^{1,a),b)}, Yukiyoshi Ohtsuki

^{1,a),c)}, Kouichi Hosaka

^{2}, Hisashi Chiba

^{2}, Hiroyuki Katsuki

^{3}and Kenji Ohmori

^{3}

### Abstract

We numerically propose a way to perform quantum computations by combining an ensemble of molecular states and weak laser pulses. A logical input state is expressed as a superposition state (a wave packet) of molecular states, which is initially prepared by a designed femtosecond laser pulse. The free propagation of the wave packet for a specified time interval leads to the specified change in the relative phases among the molecular basis states, which corresponds to a computational result. The computational results are retrieved by means of quantum interferometry. Numerical tests are implemented in the vibrational states of the state of employing controlled-NOT gate, and 2 and Fourier transforms. All the steps involved in the computational scheme, i.e., the initial preparation, gate operation, and detection steps, are achieved with extremely high precision.

We are grateful to Professor Takamasa Momose for helpful suggestions. This work was partly supported by a Grant-in-Aid from MEXT of Japan (17550005, 15204034, and Priority Area: “Control of Molecules in Intense Laser Fields”).

I. INTRODUCTION

II. THEORY

A. Gate operations

B. Input state preparation and detection

III. RESULTS AND DISCUSSION

A. Accuracy of gate operations

B. Accuracy of the logical-state preparation and detection

C. Application of QFT

IV. SUMMARY

### Key Topics

- Vibrational states
- 33.0
- Qubits
- 19.0
- Field theory
- 15.0
- Quantum computing
- 15.0
- Interferometry
- 7.0

## Figures

(a) Schematic of the experimental setup. (b) Pulse sequence for quantum computation. The times, , , and , specify the temporal peak of the shaping pulse, the initial preparation time, and the temporal peak of the reference pulse, respectively (see text). The time intervals are referred to as the input preparation time and the gate operation time .

(a) Schematic of the experimental setup. (b) Pulse sequence for quantum computation. The times, , , and , specify the temporal peak of the shaping pulse, the initial preparation time, and the temporal peak of the reference pulse, respectively (see text). The time intervals are referred to as the input preparation time and the gate operation time .

“Average” fidelity [Eq. (19)] of the quantum Fourier transform as a function of gate operation time .

“Average” fidelity [Eq. (19)] of the quantum Fourier transform as a function of gate operation time .

“Average” fidelity [Eq. (19)] of the quantum Fourier transform as a function of vibrational quantum number, in which the modified vibrational states, with to , are chosen as the basis. The gate operation time is optimized for each basis set.

“Average” fidelity [Eq. (19)] of the quantum Fourier transform as a function of vibrational quantum number, in which the modified vibrational states, with to , are chosen as the basis. The gate operation time is optimized for each basis set.

(Upper figures) Shaping and reference pulses when their temporal peaks are chosen as and , respectively. The frequency resolutions of the shaping pulses are specified by (a) , (b) , and (c) . (Lower figures) Interferograms for the populations of the vibrational states associated with the logical input state, ∣001⟩, as a function of the phase-locking angle . The short-dashed, long-dashed, dotted, and dotted-dashed lines show the populations of , , , and , respectively. The interferograms for the and state populations have maximum values around , while those for the and state populations have maximum values around . The solid lines show the interferograms for the populations of the other vibrational states.

(Upper figures) Shaping and reference pulses when their temporal peaks are chosen as and , respectively. The frequency resolutions of the shaping pulses are specified by (a) , (b) , and (c) . (Lower figures) Interferograms for the populations of the vibrational states associated with the logical input state, ∣001⟩, as a function of the phase-locking angle . The short-dashed, long-dashed, dotted, and dotted-dashed lines show the populations of , , , and , respectively. The interferograms for the and state populations have maximum values around , while those for the and state populations have maximum values around . The solid lines show the interferograms for the populations of the other vibrational states.

Interferograms for the populations of the vibrational states associated with the logical output state, , as a function of the phase-locking angle . The frequency resolutions of the shaping pulses are specified by (a) , (b) , and (c) . The short-dashed, long-dashed, dotted, and dotted-dashed lines show the populations of , , , and , the maximum values of which appear at , , , and , respectively. The solid lines show the interferograms for the populations of the other vibrational states.

Interferograms for the populations of the vibrational states associated with the logical output state, , as a function of the phase-locking angle . The frequency resolutions of the shaping pulses are specified by (a) , (b) , and (c) . The short-dashed, long-dashed, dotted, and dotted-dashed lines show the populations of , , , and , the maximum values of which appear at , , , and , respectively. The solid lines show the interferograms for the populations of the other vibrational states.

(Upper figures) Shaping and reference pulses for measuring the interferograms for populations of the vibrational states associated with (a) the logical input state and (b) the logical output state. (Middle figures) (a) Interferogram for the population of each vibrational state associated with the logical input state and (b) that associated with the logical output state as a function of the phase-locking angle of the reference pulse, . The dotted and dashed lines show the populations of and . For the other vibrational states (solid lines), the interferograms are constantly independent of . (Lower figures) Population distributions in the logical-state representation for (a) the input state and (b) the output state.

(Upper figures) Shaping and reference pulses for measuring the interferograms for populations of the vibrational states associated with (a) the logical input state and (b) the logical output state. (Middle figures) (a) Interferogram for the population of each vibrational state associated with the logical input state and (b) that associated with the logical output state as a function of the phase-locking angle of the reference pulse, . The dotted and dashed lines show the populations of and . For the other vibrational states (solid lines), the interferograms are constantly independent of . (Lower figures) Population distributions in the logical-state representation for (a) the input state and (b) the output state.

## Tables

Fidelities of CNOT and QFT in a case. The definition of fidelity is given by Eq. (18).

Fidelities of CNOT and QFT in a case. The definition of fidelity is given by Eq. (18).

Fidelities of QFT. The definition of fidelity is given by Eq. (18).

Fidelities of QFT. The definition of fidelity is given by Eq. (18).

Normalized overlap [defined by Eq. (23)] between the target logical state and the wave packet produced by the shaping pulse when the pulse shaper has a frequency resolution of .

Normalized overlap [defined by Eq. (23)] between the target logical state and the wave packet produced by the shaping pulse when the pulse shaper has a frequency resolution of .

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