//img.uscri.be/pth/d457a4f97381b5ea1930545278bdf7b8aace1b36
La lecture en ligne est gratuite
Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres
Télécharger Lire

Computer science approach to quantum control [Elektronische Ressource] / von Dominik Janzing

146 pages
Dominik JanzingComputer Science Approach to Quantum Control Computer Science Approachto Quantum Controlvon Dominik JanzingHabilitation, Universität Karlsruhe (TH)Fakultät für Informatik, 2006ImpressumUniversitätsverlag Karlsruhec/o UniversitätsbibliothekStraße am Forum 2D-76131 Karlsruhewww.uvka.deDieses Werk ist unter folgender Creative Commons-Lizenz lizenziert: http://creativecommons.org/licenses/by-nc-nd/2.0/de/Universitätsverlag Karlsruhe 2006 Print on DemandISBN-13: 978-3-86644-083-8ISBN-10: 3-86644-083-9Contents1 Quantum Information and Computation 51.1 Standard Model of a Quantum Computer . . . . . . . . . . . . . . . . . . 51.1.1 Quantum Gates and Quantum Circuits . . . . . . . . . . . . . . . 71.1.2 Readout of Quantum Registers . . . . . . . . . . . . . . . . . . . 101.1.3 Quantum Circuits and Algorithms. . . . . . . . . . . . . . . . . . 101.1.4 Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 Arbitrary Quantum Systems as Quantum Registers . . . . . . . . . . . . 131.2.1 States, Dynamics and Measurements in Textbook Quantum Me-chanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Tools of Quantum Communication Theory . . . . . . . . . . . . . . . . . 161.3.1 Entropy and Information . . . . . . . . . . . . . . . . . . . . . . . 161.3.2 Quantum and Classical Correlations . . . . . . . . . . . . . . . . 171.3.3 Secret Correlations and Entanglement . . . . . . . . . . . . . . .
Voir plus Voir moins

Dominik Janzing
Computer Science Approach to Quantum Control Computer Science Approach
to Quantum Control
von
Dominik JanzingHabilitation, Universität Karlsruhe (TH)
Fakultät für Informatik, 2006
Impressum
Universitätsverlag Karlsruhe
c/o Universitätsbibliothek
Straße am Forum 2
D-76131 Karlsruhe
www.uvka.de
Dieses Werk ist unter folgender Creative Commons-Lizenz
lizenziert: http://creativecommons.org/licenses/by-nc-nd/2.0/de/
Universitätsverlag Karlsruhe 2006
Print on Demand
ISBN-13: 978-3-86644-083-8
ISBN-10: 3-86644-083-9Contents
1 Quantum Information and Computation 5
1.1 Standard Model of a Quantum Computer . . . . . . . . . . . . . . . . . . 5
1.1.1 Quantum Gates and Quantum Circuits . . . . . . . . . . . . . . . 7
1.1.2 Readout of Quantum Registers . . . . . . . . . . . . . . . . . . . 10
1.1.3 Quantum Circuits and Algorithms. . . . . . . . . . . . . . . . . . 10
1.1.4 Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Arbitrary Quantum Systems as Quantum Registers . . . . . . . . . . . . 13
1.2.1 States, Dynamics and Measurements in Textbook Quantum Me-
chanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Tools of Quantum Communication Theory . . . . . . . . . . . . . . . . . 16
1.3.1 Entropy and Information . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.2 Quantum and Classical Correlations . . . . . . . . . . . . . . . . 17
1.3.3 Secret Correlations and Entanglement . . . . . . . . . . . . . . . 18
2 Old Fundamental Problems in Light of Modern Quantum Information
Research 21
2.1 Revisiting Quantum Mechanics from an Algorithmic Point of View . . . 21
2.1.1 Operational Meaning of States, Observables, and Unitary Maps . 22
2.1.2 Micro- and Macro-Physics and Fault-Tolerance of Quantum Gates 27
2.2 Revisiting Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 Understanding Thermodynamics by Toy Models . . . . . . . . . . 31
2.2.2 The Most Elementary Heat Engine . . . . . . . . . . . . . . . . . 33
2.2.3 The Most Elementary Refrigerator . . . . . . . . . . . . . . . . . 34
