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arXiv:quant-ph/0002082

Quantum searching with continuous variables

Arun Kumar Pati, Samuel L. Braunstein, Seth Lloyd

Published Feb 26, 2000• 19 citations
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Abstract

A fast quantum search algorithm for continuous variables is presented. The result is the quantum continuous variable analog of Grover's algorithm originally proposed for qubits. A continuous variable analog of the Hadamard (i.e., Fourier transform) operation is used in conjunction with inversion about the average of quantum states to allow the approximate identification of an unknown quantum state in a way that gives a square-root speed-up over search algorithms using classical continuous variables. Also, we show that this quantum search algorithm is robust for a generalised Fourier transformation on continuous variables.

DOI
10.48550/arxiv.quant-ph/0002082

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A fast quantum search algorithm for continuous variables is presented. The result is the quantum continuous variable analog of Grover's algorithm originally proposed for qubits. A continuous variable analog of the Hadamard (i.e., Fourier transform) operation is used in conjunction with inversion about the average of quantum states to allow the approximate identification of an unknown quantum state in a way that gives a square-root speed-up over search algorithms using classical continuous variables. Also, we show that this quantum search algorithm is robust for a generalised Fourier transformation on continuous variables.
We recast Grover's generalised search algorithm in a geometric language even when the states are not approximately orthogonal. We provide a possible search algorithm based on an arbitrary unitary transformation which can speed up the steps still further. We discuss the lower and upper bounds on the transition matrix elements when the unitary operator changes with time, thereby implying that quantum search process can not be too fast or too slow. This is a remarkable feature of quantum computation unlike classical one. Quantum mechanical uncertainty relation puts bounds on search process. Also we mention the problems of perturbation and other issues in time-dependent search operation.
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.
We give a dimension independent formulation of the quantum search algorithm introduced in [L. K. Grover, Phys. Rev. Lett. {\bf 79}, 325 (1997)]. This algorithm provides a quadratic gain when compared to its classical counterpart by manipulating quantum two--level systems, qubits. We show that this gain, already known to be optimal, is preserved, irrespectively of the dimension of the system used to encode quantum information. This is shown by adapting the protocol to Hilbert spaces of any dimension using the same sequence of operations/logical gates as its original qubit formulation. Our results are detailed and illustrated for a system described by continuous variables, where qubits can be encoded in infinitely many distinct states using the modular variable formalism.
Quantum computing is a system of computation that exploits the quantum-mechanical nature of reality at nano-scales in order to perform certain operations in a way that can scale better than is possible with classical computing. While the hardware sideof quantum computing is still in its infancy, many quantum algorithms - sets of instructions to run on a quantum computer in order to complete particular tasks - have been designed and analyzed. One of the most famous of these is Grover’s algorithm, an algorithm for searching a database. Two types of generalizations of Grover’s algorithm have been studied: spatial search algorithms (algorithms for searching discrete spaces) and “analog” search algorithms (algorithms performed by the evolution of physical systems). We develop and analyze algorithms that are generalizations of both those generalizations – an analog search algorithm for searching continuous, rather than discrete, spaces.
Grover's quantum search algorithm is an important landmark discovery in the area of quantum computing that drives a quantum computer from a known initial state(source state) to an unknown final state(target state) by using selective phase inversions of these states. The algorithm is simply a successive iteration of the Grover's search operator, which is a product of selective inversion operators of source and target states. In its original form, Grover's operator performs the selective inversions only on a unique source state and a unique target state. In the generalized version, we make our search operator to perform selective inversions on more than one source and target states. First, we find that the case of selective inversions of two source and two target states can be easily analysed somewhat similar to the case of Grover's algorithm. But unlike Grover's operator, the selective inversions of three source and three target states is not so easy to study analytically and hence we perform numerical calculations on some specific cases. We find that these generalized search operators also give us a successful quantum search as expected.
We propose a simpler and more efficient scheme for the implementation of the multivalued Grover's quantum search. The multivalued search generalizes the original Grover's search by replacing qubits with qudits---quantum systems of ddd discrete states. The qudit database is exponentially larger than the qubit database and thus it requires fewer particles to control. The Hadamard gate, which is the key elementary gate in the qubit implementation of Grover's search, is replaced by a ddd-dimensional (complex-valued) unitary matrix F\mathbf{F}F, the only condition for which is to have a column of equal moduli elements irrespective of their phases; it can be realized through any physical interaction, which achieves an equal-weight superposition state. An example of such a transformation is the ddd-dimensional discrete Fourier transform, used in earlier proposals; however, its construction is much more costly than that of the far simpler matrix F\mathbf{F}F. We present examples of how such a transform F\mathbf{F}F can be constructed in realistic qudit systems in a single interaction step.