant. Therefore, reducing the T-depth and T-Count without increasing the qubit number represents vital optimization goals for quantum circuits. In this study, we propose the fault-tolerant implementations for TR and Peres gates with optimized T-depth and T-Count. Next, we design fault-tolerant circuits for quantum arithmetic operations using the TR and Peres gates. Then, we implement cyclic and complete translations of quantum images using quantum arithmetic operations, and the scalar matrix multiplication. Comparative analysis and simulation results reveal that the proposed arithmetic and image operations are efficient. For instance, cyclic translations of a quantum image produce 50% T-depth reduction relative to the previous best-known cyclic translation.

中文翻译：

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用于量子图像处理的高效量子算术运算电路

用于算术运算的高效量子电路对于量子算法至关重要。在存在噪声的情况下，要进行可靠的量子计算，就需要有一个容错电路。基于Clifford + T门的量子电路很容易实现容错。因此，在不增加量子位数的情况下减小T深度和T计数代表了量子电路的重要优化目标。在这项研究中，我们提出了具有优化的T深度和T计数的TR和Peres门的容错实现。接下来，我们使用TR和Peres门设计用于量子算术运算的容错电路。然后，我们使用量子算术运算和标量矩阵乘法来实现量子图像的循环和完全平移。对比分析和仿真结果表明，所提出的算法和图像运算是有效的。例如，相对于先前最广为人知的循环平移，量子图像的循环平移产生T深度减少50％。