One of the major promises of quantum computing is the realization of SIMD (single instruction - multiple data) operations using the phenomenon of superposition. Since the dimension of the state space grows exponentially with the number of qubits, we can easily reach situations where we pay less than a single quantum gate per data point for data-processing instructions which would be rather expensive in classical computing. Formulating such instructions in terms of quantum gates, however, still remains a challenging task. Laying out the foundational functions for more advanced data-processing is therefore a subject of paramount importance for advancing the realm of quantum computing. In this paper, we introduce the formalism of encoding so called-semi-boolean polynomials. As it turns out, arithmetic $\mathbb{Z}/2^n\mathbb{Z}$ ring operations can be formulated as semi-boolean polynomial evaluations, which allows convenient generation of unsigned integer arithmetic quantum circuits. For arithmetic evaluations, the resulting algorithm has been known as Fourier-arithmetic. We extend this type of algorithm with additional features, such as ancilla-free in-place multiplication and integer coefficient polynomial evaluation. Furthermore, we introduce a tailor-made method for encoding signed integers succeeded by an encoding for arbitrary floating-point numbers. This representation of floating-point numbers and their processing can be applied to any quantum algorithm that performs unsigned modular integer arithmetic. We discuss some further performance enhancements of the semi boolean polynomial encoder and finally supply a complexity estimation. The application of our methods to a 32-bit unsigned integer multiplication demonstrated a 90\% circuit depth reduction compared to carry-ripple approaches.
翻译:量子计算的主要承诺之一是利用超位现象实现 SIMD( 单向指令 - 多重数据) 运行 SIMD( 单向指令 - 多重数据 ) 。 由于国家空间的维度随着Qbit 数量而成倍增长, 我们很容易地达到以下状况: 我们为数据处理指示支付的费用低于每个数据点的单量门, 而在古典计算中, 这笔费用相当昂贵。 但是, 以量子门来制定这样的指示, 仍然是一项艰巨的任务 。 因此, 为更先进的数据处理提供基础功能, 是一个对推进量子计算领域至关重要的主题 。 在本文中, 我们引入了调值如此叫的超离值供应量- 超速- 超速- 超速供应多位数的多位数位数位数的正规化。 事实证明, 计算 计算 算术的计算方法可以算得更方便地生成未指派的数级数的算法 。 计算方法可以进一步推广这种算法, 比如, 将不使用无序的直径直径直径直径直径直流的内压的内径直径直径,, 等直径直流的递递递递递递性递性递性地计算法, 。