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arXiv:2406.13211 · Physical Review A

Amplitude amplification and estimation using the atom-optics kicked rotor

V. Keshav, M. S. Santhanam

Published Mar 3, 2025• 2 citations
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Abstract

The atom-optics realization of the quantum kicked rotor represents one of the most well-studied experimental platforms for the physics of Anderson localization. In this work, we show that the same platform can be used to implement a quantum algorithm to perform quantum amplitude amplification (AA), a generalization of Grover's search algorithm. This is done using the usual kick sequence in combination with the well established technique of Raman velocity selection. We argue that this platform is well suited to implement AA, as its characteristic property of localization can be exploited to enhance the performance of the amplitude amplification algorithm by reducing its average runtime. We also introduce a kick potential which asymptotically leads to the optimal quantum speedup, constituting a complementary approach. Finally, we demonstrate that the scheme is fairly robust to noise in the kicks and discuss the experimental feasibility of each component of our proposal

DOI
10.1103/physreva.111.032601

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The atom-optics realization of the quantum kicked rotor represents one of the most well-studied experimental platforms for the physics of Anderson localization. In this work, we show that the same platform can be used to implement a quantum algorithm to perform quantum amplitude amplification (AA), a generalization of Grover's search algorithm. This is done using the usual kick sequence in combination with the well established technique of Raman velocity selection. We argue that this platform is well suited to implement AA, as its characteristic property of localization can be exploited to enhance the performance of the amplitude amplification algorithm by reducing its average runtime. We also introduce a kick potential which asymptotically leads to the optimal quantum speedup, constituting a complementary approach. Finally, we demonstrate that the scheme is fairly robust to noise in the kicks and discuss the experimental feasibility of each component of our proposal
We study analytically the effect of amplitude noise on the quantum resonances of an atom optics realization of the δ-kicked rotor. Noise is shown to add a time growth to the ‘deterministic’ energy and to induce a time increasing spreading in the momentum distribution; exact results are given for both effects. The ballistic peaks characteristic of the noiseless distribution for particular initial conditions broaden and eventually vanish, whereas the associated quadratic growth of energy persists; at large times, the survival probability decays as t −1 . Moreover, the nonexponential ‘localization’ linked to different initial conditions is gradually destroyed. Features specific to Gaussian noise, white and coloured, are analysed. The feasibility of experimental tests of these effects is discussed.
The effect of pulse train noise on the quantum resonance peaks of the atom optics kicked rotor is investigated experimentally. Quantum resonance peaks in the late time mean energy of the atoms are found to be surprisingly robust against all levels of noise applied to the kicking amplitude, while even small levels of noise on the kicking period lead to their destruction. The robustness to amplitude noise of the resonance peak and of the fall-off in mean energy to either side of this peak are explained in terms of the occurrence of stable, epsilon classical dynamics [Nonlinearity 16, 1381 (2003)]] around each quantum resonance.
The landmark Grover algorithm for amplitude amplification serves as an essential subroutine in various types of quantum algorithms, with guaranteed quantum speedup in query complexity. However, there has been no proposal to realize the original motivating application of the algorithm, i.e., the database search or more broadly the pattern matching in a practical setting, mainly due to the technical difficulty in efficiently implementing the data loading and amplitude amplification processes. In this paper, we propose a quantum algorithm that approximately executes the entire Grover database search or pattern matching algorithm. The key idea is to use the recently proposed approximate amplitude encoding method on a shallow quantum circuit, together with the easily implementable inversion-test operation for realizing the projected quantum state having similarity to the query data, followed by the amplitude amplification operation that is independent to the target data index. We provide a thorough demonstration of the algorithm in the problem of image pattern matching.
Quantum systems lose coherence upon interaction with the environment and tend towards classical states. Quantum coherence is known to exponentially decay in time so that macroscopic quantum superpositions are generally unsustainable. In this work, slower than exponential decay of coherences is experimentally realized in an atom-optics kicked rotor system subjected to nonstationary Lévy noise in the applied kick sequence. The slower coherence decay manifests in the form of quantum subdiffusion that can be controlled through the Lévy exponent. The experimental results are in good agreement with the analytical estimates and numerical simulations for the mean energy growth and momentum profiles of an atom-optics kicked rotor.
We propose an atom optics experiment to measure the stability of the quantum kicked rotor under perturbations of the Hamiltonian. We avail ourselves of the theory of Loschmidt echoes, i.e., we consider the overlap of a quantum state evolved in a perturbed and an unperturbed potential. Atom interferometry allows us to determine the overlap integral in amplitude and phase. A numerical analysis of the kicked rotor in various regimes shows that the quantum signatures of specific classical properties can be detected experimentally.
In Chs. 2 and 3 we described several quantum algorithms that are exponentially faster than classical algorithms for the same problem, under some measure of complexity. Unfortunately, the algorithms of Ch. 2 apply to rather contrived problems that do not have known direct applications. We now describe a quantum algorithm that gives only a polynomial—more specifically, quadratic—speedup with respect to classical algorithms, but it applies to a large class of problems and finds many useful applications, both directly and as an algorithmic design tool. The algorithm is known as Grover’s search [89], and its generalization is known as amplitude amplification [43]. Amplitude amplification is widely used as part of many quantum algorithms. It also serves as the basis for amplitude estimation, a way of estimating probabilities quadratically faster than with classical Monte Carlo, and for quantum minimum finding. All these topics are discussed in this chapter.