Optimal control theory for unitary transformations

By J. P. Palao, Ronnie Kosloff. 2003

243 citations recorded by the source corpus.

The dynamics of a quantum system driven by an external field is well described by a unitary transformation generated by a time-dependent Hamiltonian. The inverse problem of finding the field that generates a specific unitary transformation is the subject of study. The unitary transformation which can represent an algorithm in a quantum computation is imposed on a subset of quantum states embedded in a larger Hilbert space. Optimal control theory is used to solve the inversion problem irrespective of the initial input state. A unified formalism based on the Krotov method is developed leading to a different scheme. The schemes are compared for the inversion of a two-qubit Fourier transform using as registers the vibrational levels of the $X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ electronic state of ${\mathrm{Na}}_{2}.$ Raman-like transitions through the $A{}^{1}{\ensuremath{\Sigma}}_{u}^{+}$ electronic state induce the transitions. Light fields are found that are able to implement the Fourier transform within a picosecond time scale. Such fields can be obtained by pulse-shaping techniques of a femtosecond pulse. Of the schemes studied, the square modulus scheme converges fastest. A study of the implementation of the Q qubit Fourier transform in the ${\mathrm{Na}}_{2}$ molecule was carried out for up to five qubits. The classical computation effort required to obtain the algorithm with a given fidelity is estimated to scale exponentially with the number of levels. The observed moderate scaling of the pulse intensity with the number of qubits in the transformation is rationalized.

Subfields: Atomic and Molecular Physics, and Optics, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3af902fa4e4f2d0bc338dc055494e1caf07f2b5c

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In Atomic and Molecular Physics, and Optics

  1. Thomas Schulte-Herbrüggen
  2. Ronnie Kosloff
  3. Dominique Sugny
  4. Tommaso Calarco
  5. Christiane P. Koch
  6. Frank K. Wilhelm
  7. Steffen J. Glaser
  8. Ugo V. Boscain

In Modeling and Simulation

  1. Ronnie Kosloff

In Statistical and Nonlinear Physics

  1. Dominique Sugny
  2. Christiane P. Koch
  3. Thomas Schulte-Herbrüggen
  4. Ronnie Kosloff
  5. Tommaso Calarco
  6. Frank K. Wilhelm
  7. Steffen J. Glaser
  8. Ugo V. Boscain

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