In this work, we develop a novel mathematical framework for universal digital quantum computation using algebraic probability theory. We rigorously define quantum circuits as finite sequences of elementary quantum gates and establish their role in implementing unitary transformations. A key result demonstrates that every unitary matrix in \(\mathrm{U}(N)\) can be expressed as a product of elementary quantum gates, leading to the concept of a universal dictionary for quantum computation. We apply this framework to the construction of quantum circuits that encode probability distributions, focusing on the Grover-Rudolph algorithm. By leveraging controlled quantum gates and rotation matrices, we design a quantum circuit that approximates a given probability density function. Numerical simulations, conducted using Qiskit, confirm the theoretical predictions and validate the effectiveness of our approach. These results provide a rigorous foundation for quantum circuit synthesis within an algebraic probability framework and offer new insights into the encoding of probability distributions in quantum algorithms. Potential applications include quantum machine learning, circuit optimization, and experimental implementations on real quantum hardware.

翻译：本研究基于代数概率理论，提出了一种用于通用数字量子计算的新型数学框架。我们严格将量子电路定义为基本量子门的有限序列，并确立其在实现酉变换中的作用。关键结果表明，\(\mathrm{U}(N)\)中的每个酉矩阵均可表示为基本量子门的乘积，由此引出量子计算通用字典的概念。我们将该框架应用于编码概率分布的量子电路构建，重点关注Grover-Rudolph算法。通过利用受控量子门与旋转矩阵，我们设计了一种能近似给定概率密度函数的量子电路。使用Qiskit进行的数值模拟验证了理论预测，并证实了本方法的有效性。这些成果为代数概率框架下的量子电路综合奠定了严格的理论基础，并为量子算法中概率分布的编码提供了新见解。潜在应用包括量子机器学习、电路优化以及在真实量子硬件上的实验实现。