2.2.4 Computer Scientist’s Fridge . . . . . . . . . . . . . . . . . . . . . 34
2.2.5 Computer Scientist’s Heat Engine . . . . . . . . . . . . . . . . . . 35
2.2.6 Classifying Thermodynamic Resources . . . . . . . . . . . . . . . 38
2.2.7 Timing Information as a Thermodynamic Quantity . . . . . . . . 44
2.2.8 Coherence, Reversibility and Secrecy of Computation . . . . . . . 48
2.2.9 Computing and Quantum Control in a Closed Physical System . . 50
2.2.10 Saving Energy by Quantum Information Transfer? . . . . . . . . . 56
2.3 Complexity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.3.1 Challenging the Strong Church-Turing Thesis . . . . . . . . . . . 59
2.3.2 Quantum Complexity Classes . . . . . . . . . . . . . . . . . . . . 60
2.3.3 Complexity Measures from Nature? . . . . . . . . . . . . . . . . . 62
2.4 Quantum Communication and Causal Reasoning. . . . . . . . . . . . . . 63
2.4.1 Dense Coding and Quantification of Causal Effects . . . . . . . . 64
i2.4.2 Hidden Variable Models in Clinical Drug Testing . . . . . . . . . 66
3 Extending the Definition of Algorithms 73
3.1 Continuous Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.1.1 Languages for Continuous Algorithms . . . . . . . . . . . . . . . . 74
3.1.2 Comparing Discrete to Continuous Complexity . . . . . . . . . . . 75
3.1.3 Mutual Simulation of Hamiltonians . . . . . . . . . . . . . . . . . 77
3.1.4 Adiabatic Quantum Computing . . . . . . . . . . . . . . . . . . . 83
3.2 Non-Computational Problems . . . . . . . . . . . . . . . . . . . . . . . . 84
3.2.1 Difference between Implementing and Computing a
Boolean Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.2 Algorithms with Quantum Input or Output . . . . . . . . . . . . 87
4 Algorithmic Approach to Natural Non-Computational Problems 89
4.1 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.1.1 Von-Neumann Measurements and their Complexity . . . . . . . . 90
4.1.2 Generalized Measurements (POVMs) . . . . . . . . . . . . . . . . 98
4.2 Thermodynamic Machines . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.1 Cooling Weakly Interacting Systems . . . . . . . . . . . . . . . . 103
4.2.2 Complexity of Cooling Strongly Interacting Systems . . . . . . . . 105
4.2.3 Complexity and Efficiency of Molecular Heat
Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3 Imaging and Material Analysis . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3.1 Decoupling Strategies and their Complexity . . . . . . . . . . . . 115
4.3.2 Is an Image a Covariant POVM-Measurement? . . . . . . . . . . 117
4.3.3 Microscopy with Pre-Processing the Input Beam . . . . . . . . . . 118
4.3.4 Microscopy with Quantum Post-Processing . . . . . . . . . . . . . 119
5 Conclusions 123
iiPreface: Algorithms - not only for
Computational Problems
The term “algorithm”lies at the heart of computer science. Following asimple definition
in the web [1], an algorithm is “a finite set of well-defined instructions for accomplishing
some task which, given an initial state, will result in a corresponding recognizable end-
state”. This definition leaves the task itself unspecified and also means, as an example,
that a set ofinstructions to prepare tomato salad is an algorithm. Forobvious reasons, it
is common in computer science to consider an algorithm as a set of instructions solving a
computational problem. Following Horowitz and Sahni [2], one has further the following
criteria:
1. Input: there are zero or more quantities which are externally supplied;
2. Output: at least one quantity is produced;
3. Definiteness: if we trace out the instructions of an algorithm, then for all cases the
algorithm will terminate after a finite number of steps.
