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arXiv:0906.1033 · FCS

Quantum Pseudo-fractional Fourier Transform and its application to quantum phase estimation

Vamsi Parasa, Kumar Eswaran

Published Jun 5, 2009• 2 citations
arXivHTMLPDF

Abstract

- In this paper we present a method to compute the coefficients of the fractional Fourier transform (FrFT) on a quantum computer using quantum gates of polynomial complexity of the order O(n^3). The FrFt, a generalization of the DFT, has wide applications in signal processing and is particularly useful to implement the Pseudopolar and Radon transforms. Even though the FrFT is a non-unitary operation, to develop its quantum counterpart, we develop a unitary operator called the quantum Pseudo-fraction Fourier Transform (QPFrFT) in a higher-dimensional Hilbert space, in order to computer the coefficients of the FrFT. In this process we develop a unitary operator denoted U by which is an essential step to implement the QPFrFT. We then show the application of the operator U in the problem of quantum phase estimation.

DOI
10.48550/arxiv.0906.1033

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  • In this paper we present a method to compute the coefficients of the fractional Fourier transform (FrFT) on a quantum computer using quantum gates of polynomial complexity of the order O(n^3). The FrFt, a generalization of the DFT, has wide applications in signal processing and is particularly useful to implement the Pseudopolar and Radon transforms. Even though the FrFT is a non-unitary operation, to develop its quantum counterpart, we develop a unitary operator called the quantum Pseudo-fraction Fourier Transform (QPFrFT) in a higher-dimensional Hilbert space, in order to computer the coefficients of the FrFT. In this process we develop a unitary operator denoted U by which is an essential step to implement the QPFrFT. We then show the application of the operator U in the problem of quantum phase estimation.
This paper presents the multivalued logic version of the quantum pseudo-fractional Fourier transform (QFrFT), which is a more general transform of which the widely used quantum Fourier transform (QFT) is a special case. We also show how to efficiently implement the QPFrFT using O(n3) two qudit rotation gates. It is possible to implement an approximate QPFrFT by eliminating the rotation gates which implement exponentially decreasing rotation angles. The multivalued logic QPFrFT has improved approximation properties as the radix d of the logic used increases. The binary QPFrFT has the same circuit complexity and yet has the worst approximation properties when compared to multivalued logic implementations.
Quantum Fourier transform (QFT) transformation plays a very important role in the design of many quantum algorithms. Fractional Fourier transform (FRFT), as an extension of the Fourier transform, is particularly important due to the design of its quantum algorithm. In this paper, a new reformulation of the weighted fractional Fourier transform (WFRFT) is proposed in order to realize quantum FRFT; however, we found that this reformulation can be applied to other transformations, and therefore, this paper presents the weighted fractional Hartley transform (WFRHT). For the universality of application, we further propose a general weighted fractional-order transform (WFRT). When designing the quantum circuits, we realized the quantum WFRFT via QFT and quantum phase estimation (QPE). Moreover, after extending our design to the WFRHT, we were able to formulate the quantum WFRHT. Finally, in accordance with the research results, we designed the quantum circuit of the general WFRT, and subsequently proposed the quantum WFRT. The research in this paper has great value as a reference for the design and application of quantum algorithms.
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We present a new number theoretic definition of discrete fractional Fourier transform (DFrFT) . In this approach the DFrFT is defined as the N×NN \times NN×N dimensional unitary representation of the generator of the arithmetic rotational group SO2[ZN]SO_{2}[\mathbb{Z}_N]SO2​[ZN​], which is the finite set of  mod N\bmod NmodN integer, 2×22\times 22×2 matrices acting on the points of the discrete toroidal phase space lattice ZN×ZN\mathbb{Z}_N \times \mathbb{Z}_NZN​×ZN​, preserving the euclidean distance  mod N\bmod NmodN. We construct explicitly, using techniques of the Finite Quantum Mechanics (FQM), the pnp^npn dimensional unitary matrix representation of the group SO2[Zpn]SO_{2}[\mathbb{Z}_{p^n}]SO2​[Zpn​] and especially we work out in detail the one which corresponds to the generator. This is our definition of the arithmetic fractional Fourier transform (AFrFT). Following this definition, we proceed to the construction of efficient quantum circuits for the AFrFT, on sets of nnn ppp-dimensional qudits with ppp a prime integer, by introducing novel quantum subcircuits for diagonal operators with quadratic phases as well as new quantum subcircuits for multipliers by a constant. The quantum subcircuits that we introduce provide a set capable to construct quantum circuits for any element of a more general group, the group of Linear Canonical Transformations (LCT), SL2[ZN]SL_{2}[\mathbb{Z}_N]SL2​[ZN​] of the toroidal phase space lattice. As a byproduct, extensions of the diagonal and multiplier quantum circuits for both the qudit and qubit case are given, which are useful alone in various applications. Also, we analyze the depth, width and gate complexity of the efficient AFrFT quantum circuit and we estimate its gate complexity which is of the order O(n2)O(n^2)O(n2), its depth which is of the order O(n)O(n)O(n) with depth nnn, while at the same time it has a structure permitting local interactions between the qudits.