A Novel Finite Fractional Fourier Transform and its Quantum Circuit Implementation on Qudits

TL;DR

This work defines a finite arithmetic fractional Fourier transform (AFrFT) as the unitary generator of the rotation subgroup $SO_{2}[\mathbb{Z}_N]$ acting on the toroidal phase space, and constructs its $p^{n}$-dimensional unitary realization using Finite Quantum Mechanics. It then develops a full quantum circuit framework for the QAFrFT on multilevel qudits, decomposing the transform into a quadratic diagonal phase operator, a constant-multiplier block implemented via a modified QFT ($mQFT_{\lambda}^{(p)}$), and a QFT block, all designed for local interactions in a 1D nearest-neighbor layout. The paper provides detailed circuit topologies, parallelization strategies, and depth/gate complexity estimates (depth ~ $O(n)$, gate count ~ $O(n^{2})$) and discusses variations to accommodate qubits and constants not coprime to $p^{n}$. By exploiting the Weil metaplectic representation and the LCT structure, the QAFrFT achieves a structured, scalable implementation with potential cryptographic applications and an efficient classical analogue $O(N\log_p N)$ for AFrFT. Overall, the work advances finite-group-based fractional transforms and their practical quantum realizations on qudits.

Abstract

We present a new number theoretic definition of discrete fractional Fourier transform (DFrFT) . In this approach the DFrFT is defined as the $N \times N$ dimensional unitary representation of the generator of the arithmetic rotational group $SO_{2}[\mathbb{Z}_N]$, which is the finite set of $\bmod N$ integer, $2\times 2$ matrices acting on the points of the discrete toroidal phase space lattice $\mathbb{Z}_N \times \mathbb{Z}_N$, preserving the euclidean distance $\bmod N$. We construct explicitly, using techniques of the Finite Quantum Mechanics (FQM), the $p^n$ dimensional unitary matrix representation of the group $SO_{2}[\mathbb{Z}_{p^n}]$ 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 $n$ $p$-dimensional qudits with $p$ 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), $SL_{2}[\mathbb{Z}_N]$ 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 $O(n^2)$, its depth which is of the order $O(n)$ with depth $n$, while at the same time it has a structure permitting local interactions between the qudits.

Figures (14)

• Figure 1: QFT circuit with local interactions on 6 qudits. Single qudit Hadamard gates $H^{(p)}$ are denoted with a bullet, while two qudit rotation gates $R_{k}^{(p)}$ are denoted with two bullets connected with a vertical line. Swap gates are denoted with crossing lines. Note that the qudits are rearranged in the correct reverse order at the end of the computation.
• Figure 2: Symbols for QFT and IQFT circuits with local interactions Thick bullets at the input and ouput sides denote the most significant qudit positions.
• Figure 3: Modified Quantum Fourier Transform circuit on $n$ qudits of $p$-levels with parameter $\lambda$ where $\gcd(\lambda,p)=1$ (note that the order of the qubits must be reversed at the end of the computation). In the above notation it is assumed that the computational basis input state $\ket{j}$ is expressed with $\ket{j_{n-1}}\ket{j_{n-2}} \cdots \ket{j_{0}}$ in $p$-ary representation. The output states are $\ket{q_{r}}= \frac{1}{\sqrt{p}}\sum_{m=0}^{p-1} e^{i2\pi(0.j_{r}j_{r-1}\ldots j_{0} )m}\ket{m}$ and they are unentangled. Observe that when $\lambda=1$ this circuit coincides with the usual $n$ qudits QFT.
• Figure 4: The modified QFT circuit with local interactions on 6 qudits with parameter $\lambda$. Single qudit Hadamard gates $H^{(p)}$ and permutation $P_{\mu}^{(p)}$ gates are denoted with a bullet, while two-qudit rotation gates $R_{k}^{\lambda(p)}$ are denoted with two bullets connected with a vertical line. Swap gates are denoted with crossing lines. Note that the qudits are rearranged in the correct reverse order at the end of the computation.
• Figure 5: Symbols for $mQFT_{\lambda}^{(p)}$ and its inverse $mIQFT_{\lambda}^{(p)}$ with local interactions Thick bullets at the input and ouput sides denote the most significant qudit positions.