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arXiv:1309.1746 · Physical Review A

Quantum dynamics simulation with classical oscillators

J S Briggs, Alexander Eisfeld

Published Dec 16, 2013• 32 citations
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Abstract

In a previous paper [J. S. Briggs and A. Eisfeld, Phys. Rev. A 85, 052111 (2012)] we showed that the time development of the complex amplitudes of $N$ coupled quantum states can be mapped by the time development of positions and velocities of $N$ coupled classical oscillators. Here we examine to what extent this mapping can be realized to simulate the ``quantum,'' properties of entanglement and qubit manipulation. By working through specific examples, e.g., of quantum gate operation, we seek to illuminate quantum and classical differences which hitherto have been treated more mathematically. In addition, we show that important quantum coupled phenomena, such as the Landau-Zener transition and the occurrence of Fano resonances can be simulated by classical oscillators.

DOI
10.1103/physreva.88.062104

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In a previous paper [J. S. Briggs and A. Eisfeld, Phys. Rev. A 85, 052111 (2012)] we showed that the time development of the complex amplitudes of NNN coupled quantum states can be mapped by the time development of positions and velocities of NNN coupled classical oscillators. Here we examine to what extent this mapping can be realized to simulate the ``quantum,'' properties of entanglement and qubit manipulation. By working through specific examples, e.g., of quantum gate operation, we seek to illuminate quantum and classical differences which hitherto have been treated more mathematically. In addition, we show that important quantum coupled phenomena, such as the Landau-Zener transition and the occurrence of Fano resonances can be simulated by classical oscillators.
Quantum simulation involves using a quantum system to study another quantum system. The proposal to simulate quantum physics using quantum mechanical devices, as outlined by Richard Feynman in 1982, stems from classical computers' challenges in simulating quantum systems. The pursuit of simulating complex quantum systems has led to the intersection of quantum mechanics and quantum computing. This offers new opportunities to understand and manipulate the quantum world. Central to describing quantum dynamics is the Hamiltonian operator H^\hat{H}H^, which governs the time evolution of physical systems and serves as the mathematical foundation for quantum simulations. This thesis explores the methodologies and advancements in simulating Hamiltonian operators on quantum computers, focusing on decomposing these operators into quantum gates and the efficiency gains over classical computational approaches. Initially, the thesis explains the fundamental theory and basics of quantum computing. Further it dives into the foundations of the Hamiltonian operator within quantum mechanics, highlighting its role in describing the dynamics of quantum systems. The complexity and high-dimensional nature of Hamiltonian matrices necessitate using quantum computers to simulate quantum phenomena accurately and efficiently. The process of simulating the Hamiltonian operator on a quantum computer involves a detailed examination of the strategies for decomposing the operator into a sequence of quantum gates. This decomposition is crucial for translating the abstract mathematical description of quantum systems into the physical operations of a quantum computer. By utilizing the principles of quantum mechanics, such as superposition and entanglement, quantum gates manipulate qubits to mimic the evolution of quantum states as the Hamiltonian operator dictates. Latter portions of the thesis are dedicated to comparing the efficiency of quantum simulations against traditional classical simulations and inspecting efficient quantum simulations. Quantum computers offer a polynomial or an exponential speedup for certain classes of problems, such as those involving highly entangled quantum states or requiring the simulation of complex quantum dynamics. This advantage is due to the quantum computer's ability to naturally represent and process quantum information - a task for which classical computers require exponential resources as the system size increases. The efficiency of simulating the Hamiltonian operator on a quantum computer is further enhanced through optimization techniques and algorithmic improvements. These include error mitigation strategies and the development of quantum algorithms tailored to specific types of Hamiltonian operators. The thesis presents a comprehensive analysis of these methods, demonstrating how they make quantum simulations more practical and accessible for scientific research and technological development. In conclusion this thesis is a literature review that examines the transformative potential of quantum simulations. By efficiently simulating the Hamiltonian operator on quantum computers researchers can explore quantum phenomena with precision, paving the way for breakthroughs in quantum physics, materials science, and beyond. The advancements in quantum computing technology and algorithmic approaches discussed here highlight the current capabilities and future explorations in the quantum realm.
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We present a quantum algorithm for simulating the classical dynamics of 2 n coupled oscillators (e.g., 2 n masses coupled by springs).Our approach leverages a mapping between the Schrödinger equation and Newton's equation for harmonic potentials such that the amplitudes of the evolved quantum state encode the momenta and displacements of the classical oscillators.When individual masses and spring constants can be efficiently queried, and when the initial state can be efficiently prepared, the complexity of our quantum algorithm is polynomial in n, almost linear in the evolution time, and sublinear in the sparsity.As an example application, we apply our quantum algorithm to efficiently estimate the kinetic energy of an oscillator at any time.We show that any classical algorithm solving this same problem is inefficient and must make 2 ΩðnÞ queries to the oracle, and when the oracles are instantiated by efficient quantum circuits, the problem is bounded-error quantum polynomial time complete.Thus, our approach solves a potentially practical application with an exponential speedup over classical computers.Finally, we show that under similar conditions our approach can efficiently simulate more general classical harmonic systems with 2 n modes.
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Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.