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Chaos An Interdisciplinary Journal of Nonlinear Science

Quantum Analog Computing

Michail A. Zak

Published Sep 1, 1998• 1 citation
OpenAlex

Abstract

Quantum analog computing is based upon similarity between mathematical formalism of quantum mechanics and phenomena to be computed. It exploits a dynamical convergence of several competing phenomena to an attractor which can represent an externum of a function, an image, a solution to a system of ODE, or a stochastic process.

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Quantum computing by simulations is based upon similarity between mathematical formalism of quantum mechanics and phenomena to be computed. It exploits a dynamical convergence of several competing phenomena to an attractor which can represent an extrenum of a function, an image, a solution to a system of ODE, or a stochastic process. In this chapter, a quantum version of recurrent nets (QRN) as an analog computing device is discussed. This concept is introduced by incorporating classical feedback loops into conventional quantum networks. It is shown that the dynamical evolution of such networks, which interleave quantum evolution with measurement and reset operations, exhibit novel dynamical properties. Moreover, decoherence in quantum recurrent networks is less problematic than in conventional quantum network architectures due to the modest phase coherence times needed for network operation. It is proven that a hypothetical quantum computer can implement an exponentially larger number of the degrees of freedom within the same
This paper introduces the Quantum Stability Attractor Theory (QSAT), a novel framework proposing that quantum systems are not simply random evolving wavefunctions but are instead drawn to specific stable attractor states. This concept draws an analogy to dynamical systems, where attractors represent points or regions to which trajectories converge. QSAT posits a mechanism for the emergence of stable structures within the quantum world, offering a potential explanation for observed correlations and long-range order in quantum systems. The theory relies on analyzing long-term multi-body quantum evolution, aiming to identify patterns indicative of attractor dynamics. The exploration of QSAT has the potential to unlock a deeper understanding of fundamental quantum phenomena and could ultimately lead to advancements in quantum technologies. ---
We briefly review what a quantum computer is, what it promises to do for us and why it is so hard to build one. Among the first applications anticipated to bear fruit is the quantum simulation of quantum systems. While most quantum computation is an extension of classical digital computation, quantum simulation differs fundamentally in how the data are encoded in the quantum computer. To perform a quantum simulation, the Hilbert space of the system to be simulated is mapped directly onto the Hilbert space of the (logical) qubits in the quantum computer. This type of direct correspondence is how data are encoded in a classical analogue computer. There is no binary encoding, and increasing precision becomes exponentially costly: an extra bit of precision doubles the size of the computer. This has important consequences for both the precision and error-correction requirements of quantum simulation, and significant open questions remain about its practicality. It also means that the quantum version of analogue computers, continuous-variable quantum computers, becomes an equally efficient architecture for quantum simulation. Lessons from past use of classical analogue computers can help us to build better quantum simulators in future.
Iterated Function Systems (IFS) generate invariant measures with fractal structures through the random iteration of a family of contraction mappings, servingas a standard framework for describing scale generation phenomena in classicalnonlinear dynamics. This paper generalizes classical IFS to the quantum domain,establishing a rigorous mathematical formulation of Quantum Iterated FunctionSystems (QIFS): classical contraction mappings are replaced by families of completely positive maps acting on the space of density operators, and the quantumattractor state is defined as the fixed point satisfying a self-similarity equation. Onthis basis, we prove that the rescaled position distribution of discrete-time quantum random walks with decoherence converges to a convex combination of normaldistributions, with the convergence rate controlled by the spectral gap of the superoperator. Furthermore, we establish a quantum-classical correspondence principle:in the limit of vanishing decoherence rate, the support of the Wigner function ofthe quantum attractor state converges in distribution to the classical IFS attractor,and the convergence exponents satisfy a scaling relation with the classical contraction rates. This framework unifies fractal geometry, quantum open systems, anddecoherence theory into a coherent generative language, providing a rigorous mathematical foundation for understanding scaling behavior in quantum informationprocessing.
We briefly review what a quantum computer is, what it promises to do for us and why it is so hard to build one. Among the first applications anticipated to bear fruit is the quantum simulation of quantum systems. While most quantum computation is an extension of classical digital computation, quantum simulation differs fundamentally in how the data are encoded in the quantum computer. To perform a quantum simulation, the Hilbert space of the system to be simulated is mapped directly onto the Hilbert space of the (logical) qubits in the quantum computer. This type of direct correspondence is how data are encoded in a classical analogue computer. There is no binary encoding, and increasing precision becomes exponentially costly: an extra bit of precision doubles the size of the computer. This has important consequences for both the precision and errorcorrection requirements of quantum simulation, and significant open questions remain about its practicality. It also means that the quantum version of analogue computers, continuous-variable quantum computers, becomes an equally efficient architecture for quantum simulation. Lessons from past use of classical analogue computers can help us to build better quantum simulators in future.
Training deep neural networks remains a computationally intensive and often challenging process. Traditional optimization methods, such as stochastic gradient descent (SGD) and its variants, frequently struggle to escape local minima and converge to optimal solutions, leading to suboptimal network performance. This paper proposes a novel approach to neural network training that leverages principles inspired by quantum mechanics. Specifically, we investigate the application of quantum-inspired optimization techniques, including simulated annealing and quantum walks, to dynamically adjust the training process. The core idea is to mimic the probabilistic exploration inherent in quantum systems to effectively navigate the complex loss landscape of neural networks. We formulate the training process as a quantum-inspired optimization problem, where the network's weights are treated as quantum bits (Qubits) and the optimization algorithm seeks to find the lowest energy state, analogous to finding the minimum loss. The proposed method demonstrates the potential for accelerating convergence and improving the overall accuracy of neural networks compared to conventional optimization strategies. Our analysis focuses on the key aspects of the approach, including the mathematical formulation of the quantum-inspired optimization problem and the selection of appropriate parameters to govern the exploration and exploitation phases. We present a theoretical framework for understanding the dynamics of this approach and discuss potential avenues for further research, including hybrid quantum-classical algorithms and the integration of quantum computing hardware.