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arXiv:1303.3625

Quantum Logic Dequantization: Information Loss

Maksym Teslyk, Olena Teslyk

Published Mar 14, 2013• 0 citations
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Abstract

An attempt to ascertain reasons of the quantum logic efficiency was made. To do this, we reduce the complete set of elementary quantum logic operations to the classical one via taking the semi-classical limit (dequantization). We estimate the amount of information loss for any element of the set and obtain general expression for the loss estimation of any quantum algorithm under dequantization. We demonstrate the technique on quantum discrete fast Fourier transform and Grover search algorithms.

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An attempt to ascertain reasons of the quantum logic efficiency was made. To do this, we reduce the complete set of elementary quantum logic operations to the classical one via taking the semi-classical limit (dequantization). We estimate the amount of information loss for any element of the set and obtain general expression for the loss estimation of any quantum algorithm under dequantization. We demonstrate the technique on quantum discrete fast Fourier transform and Grover search algorithms.
A classical limit of Grover's algorithm is discussed by assuming a very rapid decoherence (or dephasing) between consecutive Grover's unitary operations, which leads pure quantum states to completely decohered mixed states. One can identify a specific element among NNN unsorted elements by a probability of the order of unity after k∼N/4k\sim N/4k∼N/4 steps of classical amplification defined by the decohered mixed states, in contrast to Grover's k∼πN/4k\sim \pi \sqrt{N}/4k∼πN​/4 steps in quantum mechanical amplification. This difference is caused by the loss of quantum coherence with or without the loss of entanglement depending on each case.
A new approach to the classical limit of Grover’s algorithm is discussed by assuming a very rapid dephasing of a system between consecutive Grover’s unitary operations, which drives pure quantum states to decohered mixed states. One can identify a specific element among [Formula: see text] unsorted elements by a probability of the order of unity after [Formula: see text] steps of classical amplification, which is realized by a combination of Grover’s unitary operation and rapid dephasing, in contrast to [Formula: see text] steps in quantum mechanical amplification. The initial two-state system with enormously unbalanced existence probabilities, which is realized by a chosen specific state and a superposition of all the rest of the states among [Formula: see text] unsorted states, is crucial in the present analysis of classical amplification. This analysis illustrates Grover’s algorithm in extremely noisy circumstances. A similar increase from [Formula: see text] to [Formula: see text] steps due to the loss of quantum coherence takes place in the analog model of Farhi and Gutmann where the entanglement does not play an obvious role. This supports a view that entanglement is crucial in quantum computation to describe quantum states by a set of qubits, but the actual speedup of the quantum computation is based on quantum coherence.
Quantum computation has shown much promise at providing, at least in some cases, a significant advantage over classical computation. However, the nature of quantum computation is still far from being well understood. In order to develop quantum algorithms effectively, it is important to understand the true nature of the differences between classical and quantum computation. We investigate these differences more closely by looking at de-quantising quantum algorithms into classical counterparts which retain the benefit provided by, and thought to be intrinsic to, the quantum algorithms. We extend a previous de-quantisation of Deutsch’s problem to show that in some situations the quantum algorithm solving the Deutsch-Jozsa problem can be de-quantised into an equivalent classical one. We quantify the entanglement in this problem and show that the inability to easily extend the de-quantisation to the general case is a result of the entanglement destroying the concise classical state description needed to de-quantise such a black-box algorithm. We further show that the quantum Fourier transform, an important process in many quantum algorithms, in its standard form is de-quantisable. This highlights key misconceptions about both the quantum Fourier transform and quantum computation itself. In such cases it is the linearity of quantum mechanics which allows constructing quantum algorithms which offer advantage over classical algorithms. The careful investigation of de-quantisation in relation to these problems allows a deeper insight into the nature of quantum computation.
The discrete Fourier transform (DFT) is the base of modern signal processing. 1-dimensional fast Fourier transform (ID FFT) and 2D FFT have time complexity O ( N log N ) and O ( N 2 log N ) respectively. Since 1965, there has been no more essential breakthrough for the design of fast DFT algorithm. DFT has two properties. One property is that DFT is energy conservation transform. The other property is that many DFT coefficients are close to zero. The basic idea of this paper is that the generalized Grover's iteration can perform the computation of DFT which acts on the entangled states to search the big DFT coefficients until these big coefficients contain nearly all energy. One-dimensional quantum DFT (ID QDFT) and two-dimensional quantum DFT (2D QDFT) are presented in this paper. The quantum algorithm for convolution estimation is also presented in this paper. Compared with FFT, ID and 2D QDFT have time complexity O(√ N ) and O ( N ) respectively. QDFT and quantum convolution demonstrate that quantum computation to process classical signal is possible.
Background: The future computations need parallelism, and a way to achieve the same is through quantum-based specific properties. Quantum computation supports exponential operation to be performed in parallel over a single execution step and hence achieves computational speedup. Objective: One of the quantum-based algorithms allows to search the key element over an index based unstructured database and succeeds to obtain the speedup. In the whole context of article writing, our approach reviews the significance of quantum computing, the basics required for quantum computation and their quantum logic-based circuit operations. Method: The article focuses on Grover’s search algorithm and its variations, and then illustrates the relevant amplitude amplification & phase matching techniques in accordance with the advantages and limitation to the specific perspective. Results: We made the comparative analysis between the amplitude amplification-based static & dynamic Grover’s searching and phase estimation technique. Along with this, we justified the possibilities of the empirical results and observed facts over Grover’s variant approaches. Conclusion: Finally, we conclude the analytical comparison between quantum-based searching techniques along with the applicability of Grover’s algorithm in practical computational science.
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.