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arXiv:quant-ph/0611060

Some Possibility and Impossibility Results related to Discrete Fourier type transforms in Quantum Information

Arpita Maitra

Published Nov 6, 2006• 0 citations
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Abstract

Very recently the most general ensemble of qubits are identified using the notion of linearity; any of these qubits gets accepted by a Hadamard gate to generate the equal superposition of the qubit and its orthogonal. Towards more generalization, we investigate the possibility and impossibility results related to Discrete Fourier Transform (DFT) type of operations for a more general set up of qutrits.

DOI
10.48550/arxiv.quant-ph/0611060

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Very recently the most general ensemble of qubits are identified using the notion of linearity; any of these qubits gets accepted by a Hadamard gate to generate the equal superposition of the qubit and its orthogonal. Towards more generalization, we investigate the possibility and impossibility results related to Discrete Fourier Transform (DFT) type of operations for a more general set up of qutrits.
A methodology is presented to obtain the basis of qudits which are admissible to quantum Fourier transform (QFT) in the sense that the set of such kets are related by the QFT in the same way as the kets of the computational basis. We first study this method for qubits to characterize the ensemble that works for the Hadamard transformation (QFT for two dimension). In this regard we identify certain incompleteness in the result of Maitra and Parashar (Int. J. Quantum Inform. 4 (2006) 653). Next we characterize the ensemble of qutrits for which QFT is possible. Further, some theoretical results related to higher dimensions are also discussed. Considering the unitary matrix U n related to QFT, the issue boils down to the problem of characterizing matrices that commute with U n .
Consider the problem: Alice wishes to send the same key to n−1n-1n−1 users (Bob, Carol,. . . , Nathan), while preventing eavesdropper Eve from acquiring information without being detected. The problem has no solution in the classical cryptography but in quantum telecommunication there are some codes to solve the problem. In the paper \cite{zengetall}, Guo-Jyun Zeng, Kuan-Hung Chen, Zhe-Hua Chang, Yu-Shan Yang, and Yao-Hsin Chou from one side and Cabello in \cite{cabello} from other side, used Hadamard gates, Pauli gates in providing the quantum communication code for two-partity telecommunication with 3 persons and then generalized it to the case of arbitrary number of participants, indicating the position of measurements of participants. We remark that the Hadamard gate with precising the position of measurement is the same as Fourier transform for two qubits and hence use the general Fourier transform for nnn entangled qubits, in place of Hadamard gates. The result is more natural for arbitrary nnn qudits.
In this work, we introduce a definition of the Discrete Fourier Transform (DFT) on Euclidean lattices in Rn\R^nRn, that generalizes the nnn-th fold DFT of the integer lattice Zn\Z^nZn to arbitrary lattices. This definition is not applicable for every lattice, but can be defined on lattices known as Systematic Normal Form (SysNF) introduced in \cite{ES16}. Systematic Normal Form lattices are sets of integer vectors that satisfy a single homogeneous modular equation, which itself satisfies a certain number-theoretic property. Such lattices form a dense set in the space of nnn-dimensional lattices, and can be used to approximate efficiently any lattice. This implies that for every lattice LLL a DFT can be computed efficiently on a lattice near LLL. Our proof of the statement above uses arguments from quantum computing, and as an application of our definition we show a quantum algorithm for sampling from discrete distributions on lattices, that extends our ability to sample efficiently from the discrete Gaussian distribution \cite{GPV08} to any distribution that is sufficiently "smooth". We conjecture that studying the eigenvectors of the newly-defined lattice DFT may provide new insights into the structure of lattices, especially regarding hard computational problems, like the shortest vector problem.
We advocate the step change in properties of discrete d -level quantum systems, between d = 2 and d ≥ 3. Qubit systems, or multipartite systems containing qubit subsystem, are exceptional in their relative simplicity. One faces a step in complexity in valuating measures of quantum correlations for qutrits and then other higher dimensional qudits. There is a growing number of arguments leading to such conclusion: recently found no-go theorem for generalization of the Peres-Horodecki's PPT criterion [1], change in geometry of state spaces of qubit and higher degree qudits (the so called 'generalized Bloch ball' is not a ball anymore), restricted possibilities for diagonalization of correlation matrices for bipartite systems, more difficult way for handling the set of relevant families of orthogonal projectors.
The quantum Fourier transform for discrete variable (dvQFT) is an efficient algorithm for several applications. It is usually considered for the processing of quantum bits (qubits) and its efficient implementation is obtained with two elementary components: the Hadamard gate and the controlled--phase gate. In this paper, the quantum Fourier transform operating with continuous variables (cvQFT) is considered. Thus, the environment becomes the Hilbert space, where the natural definition of the cvQFT will be related to rotation operators, which in the NNN--mode are completely specified by unitary matrices of order NNN. Then the cvQFT is defined as the rotation operator whose rotation matrix is given by the discrete Fourier transform (DFT) matrix. For the implementation of rotation operators with primitive components (single--mode rotations and beam splitters), we follow the well known Murnaghan procedure, with appropriate modifications. Moreover, algorithms related to the fast Fourier transform (FFT) are applied to reduce drastically the implementation complexity. The final part is concerned with the application of the cvQFT to general Gaussian states. In particular, we show that cvQFT has the simple effect of transforming the displacement vector by a one-dimensional DFT, the squeeze matrix by a two-dimensional DFT, and the rotation matrix by a Fourier-like similarity transform.