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arXiv:quant-ph/0005007 · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Contact between laboratory instruments and equations of quantum mechanics

John M. Myers, F. Hadi Madjid

Published Jul 13, 2000• 1 citation
arXivHTMLPDF

Abstract

Ambiguity int he contact between laboratory instruments and equations of quantum mechanics is formulated in terms of responses of the instruments to commands transmitted to them by a Classical digital Process-control Computer (CPC); in this way instruments are distinguished from quantum- mechanical models (sets of equations) that specify what is desired of the instruments. Results include: (1) a formulation of quantum mechanics adapted to computer- controlled instruments; (2) a lower bound on the precision of unitary transforms required for quantum searching and a lower bound on sample size needed to show that instruments implement a desired model at that precision; (3) a lower bound on precision of timing required of a CPC in directing instruments; (4) a demonstration that guesswork is necessary in ratcheting up the precision of commands.

DOI
10.1117/12.391960

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Ambiguity int he contact between laboratory instruments and equations of quantum mechanics is formulated in terms of responses of the instruments to commands transmitted to them by a Classical digital Process-control Computer (CPC); in this way instruments are distinguished from quantum- mechanical models (sets of equations) that specify what is desired of the instruments. Results include: (1) a formulation of quantum mechanics adapted to computer- controlled instruments; (2) a lower bound on the precision of unitary transforms required for quantum searching and a lower bound on sample size needed to show that instruments implement a desired model at that precision; (3) a lower bound on precision of timing required of a CPC in directing instruments; (4) a demonstration that guesswork is necessary in ratcheting up the precision of commands.
The design and operation of a quantum-mechanical device as a laboratory instrument puts models written in equations of quantum mechanics in contact with instruments. This contact is recordable in files of a Classical Digital Process-control Computer (CPC) used both to calculate with the equations and to manage the instruments. By noticing that equations and instruments make contact in a CPC, we rewrite equations of quantum mechanics to explicitly include functions of CPC-commands to the instruments. This sets up a proof that a scientist's choice in linking mathematical models to instruments is unresolvable without guesswork to narrow the set of models from which one is to be chosen. As for implications of the proof, scientists inherit choices from the past and frame choices for the future, choices open to guesswork and visible in CPC files. To picture these choices, we adapt colored Petri nets, and the availability of these net fragments makes choice and guesswork part and parcel of physics. Net fragments as a means of expressing guess-demanding choices are applied to portray guesswork needed in testing and calibrating a quantum computer. The sample size required to test a quantum gate in a quantum computer is shown to grow as the inverse square of the error allowed in implementing the gate.
This paper presents a general self-guided algorithm for direct laboratory learning of controls to manipulate quantum-mechanical systems. The primary focus is on an algorithm based on the learning of a linear laboratory input–output map from a sequence of controls, and their observed impact on the quantum-mechanical system. This map is then employed in an iterative fashion, to sequentially home in on the desired objective. The objective may be a target state at a final time, or a continuously weighted observational trajectory. The self-guided aspects of the algorithm are based on implementing a cost functional that only contains laboratory-accessible information. Through choice of the weights in this functional, the algorithm can automatically stay within the bounds of each local linear map and indicate when a new map is necessary for additional iterative improvement. Finally, these concepts can be generalized to include the possibility of employing nonlinear maps, as well as just the laboratory control instrument settings, rather than observation of the control itself. An illustrative simulation of the concepts is presented for the control of a four-level quantum system.
We relate quantum mechanics to the concept of an experimental trial, borrowed from statistics, in order to make clear that devices designed using quantum-mechanical equations require, in effect, management by a classical digital process-control computer (CPC). Energy dissipation essential for tolerance to component imperfections in classical digital computers is shown not to mix with quantum superposition expressed by the quantum dynamics of a Schrodinger equation. This precludes so-called quantum computers from producing super-linear speed-up at a scale large enough to be useful, and at the same clarifies a conceptual separation between digital computing and devices exhibiting quantum phase sensitivity. Quantum effects, such as random-number generation and photon correlations, are know to be useful and show further great potential for devices for computational and other purpose. Several such quantum-based devices are discussed that require or can be made more effective by bridging between these devices and CPCs that manage them. While the interdisciplinary promise of 'quantum computing' has faded, the area of quantum-to- classical bridging, once liberated from the misleading goal of super-linear speed-up, invites vigorous study and invention.
The scope of this work is to provide a self-contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well as a brief survey on the main approaches to control synthesis. While part of the existing theory, especially in the open-loop setting, stems directly from classical control theory (most notably geometric control and optimal control), a number of tools specifically tailored for quantum systems have been developed since the 1980s, in order to take into account their distinctive features: the probabilistic nature of atomic-scale physical systems, the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback-design problems. The relevant dynamical models for both closed and open quantum systems are presented, along with the main results on their controllability and stability. A brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies.