Optimal control theory for unitary transformations

Journal Article · Mon Dec 01 04:00:00 EST 2003 · Physical Review. A

DOI:https://doi.org/10.1103/PhysRevA.68.062308· OSTI ID:20640459

Palao, Jose P ^[1]; Kosloff, Ronnie ^[1]

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1. Department of Physical Chemistry and the Fritz Haber Research Center for Molecular Dynamics, Hebrew University, Jerusalem 91904 (Israel)

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 {sup 1}{sigma}{sub g}{sup +} electronic state of Na{sub 2}. Raman-like transitions through the A {sup 1}{sigma}{sub u}{sup +} 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 Na{sub 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.

• OSTI ID: 20640459
• Journal Information: Physical Review. A, Journal Name: Physical Review. A Journal Issue: 6 Vol. 68; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY
74 ATOMIC AND MOLECULAR PHYSICS
ALGORITHMS
FOURIER TRANSFORMATION
HAMILTONIANS
HILBERT SPACE
IMPLEMENTATION
INFORMATION THEORY
MOLECULES
OPTIMAL CONTROL
PULSES
QUANTUM MECHANICS
RAMAN SPECTRA
SODIUM
TIME DEPENDENCE
VIBRATIONAL STATES
VISIBLE RADIATION