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arXiv:quant-ph/0611265 · Physical Review A

Quantum optical random walk: Quantization rules and quantum simulation of asymptotics

Demosthenes Ellinas, Ioannis Smyrnakis

Published Aug 28, 2007• 10 citations
arXivHTMLPDF

Abstract

Rules for quantizing the walker-coin parts of a classical random walk are provided by treating them as interacting quantum systems. A quantum optical walk (QOW) is introduced by means of a rule that treats the quantum or classical noise affecting the coin's state as a source of quantization. The long-term asymptotic statistics of the QO walker's position, which shows enhanced diffusion rates as compared to the classical case, is exactly solved. A quantum optical implementation of the walk provides a physical framework for quantum simulation of its asymptotic statistics. The simulation utilizes interacting two-level atoms and/or randomly pulsating laser fields with fluctuating parameters.

DOI
10.1103/physreva.76.022333

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Rules for quantizing the walker-coin parts of a classical random walk are provided by treating them as interacting quantum systems. A quantum optical walk (QOW) is introduced by means of a rule that treats the quantum or classical noise affecting the coin's state as a source of quantization. The long-term asymptotic statistics of the QO walker's position, which shows enhanced diffusion rates as compared to the classical case, is exactly solved. A quantum optical implementation of the walk provides a physical framework for quantum simulation of its asymptotic statistics. The simulation utilizes interacting two-level atoms and/or randomly pulsating laser fields with fluctuating parameters.
We investigate a novel quantum random walk (QRW) model, possibly useful in quantum algorithm implementation, that achieves a quadratically faster diffusion rate compared to its classical counterpart. We evaluate its asymptotic behavior expressed in the form of a limit probability distribution of a double horn shape. Questions of robustness and control of that limit distribution are addressed by introducing a quantum optical cavity in which a resonant Jaynes-Cummings type of interaction between the quantum walk coin system realized in the form of a two-level atom and a laser field is taking place. Driving the optical cavity by means of the coin-field interaction time and the initial quantum coin state, we determine two types of modification of the asymptotic behavior of the QRW. In the first one the limit distribution is robustly reproduced up to a scaling, while in the second one the quantum features of the walk, exemplified by enhanced diffusion rate, are washed out and Gaussian asymptotics prevail. Verification of these findings in an experimental setup that involves two quantum optical cavities that implement the driven QRW and its quantum to classical transition is discussed.
We investigate a novel quantum random walk (QRW) model, of possible use in quantum algorithm implementation, that achieves a quadratically faster diffusion rate compared to its classical counterpart. We evaluate its asymptotic behaviour expressed in the form of a limit probability distribution of a double-horn shape. Questions of robustness and control of that limit distribution are addressed by introducing a quantum optical cavity in which a resonant Jaynes–Cummings type of interaction between the quantum walk coin system realized in the form of a two-level atom and a laser field is taking place. Driving the optical cavity by means of the coin–field interaction time and the initial quantum coin state, we determine two types of modification of the asymptotic behaviour of the QRW. In the first one the limit distribution is robustly reproduced up to a scaling, while in the second one the quantum features of the walk, exemplified by an enhanced diffusion rate, are washed out and Gaussian asymptotics prevail. Verification of these findings in an experimental set-up that involves two quantum optical cavities that implement the driven QRW and its quantum to classical transition is discussed.
Open quantum walks (OQWs) are a new type of quantum walks which are entirely driven by the dissipative interaction with external environments and are formulated as completely positive trace-preserving maps on graphs. A generalized quantum optical scheme for implementing OQWs that includes non-zero temperature of the environment is suggested. In the proposed quantum optical scheme, a two-level atom plays the role of the "walker", and the Fock states of the cavity mode correspond to the lattice sites for the "walker". Using the small unitary rotations approach the effective dynamics of the system is shown to be an OQW. For the chosen set of parameters, an increase in the temperature of the environment causes the system to reach the asymptotic distribution much faster compared to the scheme proposed earlier where the temperature of the environment is zero. For this case the asymptotic distribution is given by a steady Gaussian distribution.
Quantum walks constitute a rich area of quantum information science, where multipartite entanglement plays a central role in the dynamics and scalability of quantum advantage over classical simulators. In this work, we study the multipartite entanglement of quantum walks in optical settings. We present methods for computing a geometric measure of entanglement for arbitrary partitions of a single-walker quantum walk and for analyzing the entanglement in multi-walker scenarios. These techniques are used for numerical studies on the entanglement dynamics of quantum walks in large systems and under various initial conditions. For a given bipartition, based on the coin degrees of freedom, we derive exact expressions describing the complete entanglement dynamics for arbitrary localized initial conditions. We use these expressions for analytic statements about the asymptotic behavior of the system. Furthermore, we demonstrate the emergence of entanglement typicality in statistical ensembles of random optical networks.
Quantum walk (QW), which is considered as the quantum counterpart of the classical random walk (CRW), is actually the quantum extension of CRW from the single-coin interpretation. The sequential unitary evolution engenders correlation between different steps in QW and leads to a non-binomial position distribution. In this paper, we propose an alternative quantum extension of CRW from the ensemble interpretation, named quantum random walk (QRW), where the walker has many unrelated coins, modeled as two-level systems, initially prepared in the same state. We calculate the walker’s position distribution in QRW for different initial coin states with the coin operator chosen as Hadamard matrix. In one-dimensional case, the walker’s position is the asymmetric binomial distribution. We further demonstrate that in QRW, coherence leads the walker to perform directional movement. For an initially decoherenced coin state, the walker’s position distribution is exactly the same as that of CRW. Moreover, we study QRW in 2D lattice, where the coherence plays a more diversified role in the walker’s position distribution.
We present the first robust implementation of a coined quantum walk over five steps using only passive optical elements. By employing a fiber network loop we keep the amount of required resources constant as the walker's position Hilbert space is increased. We observed a non-Gaussian distribution of the walker's final position, thus characterizing a faster spread of the photon wave packet in comparison to the classical random walk. The walk is realized for many different coin settings and initial states, opening the way for the implementation of a quantum-walk-based search algorithm.