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arXiv:quant-ph/0003144

A proof that measured data and equations of quantum mechanics can be linked only by guesswork

John M. Myers, F. Hadi Madjid

Published Mar 30, 2000• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0003144

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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.
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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.
Experimenting with metastability in recording devices leads us to wonder about an interface between equations of motion and the stillness of experimental records. Here we delineate an interface between wave functions as language to describe motion and Turing tapes as language to describe experimental records. After extending quantum formalism to make this interface explicit, we report on constraints and freedoms in choosing quantum-mechanical equations to model experiments with devices. We prove that choosing equations of wave functions and operators to achieve a fit between calculated probabilities and experimental records requires reaching beyond both logic and the experimental records. Although informed by experience, a "reach beyond" can fairly be called a guess. Recognizing that particles as features of wave functions depend on guesswork, we introduce their use not as objects of physical investigation but as elements of thought in quantum-mechanical models of devices. We make informed guesses to offer a quantum-mechanical model of a 1-bit recording device in a metastable condition. Probabilities calculated from the model fit an experimental record of an oscillation in a time-varying probability, showing a temperature-independent role for Planck's constant in what heretofore was viewed as a "classical" electronic device.
Proofs of the security of quantum key distribution are propositions about models written in the mathematical language of quantum mechanics, and the issue is the linking of such models to actual devices in an experiment on security. To explore this issue, we adapt Wittgenstein's method of language games to view quantum language in its application to experimental activity involving transmitting and receiving devices. We sketch concepts with which to think about models in relation to experiments, without assuming the experiments accord with any model included is a concept of one quantum-mechanical model enveloping another. For any model that agrees with given experimental results and implies the security of a key, there is an enveloping model that agrees with the same results while denying that security. As a result there is a gap between equations and the behavior recorded from devices in an experiment, a gap bridged only by resort to something beyond the reach of logic and measured data, well named by the word guesswork. While this recognition of guesswork encourages eavesdropping, a related recognition of guesswork in the design of feedback loops can help a transmitter and receiver to reduce their vulnerability to eavesdropping.
Traditional methods of cryptographic key distribution rest on judgments about an attacker.With the advent of quantum key distribution (QKD) came proofs of security for the mathematical models that define the protocols BB84 and B92; however, applying such proofs to actual transmitting and receiving devices has been questioned.Proofs of QKD security are propositions about models written in the mathematical language of quantum mechanics, and the issue is the linking of such models to actual devices in an experiment on security.To explore this issue, we adapt Wittgenstein's method of language games to view quantum language in its application to experimental activity involving transmitting and receiving devices.We sketch concepts with which to think about models in relation to experiments, without assuming the experiments accord with any model; included is a concept of one quantummechanical model enveloping another.For any model that agrees with given experimental results and implies the security of a key, there is an enveloping model that agrees with the same results while denying that security.As a result there is a gap between equations and the behavior recorded from devices in an experiment, a gap bridged only by resort to something beyond the reach of logic and measured data, well named by the word guesswork.While this recognition of guesswork encourages eavesdropping, a related recognition of guesswork in the design of feedback loops can help a transmitter and receiver to reduce their vulnerability to eavesdropping.
A mathematical model of computation is an idealized abstraction. We design algorithms and perform analysis on the assumption that the mathematical operations we specify will be carried out exactly, and without error. Physical devices that implement an abstract model of computation are imperfect and of limited precision. For example, when a digital circuit is implemented on a physical circuit board, unwanted electrical noise in the environment may cause components to behave differently than expected, and may cause voltage levels (bit-values) to change. These sources of error must be controlled or compensated for, or else the resulting loss of efficiency may reduce the power of the information-processing device. If individual steps in a computation succeed with probability p, then a computation involving t sequential steps will have a success probability that decreases exponentially as pt. Although it may be impossible to eliminate the sources of errors, we can devise schemes to allow us to recover from errors using a reasonable amount of additional resources. Many classical digital computing devices use error-correcting codes to perform detection of and recovery from errors. The theory of error-correcting codes is itself a mathematical abstraction, but it is one that explicitly accounts for errors introduced by the imperfection and imprecision of realistic devices. This theory has proven extremely effective in allowing engineers to build computing devices that are resilient against errors. Quantum computers are more susceptible to errors than classical digital computers, because quantum mechanical systems are more delicate and more difficult to control. If large-scale quantum computers are to be possible, a theory of quantum error correction is needed. The discovery of quantum error correction has given researchers confidence that realistic large-scale quantum computing devices can be built despite the presence of errors. We begin by considering fundamental concepts for error correction in a classical setting. We will focus on three of these concepts: (a) the characterization of the error model, (b) the introduction of redundancy through encoding, and (c) an error recovery procedure. We will later see that these concepts generalize quite naturally for quantum error correction.