4. Effectiveness: every instruction must be sufficiently basic that it can in principle
be carried out by a person using only pencil and paper. It is not enough that each
operation be definite as in (3) but it must also be feasible.
Duringthe90stheideaofbuildingaquantumcomputerattractedbroadinterdisciplinary
interestincomputerscience, physics, andmathematics. Suchamachineworkswithquan-
tum superpositions of logically different states, and a completely new type of algorithms,
so-called quantum algorithms, were invented. The elementary operations in quantum
algorithms cannot be described as classical logical computation steps. These algorithms
contain instructions which are not“sufficiently basic that they can in principle be carried
out by a person using only pencil and paper” (see item 4.). Instead, they would require
special ‘quantum paper’ on which one could write superpositions of different numbers. A
quantum computation can only be performed with hardware which enables such super-
positions between logically different states. This works only in physical systems which
are sufficiently isolated from their environment to show typical quantum phenomena,
and it is more likely feasible for small systems than for macroscopic systems. Therefore
an important part of quantum computing research is the development of methods for
controlling physical systems on the nanoscopic, microscopic, or mesoscopic scale. These
systems can, for instance, be ions, atoms, molecules, or electromagnetic fields. Typical
tools to manipulate their physical states are laser beams or other kind of electromagnetic
waves like high-frequency radiation.
12
Apart from the goal of developing hardware for the quantum computer, it is clear
that controlling tiny objects is a challenging task in its own right, having many poten-
tial applications. Focussing electron or light beams in a microscope or building smaller
transistors which are triggered by single electrons for future electronic chips are only
two examples of applications where states of quantum systems have to be controlled
and measured. The idea of quantum computing encouraged theoreticians and experi-
mentalists to further develop existing control technologies. On the other hand, it seems
as if theoretical and experimental tools created for quantum computing could be also
useful for non-computational problems. Among others, the following non-computational
applications of quantum control techniques are closely related to quantum computing:
1. NMR Spectroscopy: Nuclear magnetic resonance (NMR) experiments are used
inmedicine, biology, chemistry, and physics toanalyze organicand an-organicmat-
ter. Roughly speaking, the spins of the atoms in the analyzed material can be
considered as little magnets. These ‘magnets’ are rotated by an electromagnetic
pulse. While they turn back to their original position, they emit electro-magnetic
pulses which give insight in the chemical structure. In many experiments the mat-
ter is subjected to rather complex sequences of electromagnetic pulses since the
emitted radiation after this treatment is characteristic for the specific material and
the interactions among its nuclear spins. The design of these pulse sequences is
analogous to algorithmic problems in quantum computing (QC) even though these
methods existed in NMR long before they were rediscovered for QC.
2. Quantum Lithography and Microscopy: Optical lithography is aprimary tool
in the microchip industry for transferring circuit images onto substrates. Similiarly
asinmicroscopy, theresolutionoftheimageislimitedbythewavelengthofthelight
beam, the so-called diffraction limit. Recent research suggests that the diffraction
limit can in principle be beaten using non-classical light with quantum correlated
photons. The proposals to prepare this type of light use a sequence of optical
devices which is designed in strong analogy to the design of quantum circuits from
elementary gates.
3. Cooling Molecular Systems: Modern cooling techniques which lower the tem-
perature of atomic or molecular systems (in order to prepare it for further exper-
iments) use rather sophisticated sequences of control operations. In the context
of NMR experiments, there exists even a proposal for a cooling algorithm which
can at the same time be considered as a data compression algorithm if the atomic
states are interpreted as logical values of bits. Even though the original intention
of this proposal was to initialize the system for quantum computing, it shows that
the non-computing task of cooling a system can be treated algorithmically.
This work should elucidate what we can learn from quantum computing and communi-
cation research for non-computational problems:
• Quantum Algorithmic Thinking and Complexity Theory: The principle of
concatenatingelementarytransformationsthatgeneratecomplexphysicalprocesses
is not restricted to computational processes. Present research results indicate that