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Journal of Applied Computer Science

Simulation of Quantum Resonance in the Quantum Register

Marcin Ostrowski

Published Oct 31, 2019• 0 citations
Publisher

Abstract

In this paper, we examine whether a quantum computer can efficiently simulate resonant interaction between a pair of two-level quantum systems. We present an algorithm for simulating time evolution of such a system, implemented on standard two-input gates. We study the influence of accuracy of gates and decoherence on the quality of results.

DOI
10.34658/jacs.2018.26.1.63-72

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In this paper, we examine whether a quantum computer can efficiently simulate resonant interaction between a pair of two-level quantum systems. We present an algorithm for simulating time evolution of such a system, implemented on standard two-input gates. We study the influence of accuracy of gates and decoherence on the quality of results.
This paper examines whether a quantum computer can efficiently simulate the evolution of the Schrödinger particle in a one-dimensional infinite potential well.The present study involves simulation of the change of the particle's state due to instant enlarging of the well's size.In order to solve the Schrödinger equation in the quantum register, an algorithm based on the Quantum Fourier Transform is applied.The paper presents an algorithm for simulating time evolution of such a system, implemented on standard two-input gates.
Deutsch formulated in 1985 a new computer based on quantummechanics. In quantum computers, the ones and zeros of classical digital computers are replaced by the quantum state of a two-level system which is named qubit. Computation, i.e., a sequence of unitary transformations, simultaneously affects each element of the superposition, generating a massive parallel data processing, albeit within one piece of quantum hardware. Practical quantum computer requires preventing decoherence (uncontrolled interaction of a quantum system with its surrounding environment) in order to maintain the superposition. In 1998, Kane proposed the scheme of implementing a quantum computer on an array of nuclear spins located on donors in silicon. Because nuclear spins are extremely well isolated from environment, operations on nuclear-spin qubits could have low error rates. Quantum logic gate is performed by transitions between the different energy levels of the electron-nuclear spin states, which is induced by a radio-frequency magnetic field B applied at a frequency resonant with the energy-level difference. In this study, we numerically simulated the quantum logic gate of Kane’s quantum computer, by neglecting the complications generated from the band structure in Si. For the calculation of the time evolution of the quantum logic gate, we compute the transition probability between the energy levels of electron-nuclear spin states under the influence of a time-dependent perturbation.
Methods and results for numerical simulations of one and two interacting rf-SQUID systems suitable for adiabatic quantum gates are presented. These are based on high accuracy numerical solutions to the static and time dependent Schr\"odinger equation for the full SQUID Hamiltonian in one and two variables. Among the points examined in the static analysis is the range of validity of the effective two-state or ``spin-12\frac{1}{2}21​'' picture. A range of parameters is determined where the picture holds to good accuracy as the energy levels undergo gate manipulations. Some general points are presented concerning the relations between device parameters and ``good'' quantum mechanical state spaces. The time dependent simulations allow the examination of suitable conditions for adiabatic behavior, and permits the introduction of a random noise to simulate the effects of decoherence. A formula is derived and tested relating the random noise to the decoherence rate. Sensitivity to device and operating parameters for the logical gates NOT and CNOT are examined, with particular attention to values of the tunnel parameter $\ensuremath{\beta}$ slightly above one. It appears that with values of $\ensuremath{\beta}$ close to 1, a quantum CNOT gate is possible even with rather short decoherence times. Many of the methods and results will apply to coupled double-potential well systems in general.
We carry out a systematic analysis of a pair of coupled qubits, each of which is subject to its own dissipative environment, and argue that a combination of the interqubit couplings which provides for the lowest possible decoherence rates corresponds to the incidence of a double spectral degeneracy in the two-qubit system. We support this general argument by the results of an evolutionary genetic algorithm which can also be used for optimizing time-dependent processes (gates) and their sequences that implement various quantum computing protocols.