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arXiv quant-ph/0006004quant-ph
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量子傅里叶变换的快速并行电路

Fast parallel circuits for the quantum Fourier transform

Richard Cleve, John Watrous

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22 pages, LaTeX2e, 2 figures
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摘要：

我们给出了量子傅里叶变换（QFT）的电路复杂度的新的界限。我们给出了计算QFT关于模2^n的近似的电路深度的O（log n + log log（1/n））的上界，误差由n限制。因此，即使是指数小的错误，我们的电路有深度O（log n）。最好的深度边界是O（n），即使对于具有常数误差的近似。时间复杂度为O（n log（n/n））。我们还给出了模2^n的精确QFT的电路大小的O（n（log n）^2 log log n）的上界，其最佳先前界为O（n^2）。作为上述深度界的应用，我们表明，Shor的因子分解算法可以基于量子线路的深度只有O（log n）和多项式的大小，结合经典的多项式时间的预处理和后处理。在计算复杂性的语言中，这意味着因式分解属于复杂性类ZPP^BQNC，其中BQNC是可通过具有多对数深度和多项式大小的量子线路以有界错误概率计算的一类问题。最后，我们证明了一个欧米茄（log n）下界的深度复杂性近似的QFT与常数误差。这意味着上面的上限是渐近最优的（对于合理范围的值）。

英文摘要：

We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log (1/epsilon)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2^n with error bounded by epsilon. Thus, even for exponentially small error, our circuits have depth O(log n). The best previous depth bound was O(n), even for approximations with constant error. Moreover, our circuits have size O(n log (n/epsilon)). We also give an upper bound of O(n (log n)^2 log log n) on the circuit size of the exact QFT modulo 2^n, for which the best previous bound was O(n^2). As an application of the above depth bound, we show that Shor's factoring algorithm may be based on quantum circuits with depth only O(log n) and polynomial-size, in combination with classical polynomial-time pre- and post-processing. In the language of computational complexity, this implies that factoring is in the complexity class ZPP^BQNC, where BQNC is the class of problems computable with bounded-error probability by quantum circuits with poly-logarithmic depth and polynomial size. Finally, we prove an Omega(log n) lower bound on the depth complexity of approximations of the QFT with constant error. This implies that the above upper bound is asymptotically optimal (for a reasonable range of values of epsilon).

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