The doctoral dissertations of the former Helsinki University of Technology (TKK) and Aalto University Schools of Technology (CHEM, ELEC, ENG, SCI) published in electronic format are available in the electronic publications archive of Aalto University - Aaltodoc.
Aalto

From the Control of Quantum Systems to Multiqubit Logic

Ville Bergholm

Dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Department of Engineering Physics and Mathematics for public examination and debate in Auditorium TU2 at Helsinki University of Technology (Espoo, Finland) on the 4th of December, 2007, at 12 o'clock noon.

Overview in PDF format (ISBN 978-951-22-9091-8)   [599 KB]
Dissertation is also available in print (ISBN 978-951-22-9090-1)

Abstract

Quantum computing and quantum information science are two recently discovered and rapidly growing fields of physics that show substantial promise in providing new and valuable technologies in the foreseeable future. Large-scale quantum computers, if ever realized experimentally, are likely to outperform their classical counterparts in a number of important computational tasks, the most important of which may be the accurate simulation of many-body quantum systems such as the ones encountered in physics, chemistry and life sciences.

This thesis investigates the problem of controlling quantum systems for the purpose of performing quantum information processing tasks. The problem is approached from a theoretical and simulational viewpoint. The work contained here encompasses a range of levels of abstraction. Firstly, we discuss the decomposition of abstract multiqubit logic gates into sequences of simple elementary gates. Secondly, we study the local commutational properties of two-qubit gates using local gate invariants. Thirdly, we develop methods for the physical implementation of the elementary gates through the control of specific quantum systems, possibly in the presence of noise and decoherence.

We present a new, almost optimal n-qubit gate decomposition based on the cosine-sine decomposition, which utilizes a likewise new intermediate quantum circuit structure we call a uniformly controlled gate. We then show how they can be used in constructing a general state transformation circuit. Both of the resulting circuits can be efficiently implemented using nearest-neighbor gates which makes their physical realization simpler. A local gate invariant is introduced which can be used to assess the suitability of two-qubit gates for serving as the entangling gate in elementary gate libraries. Finally, we develop numerical optimization methods for finding near-optimal control sequences for generating one- and two-qubit gates, both in closed quantum systems and in the presence of Markovian noise.

This thesis consists of an overview and of the following 6 publications:

  1. M. Möttönen, J. J. Vartiainen, V. Bergholm, M. M. Salomaa, Quantum circuits for general multiqubit gates, Physical Review Letters 93, 130502 (2004). © 2004 American Physical Society. By permission.
  2. M. Möttönen, J. J. Vartiainen, V. Bergholm, M. M. Salomaa, Transformation of quantum states using uniformly controlled rotations, Quantum Information and Computation 5, 467 (2005). © 2005 Rinton Press. By permission.
  3. V. Bergholm, J. J. Vartiainen, M. Möttönen, M. M. Salomaa, Quantum circuits with uniformly controlled one-qubit gates, Physical Review A 71, 052330 (2005). © 2005 American Physical Society. By permission.
  4. L. Koponen, V. Bergholm, M. M. Salomaa, A discrete local invariant for quantum gates, Quantum Information and Computation 6, 58 (2006). © 2006 Rinton Press. By permission.
  5. A. K. Spörl, T. Schulte-Herbrüggen, S. J. Glaser, V. Bergholm, M. J. Storcz, J. Ferber, F. K. Wilhelm, Optimal control of coupled Josephson qubits, Physical Review A 75, 012302 (2007). © 2007 American Physical Society. By permission.
  6. O.-P. Saira, V. Bergholm, T. Ojanen, M. Möttönen, Equivalent qubit dynamics under classical and quantum noise, Physical Review A 75, 012308 (2007). © 2007 American Physical Society. By permission.

Keywords: quantum computing, quantum gate, entanglement, decoherence, quantum control

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© 2007 Helsinki University of Technology

Last update 2011-05-26