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Computer science approach to quantum control

Dominik Janzing

Published Jan 1, 2006• 6 citations
Publisher

Abstract

This work considers several hypothetical control processes on the nanoscopic level and show their analogy to computation processes. It shows that measuring certain types of quantum observables is such a complex task that every instrument that is able to perform it would necessarily be an extremely powerful computer.
DOI
10.5445/ksp/1000005188

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Abstract The theory of computational complexity in computer science investigates the time and space resources required to solve a computational problem. We show that an analogous complexity theory can be developed for quantum control problems which do not stem from computational problems. First we use the same mathematical concept to show similar lower bounds on the complexity of the following tasks: (1) Computing the boolean function “Majority”, (1) Optimal algorithmic cooling on n> two‐level systems, (2) Optimal work extraction by a heat engine acting on 2n> hot and 1 cold two‐level system, (3) Preparing appropriate n‐qubit entanglement for improving frequency standards in Ramsey spectroscopy, (4) Implementing a von‐Neumann measurement that is strongly incompatible with macroscopic observables. We describe furthermore instances of the problems “optimal heat engines” and “precise measurements” which are likely to be complex since NP‐ or even PSPACE‐complete computational problems can be reduced to them. In order to define implementation complexity also for other systems than n qubit registers we rephrase some approaches to define complexity measures using a set of available interactions and explore the complexity of analogue to digital conversion using a Jaynes Cumming interaction as an example. Furthermore we argue that, from a physical perspective of complexity, operations should only considered to be simple if there is a physically reasonable Hamiltonian that can implement them by its autonomous time evolution within a short time interval. Therefore we discuss a Hamiltonian whose natural time evolution can implement computation processes.

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Abstract The theory of computational complexity in computer science investigates the time and space resources required to solve a computational problem. We show that an analogous complexity theory can be developed for quantum control problems which do not stem from computational problems. First we use the same mathematical concept to show similar lower bounds on the complexity of the following tasks: (1) Computing the boolean function “Majority”, (1) Optimal algorithmic cooling on n> two‐level systems, (2) Optimal work extraction by a heat engine acting on 2n> hot and 1 cold two‐level system, (3) Preparing appropriate n‐qubit entanglement for improving frequency standards in Ramsey spectroscopy, (4) Implementing a von‐Neumann measurement that is strongly incompatible with macroscopic observables. We describe furthermore instances of the problems “optimal heat engines” and “precise measurements” which are likely to be complex since NP‐ or even PSPACE‐complete computational problems can be reduced to them. In order to define implementation complexity also for other systems than n qubit registers we rephrase some approaches to define complexity measures using a set of available interactions and explore the complexity of analogue to digital conversion using a Jaynes Cumming interaction as an example. Furthermore we argue that, from a physical perspective of complexity, operations should only considered to be simple if there is a physically reasonable Hamiltonian that can implement them by its autonomous time evolution within a short time interval. Therefore we discuss a Hamiltonian whose natural time evolution can implement computation processes.
Modern and perspective tasks of robotics with control from artificial intelligence systems require the use of small-sized measuring devices. In this case, the intensively developed quantum sensors and quantum computers have a bright prospect. Their main advantage is the ability to successfully process the data of random processes with decomposition of complex functions into simple multipliers, as well as their small size and the ability to transmit data over long distances without wires. Data transmitted over quantum communication lines cannot be copied or intercepted, which is very useful for remote control of complex technical systems. Based on the results of the analysis of probabilistic noisy data quantum computer is able to quickly develop an assessment of the technical condition of the complex system. At the same time, there is no need to go through all the possible solutions to the evaluation problem with a huge amount of input data, some of which can sometimes be undefined. The main problem in the research of quantum processes is that researchers study the processes occurring in materials, but they do not indicate the ways in which quantum sensors and quantum computers are used in practical applications. This article explains how to form a measuring transformer that will be compatible with a quantum computer. The main objective of the study was to bring the results of basic research in the field of quantum computing closer to their application in applied tasks. It is shown how quantum processes can be shifted to the field of technical measurements of physical quantities used in complex systems. In the process of obtaining the results of the study, the hypothetical deductive method and the method of ascent from the abstract to the concrete within the framework of a systematic approach to the development of elements of technical systems were used. The result is a description of the processes of designing of tension sensing element made of modified diamond. The main findings of the study include the fact that quantum sensors implemented in the form of a modified diamond crystal are well described by the theory of measuring transducers with frequency output and can be used to get data about the state of an object.
Instruments are necessarily of macroscopic size because our sense organs—which can be considered as instruments occupying a certain rank in the von Neumann hierarchy—are of macroscopic size. It is only natural, therefore, that physicists should have attempted to make use of this property for constructing quantum measurement theories. Moreover, one of the remarkable properties possessed by most macroscopic systems, including all instruments, but lacking in microscopic systems is irreversibility. This simple observation suggests very strongly that the phenomenon of irreversibility and the phenomenon of "wave packet reduction," which is associated in some way or other with the measurement process, should somehow be correlated. The theories in the chapter show that it is in practice always possible to attribute definite macroscopic properties to the measuring instruments. The danger that the error we thus make will ever be detected has been proved to be vanishingly small.