^{1}and Dmitri Babikov

^{1,a)}

### Abstract

We demonstrate theoretically that it may be possible to encode states of a multi-qubit system into the progression of quantized motional/vibrational levels of an ion trapped in a weakly anharmonic potential. Control over such register of quantum information is achieved by applying oscillatory radio-frequency fields shaped optimally for excitation of the desired state-to-state transitions. Anharmonicity of the vibrational spectrum plays a key role in this approach to the control and quantum computation, since it allows resolving different state-to-state transitions and addressing them selectively. Optimal control theory is used to derive pulses for implementing the four-qubit version of Shor's algorithm in a single step. Accuracy of the qubit-state transformations, reached in the numerical simulations, is around 0.999. Very detailed insight is obtained by analysis of the time-evolution of state populations and by spectral analysis of the optimized pulse.

This work was supported by the National Science Foundation, Grant No. CHE-1012075. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Professor Martine Gruebele at the University of Illinois at Urbana-Champaign is acknowledged for fruitful discussions.

I. INTRODUCTION

II. THE MODEL SYSTEM

III. THE QUANTUM ALGORITHM

IV. OCT PULSE SHAPING

V. RESULTS AND DISCUSSION

VI. CONCLUSIONS

### Key Topics

- Qubits
- 29.0
- Vibrational states
- 15.0
- Encoding
- 14.0
- Ion trapping
- 10.0
- Optical coherence tomography
- 10.0

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## Figures

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.

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.

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.

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 *μ*s 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.

(a) Optimally shaped 50 *μ*s 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.

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.

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.

## Tables

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.

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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