Fourier Method

Chao Lei, Apostol Vourdas

Abstract Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space $$D=d^n$$ D = d n with d an odd integer, and is inspired by the Cooley-Tukey formalism. The ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (together with some other transforms) in quantum systems with d-dimensional Hilbert space. Limitations of the method are discussed. In some special cases, the n Fourier transforms can be performed in parallel. The second method is for systems with dimension of the Hilbert space $$D=d_0...d_{n-1}$$ D = d 0 . . . d n - 1 with $$d_0,...,d_{n-1}$$ d 0 , . . . , d n - 1 odd integers coprime to each other. It is inspired by the Good formalism, which in turn is based on the Chinese reminder theorem. In this case also the ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (that involve some constants related to the number theory that describes the formalism). The ‘small Fourier transforms’ can be performed in a classical computer or in a quantum computer (in which case we have the additional well known advantages of quantum Fourier transform circuits). In the case that the small Fourier transforms are performed with a classical computer, complexity arguments for both methods show the reduction in computational time from $$\mathcal{O}(D^2)$$ O ( D 2 ) to $$\mathcal{O}(D\log D)$$ O ( D log D ) . The second method is also used for the fast calculation of Wigner and Weyl functions, in quantum systems with large finite dimension of the Hilbert space.

Bernd Thaller

Kurt Bernardo Wolf

Pieter Kok, Samuel L. Braunstein, Agedi N. Boto +3 more

As demonstrated in Boto et al. [quant-ph/9912052], quantum lithography offers an increase in resolution without an upper bound. This procedure makes use of the entangled state (| n ,0〉+ e inϕ |0, n 〉)/√2. It allows us to write evenly spaced lines with sub-wavelength resolution. Here we generalise this procedure in such a way that it enables us to create arbitrary patterns in one dimension. We distinguish two methods: the ‘Fourier method’ and ‘the superposition method’. The Fourier method is conceptually easier since it depends on Fourier series, but it also involves a minimal finite amount of unwanted exposure of the substrate. The superposition method gets around this problem and gives generally better results, but lacks the intuitive clarity of the Fourier method. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Ajoy Kumar Ghatak, S. Lokanathan

Yifan Sun, Ran Tao, Xiangdong Zhang

We have extended Fourier transform of quantum light to a fractional Fourier processing, and demonstrated that a classical optical fractional Fourier processor can be used for the shaping of quantum correlations between two or more photons. Comparing the present method with that of Fourier processing, we find that fractional Fourier processing for quantum light possesses many advantages. Based on such a method, not only quantum correlations can be shaped more rich, but also the initial states can be easily identified. Moreover, the twisted phase information can be recovered and quantum states are easily controlled in performing quantum information experiments. Our findings open up new avenues for the manipulation of correlations between photons in optical quantum information processing.