Weakly anharmonic trapping potential in the model system. Energies of 32 quantized motional states of one trapped ion are indicated by horizontal lines. Effect of anharmonicity is clearly seen. Sixteen lower states (used to encode qubits) are indicated by solid lines. Upper states (included for completeness) are shown by dashed lines. Assignment of states of the four-qubit system is indicated in brackets.
Transition moment matrix for 16 lower vibrational states in a model system. Color indicates magnitudes of matrix elements in the logarithmic scale. See text for further details.
Quantum circuit diagram for the phase estimation part of Shor's algorithm for factorizing number 15 using four qubits (read from left to right). See text for details.
(a) Optimally shaped 50 μs pulse for Shor's algorithm and (b) windowed Fourier transform of the pulse. Horizontal dashed lines indicate frequencies of the state-to-state transitions. Dotted curves encircle two spectral features that correspond to the ladder climbing. See text for details.
Time evolution of state populations induced by the pulse optimized for Shor's algorithm in three representative cases: (a) Transformation #1 in Table I; (b) Transformation #3 in Table I; and (c) Transformation #11 in Table I. Thicker color lines indicate population of the initial and final states. Thinner black lines indicate population of intermediate states.
Fourier spectrum of the pulse optimized for Shor's algorithm: (a) broad range of frequencies (up to ω n,n+ 9); (b) the focus on frequency range of the main spectral structure (ω n,n+ 1 transitions); and (c) the focus on the frequency range of the overtone spectral structure (ω n,n+ 3 transitions). Arrows indicate frequencies of the state-to-state transitions from Table I.
Eigenvalues, transition frequencies, qubit state assignments, optimized transformations, and their characteristic probabilities for 16 lower vibrational states in a model of one ion in a weakly anharmonic Paul trap.
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